Yuchen Pei's Bloghttps://ypei.me/blog-feed.xml2019-03-14T00:00:00ZYuchen PeiPyAtomGreat but Manageable Expectationsposts/2019-03-14-great-but-manageable-expectations.html2019-03-14T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<div id="TOC">
<ul>
<li><a href="#rényi-divergence-and-differential-privacy">Rényi divergence and differential privacy</a><ul>
<li><a href="#moment-composition">Moment Composition</a></li>
<li><a href="#subsampling">Subsampling</a></li>
</ul></li>
<li><a href="#acgmmtz16">ACGMMTZ16</a></li>
<li><a href="#tensorflow-implementation">Tensorflow implementation</a></li>
<li><a href="#comparison-among-different-methods">Comparison among different methods</a></li>
<li><a href="#further-questions">Further questions</a></li>
<li><a href="#references">References</a></li>
</ul>
</div>
<p>This is Part 2 of a two-part blog post on differential privacy. Continuing from <a href="/posts/2019-03-13-a-tail-of-two-densities.html">Part 1</a>, I discuss the Rényi differential privacy, corresponding to the Rényi divergence, a study of the <a href="https://en.wikipedia.org/wiki/Moment-generating_function">moment generating functions</a> of the divergence between probability measures in order to derive the tail bounds.</p>
<p>Like in Part 1, I prove a composition theorem and a subsampling theorem.</p>
<p>I also attempt to reproduce a seemingly better moment bound for the Gaussian mechanism with subsampling, with one intermediate step which I am not able to prove.</p>
<p>After that I explain the Tensorflow implementation of differential privacy in its <a href="https://github.com/tensorflow/privacy/tree/master/privacy">Privacy</a> module, which focuses on the differentially private stochastic gradient descent algorithm (DP-SGD).</p>
<p>Finally I use the results from both Part 1 and Part 2 to obtain some privacy guarantees for composed subsampling queries in general, and for DP-SGD in particular. I also compare these privacy guarantees.</p>
<p><em>If you are confused by any notations, ask me or try <a href="/notations.html">this</a>.</em></p>
<h2 id="rényi-divergence-and-differential-privacy">Rényi divergence and differential privacy</h2>
<p>Recall in the proof of Gaussian mechanism privacy guarantee (Claim 8) we used the Chernoff bound for the Gaussian noise. Why not use the Chernoff bound for the divergence variable / privacy loss directly, since the latter is closer to the core subject than the noise? This leads us to the study of Rényi divergence.</p>
<p>So far we have seen several notions of divergence used in differential privacy: the max divergence which is <span class="math inline">\(\epsilon\)</span>-ind in disguise:</p>
<p><span class="math display">\[D_\infty(p || q) := \max_y \log {p(y) \over q(y)},\]</span></p>
<p>the <span class="math inline">\(\delta\)</span>-approximate max divergence that defines the <span class="math inline">\((\epsilon, \delta)\)</span>-ind:</p>
<p><span class="math display">\[D_\infty^\delta(p || q) := \max_y \log{p(y) - \delta \over q(y)},\]</span></p>
<p>and the KL-divergence which is the expectation of the divergence variable:</p>
<p><span class="math display">\[D(p || q) = \mathbb E L(p || q) = \int \log {p(y) \over q(y)} p(y) dy.\]</span></p>
<p>The Rényi divergence is an interpolation between the max divergence and the KL-divergence, defined as the log moment generating function / cumulants of the divergence variable:</p>
<p><span class="math display">\[D_\lambda(p || q) = (\lambda - 1)^{-1} \log \mathbb E \exp((\lambda - 1) L(p || q)) = (\lambda - 1)^{-1} \log \int {p(y)^\lambda \over q(y)^{\lambda - 1}} dx.\]</span></p>
<p>Indeed, when <span class="math inline">\(\lambda \to \infty\)</span> we recover the max divergence, and when <span class="math inline">\(\lambda \to 1\)</span>, by recognising <span class="math inline">\(D_\lambda\)</span> as a derivative in <span class="math inline">\(\lambda\)</span> at <span class="math inline">\(\lambda = 1\)</span>, we recover the KL divergence. In this post we only consider <span class="math inline">\(\lambda > 1\)</span>.</p>
<p>Using the Rényi divergence we may define:</p>
<p><strong>Definition (Rényi differential privacy)</strong> (Mironov 2017). An mechanism <span class="math inline">\(M\)</span> is <span class="math inline">\((\lambda, \rho)\)</span><em>-Rényi differentially private</em> (<span class="math inline">\((\lambda, \rho)\)</span>-rdp) if for all <span class="math inline">\(x\)</span> and <span class="math inline">\(x'\)</span> with distance <span class="math inline">\(1\)</span>,</p>
<p><span class="math display">\[D_\lambda(M(x) || M(x')) \le \rho.\]</span></p>
<p>For convenience we also define two related notions, <span class="math inline">\(G_\lambda (f || g)\)</span> and <span class="math inline">\(\kappa_{f, g} (t)\)</span> for <span class="math inline">\(\lambda > 1\)</span>, <span class="math inline">\(t > 0\)</span> and positive functions <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span>:</p>
<p><span class="math display">\[G_\lambda(f || g) = \int f(y)^{\lambda} g(y)^{1 - \lambda} dy; \qquad \kappa_{f, g} (t) = \log G_{t + 1}(f || g).\]</span></p>
<p>For probability densities <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span>, <span class="math inline">\(G_{t + 1}(p || q)\)</span> and <span class="math inline">\(\kappa_{p, q}(t)\)</span> are the <span class="math inline">\(t\)</span>th moment generating function and <a href="https://en.wikipedia.org/wiki/Cumulant">cumulant</a> of the divergence variable <span class="math inline">\(L(p || q)\)</span>, and</p>
<p><span class="math display">\[D_\lambda(p || q) = (\lambda - 1)^{-1} \kappa_{p, q}(\lambda - 1).\]</span></p>
<p>In the following, whenever you see <span class="math inline">\(t\)</span>, think of it as <span class="math inline">\(\lambda - 1\)</span>.</p>
<p><strong>Example 1 (RDP for the Gaussian mechanism)</strong>. Using the scaling and translation invariance of <span class="math inline">\(L\)</span> (6.1), we have that the divergence variable for two Gaussians with the same variance is</p>
<p><span class="math display">\[L(N(\mu_1, \sigma^2 I) || N(\mu_2, \sigma^2 I)) \overset{d}{=} L(N(0, I) || N((\mu_2 - \mu_1) / \sigma, I)).\]</span></p>
<p>With this we get</p>
<p><span class="math display">\[D_\lambda(N(\mu_1, \sigma^2 I) || N(\mu_2, \sigma^2 I)) = {\lambda \|\mu_2 - \mu_1\|_2^2 \over 2 \sigma^2} = D_\lambda(N(\mu_2, \sigma^2 I) || N(\mu_1, \sigma^2 I)).\]</span></p>
<p>Again due to the scaling invariance of <span class="math inline">\(L\)</span>, we only need to consider <span class="math inline">\(f\)</span> with sensitivity <span class="math inline">\(1\)</span>, see the discussion under (6.1). The Gaussian mechanism on query <span class="math inline">\(f\)</span> is thus <span class="math inline">\((\lambda, \lambda / 2 \sigma^2)\)</span>-rdp for any <span class="math inline">\(\lambda > 1\)</span>.</p>
<p>From the example of Gaussian mechanism, we see that the relation between <span class="math inline">\(\lambda\)</span> and <span class="math inline">\(\rho\)</span> is like that between <span class="math inline">\(\epsilon\)</span> and <span class="math inline">\(\delta\)</span>. Given <span class="math inline">\(\lambda\)</span> (resp. <span class="math inline">\(\rho\)</span>) and parameters like variance of the noise and the sensitivity of the query, we can write <span class="math inline">\(\rho = \rho(\lambda)\)</span> (resp. <span class="math inline">\(\lambda = \lambda(\rho)\)</span>).</p>
<p>Using the Chernoff bound (6.7), we can bound the divergence variable:</p>
<p><span class="math display">\[\mathbb P(L(p || q) \ge \epsilon) \le {\mathbb E \exp(t L(p || q)) \over \exp(t \epsilon))} = \exp (\kappa_{p, q}(t) - \epsilon t). \qquad (7.7)\]</span></p>
<p>For a function <span class="math inline">\(f: I \to \mathbb R\)</span>, denote its <a href="https://en.wikipedia.org/wiki/Legendre_transformation">Legendre transform</a> by</p>
<p><span class="math display">\[f^*(\epsilon) := \sup_{t \in I} (\epsilon t - f(t)).\]</span></p>
<p>By taking infimum on the RHS of (7.7), we obtain</p>
<p><strong>Claim 20</strong>. Two probability densities <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\((\epsilon, \exp(-\kappa_{p, q}^*(\epsilon)))\)</span>-ind.</p>
<p>Given a mechanism <span class="math inline">\(M\)</span>, let <span class="math inline">\(\kappa_M(t)\)</span> denote an upper bound for the cumulant of its privacy loss:</p>
<p><span class="math display">\[\log \mathbb E \exp(t L(M(x) || M(x'))) \le \kappa_M(t), \qquad \forall x, x'\text{ with } d(x, x') = 1.\]</span></p>
<p>For example, we can set <span class="math inline">\(\kappa_M(t) = t \rho(t + 1)\)</span>. Using the same argument we have the following:</p>
<p><strong>Claim 21</strong>. If <span class="math inline">\(M\)</span> is <span class="math inline">\((\lambda, \rho)\)</span>-rdp, then</p>
<ol style="list-style-type: decimal">
<li>it is also <span class="math inline">\((\epsilon, \exp((\lambda - 1) (\rho - \epsilon)))\)</span>-dp for any <span class="math inline">\(\epsilon \ge \rho\)</span>.</li>
<li>Alternatively, <span class="math inline">\(M\)</span> is <span class="math inline">\((\epsilon, - \exp(\kappa_M^*(\epsilon)))\)</span>-dp for any <span class="math inline">\(\epsilon > 0\)</span>.</li>
<li>Alternatively, for any <span class="math inline">\(0 < \delta \le 1\)</span>, <span class="math inline">\(M\)</span> is <span class="math inline">\((\rho + (\lambda - 1)^{-1} \log \delta^{-1}, \delta)\)</span>-dp.</li>
</ol>
<p><strong>Example 2 (Gaussian mechanism)</strong>. We can apply the above argument to the Gaussian mechanism on query <span class="math inline">\(f\)</span> and get:</p>
<p><span class="math display">\[\delta \le \inf_{\lambda > 1} \exp((\lambda - 1) ({\lambda \over 2 \sigma^2} - \epsilon))\]</span></p>
<p>By assuming <span class="math inline">\(\sigma^2 > (2 \epsilon)^{-1}\)</span> we have that the infimum is achieved when <span class="math inline">\(\lambda = (1 + 2 \epsilon / \sigma^2) / 2\)</span> and</p>
<p><span class="math display">\[\delta \le \exp(- ((2 \sigma)^{-1} - \epsilon \sigma)^2 / 2)\]</span></p>
<p>which is the same result as (6.8), obtained using the Chernoff bound of the noise.</p>
<p>However, as we will see later, compositions will yield different results from those obtained from methods in <a href="/posts/2019-03-13-a-tail-of-two-densities.html">Part 1</a> when considering Rényi dp.</p>
<h3 id="moment-composition">Moment Composition</h3>
<p><strong>Claim 22 (Moment Composition Theorem)</strong>. Let <span class="math inline">\(M\)</span> be the adaptive composition of <span class="math inline">\(M_{1 : k}\)</span>. Suppose for any <span class="math inline">\(y_{< i}\)</span>, <span class="math inline">\(M_i(y_{< i})\)</span> is <span class="math inline">\((\lambda, \rho)\)</span>-rdp. Then <span class="math inline">\(M\)</span> is <span class="math inline">\((\lambda, k\rho)\)</span>-rdp.</p>
<p><strong>Proof</strong>. Rather straightforward. As before let <span class="math inline">\(p_i\)</span> and <span class="math inline">\(q_i\)</span> be the conditional laws of adpative composition of <span class="math inline">\(M_{1 : i}\)</span> at <span class="math inline">\(x\)</span> and <span class="math inline">\(x'\)</span> respectively, and <span class="math inline">\(p^i\)</span> and <span class="math inline">\(q^i\)</span> be the joint laws of <span class="math inline">\(M_{1 : i}\)</span> at <span class="math inline">\(x\)</span> and <span class="math inline">\(x'\)</span> respectively. Denote</p>
<p><span class="math display">\[D_i = \mathbb E \exp((\lambda - 1)\log {p^i(\xi_{1 : i}) \over q^i(\xi_{1 : i})})\]</span></p>
<p>Then</p>
<p><span class="math display">\[\begin{aligned}
D_i &= \mathbb E\mathbb E \left(\exp((\lambda - 1)\log {p_i(\xi_i | \xi_{< i}) \over q_i(\xi_i | \xi_{< i})}) \exp((\lambda - 1)\log {p^{i - 1}(\xi_{< i}) \over q^{i - 1}(\xi_{< i})}) \big| \xi_{< i}\right) \\
&= \mathbb E \mathbb E \left(\exp((\lambda - 1)\log {p_i(\xi_i | \xi_{< i}) \over q_i(\xi_i | \xi_{< i})}) | \xi_{< i}\right) \exp\left((\lambda - 1)\log {p^{i - 1}(\xi_{< i}) \over q^{i - 1}(\xi_{< i})}\right)\\
&\le \mathbb E \exp((\lambda - 1) \rho) \exp\left((\lambda - 1)\log {p^{i - 1}(\xi_{< i}) \over q^{i - 1}(\xi_{< i})}\right)\\
&= \exp((\lambda - 1) \rho) D_{i - 1}.
\end{aligned}\]</span></p>
<p>Applying this recursively we have</p>
<p><span class="math display">\[D_k \le \exp(k(\lambda - 1) \rho),\]</span></p>
<p>and so</p>
<p><span class="math display">\[(\lambda - 1)^{-1} \log \mathbb E \exp((\lambda - 1)\log {p^k(\xi_{1 : i}) \over q^k(\xi_{1 : i})}) = (\lambda - 1)^{-1} \log D_k \le k \rho.\]</span></p>
<p>Since this holds for all <span class="math inline">\(x\)</span> and <span class="math inline">\(x'\)</span>, we are done. <span class="math inline">\(\square\)</span></p>
<p>This, together with the scaling property of the legendre transformation:</p>
<p><span class="math display">\[(k f)^*(x) = k f^*(x / k)\]</span></p>
<p>yields</p>
<p><strong>Claim 23</strong>. The <span class="math inline">\(k\)</span>-fold adaptive composition of <span class="math inline">\((\lambda, \rho(\lambda))\)</span>-rdp mechanisms is <span class="math inline">\((\epsilon, \exp(- k \kappa^*(\epsilon / k)))\)</span>-dp, where <span class="math inline">\(\kappa(t) := t \rho(t + 1)\)</span>.</p>
<p><strong>Example 3 (Gaussian mechanism)</strong>. We can apply the above claim to Gaussian mechanism. Again, without loss of generality we assume <span class="math inline">\(S_f = 1\)</span>. But let us do it manually to get the same results. If we apply the Moment Composition Theorem to the an adaptive composition of Gaussian mechanisms on the same query, then since each <span class="math inline">\(M_i\)</span> is <span class="math inline">\((\lambda, (2 \sigma^2)^{-1} \lambda)\)</span>-rdp, the composition <span class="math inline">\(M\)</span> is <span class="math inline">\((\lambda, (2 \sigma^2)^{-1} k \lambda)\)</span>-rdp. Processing this using the Chernoff bound as in the previous example, we have</p>
<p><span class="math display">\[\delta = \exp(- ((2 \sigma / \sqrt k)^{-1} - \epsilon \sigma / \sqrt k)^2 / 2),\]</span></p>
<p>Substituting <span class="math inline">\(\sigma\)</span> with <span class="math inline">\(\sigma / \sqrt k\)</span> in (6.81), we conclude that if</p>
<p><span class="math display">\[\sigma > \sqrt k \left(\epsilon^{-1} \sqrt{2 \log \delta^{-1}} + (2 \epsilon)^{- {1 \over 2}}\right)\]</span></p>
<p>then the composition <span class="math inline">\(M\)</span> is <span class="math inline">\((\epsilon, \delta)\)</span>-dp.</p>
<p>As we will see in the discussions at the end of this post, this result is different from (and probably better than) the one obtained by using the Advanced Composition Theorem (Claim 18).</p>
<h3 id="subsampling">Subsampling</h3>
<p>We also have a subsampling theorem for the Rényi dp.</p>
<p><strong>Claim 24</strong>. Fix <span class="math inline">\(r \in [0, 1]\)</span>. Let <span class="math inline">\(m \le n\)</span> be two nonnegative integers with <span class="math inline">\(m = r n\)</span>. Let <span class="math inline">\(N\)</span> be a <span class="math inline">\((\lambda, \rho)\)</span>-rdp machanism on <span class="math inline">\(X^m\)</span>. Let <span class="math inline">\(\mathcal I := \{J \subset [n]: |J| = m\}\)</span> be the set of subsets of <span class="math inline">\([n]\)</span> of size <span class="math inline">\(m\)</span>. Define mechanism <span class="math inline">\(M\)</span> on <span class="math inline">\(X^n\)</span> by</p>
<p><span class="math display">\[M(x) = N(x_\gamma)\]</span></p>
<p>where <span class="math inline">\(\gamma\)</span> is sampled uniformly from <span class="math inline">\(\mathcal I\)</span>. Then <span class="math inline">\(M\)</span> is <span class="math inline">\((\lambda, {1 \over \lambda - 1} \log (1 + r(e^{(\lambda - 1) \rho} - 1)))\)</span>-rdp.</p>
<p>To prove Claim 24, we need a useful lemma:</p>
<p><strong>Claim 25</strong>. Let <span class="math inline">\(p_{1 : n}\)</span> and <span class="math inline">\(q_{1 : n}\)</span> be nonnegative integers, and <span class="math inline">\(\lambda > 1\)</span>. Then</p>
<p><span class="math display">\[{(\sum p_i)^\lambda \over (\sum q_i)^{\lambda - 1}} \le \sum_i {p_i^\lambda \over q_i^{\lambda - 1}}. \qquad (8)\]</span></p>
<p><strong>Proof</strong>. Let</p>
<p><span class="math display">\[r(i) := p_i / P, \qquad u(i) := q_i / Q\]</span></p>
<p>where</p>
<p><span class="math display">\[P := \sum p_i, \qquad Q := \sum q_i\]</span></p>
<p>then <span class="math inline">\(r\)</span> and <span class="math inline">\(u\)</span> are probability mass functions. Plugging in <span class="math inline">\(p_i = r(i) P\)</span> and <span class="math inline">\(q_i = u(i) Q\)</span> into the objective (8), it suffices to show</p>
<p><span class="math display">\[1 \le \sum_i {r(i)^\lambda \over u(i)^{\lambda - 1}} = \mathbb E_{\xi \sim u} \left({r(\xi) \over u(\xi)}\right)^\lambda\]</span></p>
<p>This is true due to Jensen's Inequality:</p>
<p><span class="math display">\[\mathbb E_{\xi \sim u} \left({r(\xi) \over u(\xi)}\right)^\lambda \ge \left(\mathbb E_{\xi \sim u} {r(\xi) \over u(\xi)} \right)^\lambda = 1.\]</span></p>
<p><span class="math inline">\(\square\)</span></p>
<p><strong>Proof of Claim 24</strong>. Define <span class="math inline">\(\mathcal I\)</span> as before.</p>
<p>Let <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> be the laws of <span class="math inline">\(M(x)\)</span> and <span class="math inline">\(M(x')\)</span> respectively. For any <span class="math inline">\(I \in \mathcal I\)</span>, let <span class="math inline">\(p_I\)</span> and <span class="math inline">\(q_I\)</span> be the laws of <span class="math inline">\(N(x_I)\)</span> and <span class="math inline">\(N(x_I')\)</span> respectively. Then we have</p>
<p><span class="math display">\[\begin{aligned}
p(y) &= n^{-1} \sum_{I \in \mathcal I} p_I(y) \\
q(y) &= n^{-1} \sum_{I \in \mathcal I} q_I(y),
\end{aligned}\]</span></p>
<p>where <span class="math inline">\(n = |\mathcal I|\)</span>.</p>
<p>The MGF of <span class="math inline">\(L(p || q)\)</span> is thus</p>
<p><span class="math display">\[\mathbb E((\lambda - 1) L(p || q)) = n^{-1} \int {(\sum_I p_I(y))^\lambda \over (\sum_I q_I(y))^{\lambda - 1}} dy \le n^{-1} \sum_I \int {p_I(y)^\lambda \over q_I(y)^{\lambda - 1}} dy \qquad (9)\]</span></p>
<p>where in the last step we used Claim 25. As in the proof of Claim 19, we divide <span class="math inline">\(\mathcal I\)</span> into disjoint sets <span class="math inline">\(\mathcal I_\in\)</span> and <span class="math inline">\(\mathcal I_\notin\)</span>. Furthermore we denote by <span class="math inline">\(n_\in\)</span> and <span class="math inline">\(n_\notin\)</span> their cardinalities. Then the right hand side of (9) becomes</p>
<p><span class="math display">\[n^{-1} \sum_{I \in \mathcal I_\in} \int {p_I(y)^\lambda \over q_I(y)^{\lambda - 1}} dy + n^{-1} \sum_{I \in \mathcal I_\notin} \int {p_I(y)^\lambda \over q_I(y)^{\lambda - 1}} dy\]</span></p>
<p>The summands in the first are the MGF of <span class="math inline">\(L(p_I || q_I)\)</span>, and the summands in the second term are <span class="math inline">\(1\)</span>, so</p>
<p><span class="math display">\[\begin{aligned}
\mathbb E((\lambda - 1) L(p || q)) &\le n^{-1} \sum_{I \in \mathcal I_\in} \mathbb E \exp((\lambda - 1) L(p_I || q_I)) + (1 - r) \\
&\le n^{-1} \sum_{I \in \mathcal I_\in} \exp((\lambda - 1) D_\lambda(p_I || q_I)) + (1 - r) \\
&\le r \exp((\lambda - 1) \rho) + (1 - r).
\end{aligned}\]</span></p>
<p>Taking log and dividing by <span class="math inline">\((\lambda - 1)\)</span> on both sides we have</p>
<p><span class="math display">\[D_\lambda(p || q) \le (\lambda - 1)^{-1} \log (1 + r(\exp((\lambda - 1) \rho) - 1)).\]</span></p>
<p><span class="math inline">\(\square\)</span></p>
<p>As before, we can rewrite the conclusion of Lemma 6 using <span class="math inline">\(1 + z \le e^z\)</span> and obtain <span class="math inline">\((\lambda, (\lambda - 1)^{-1} r (e^{(\lambda - 1) \rho} - 1))\)</span>-rdp, which further gives <span class="math inline">\((\lambda, \alpha^{-1} (e^\alpha - 1) r \rho)\)</span>-rdp (or <span class="math inline">\((\lambda, O(r \rho))\)</span>-rdp) if <span class="math inline">\((\lambda - 1) \rho < \alpha\)</span> for some <span class="math inline">\(\alpha\)</span>.</p>
<p>It is not hard to see that the subsampling theorem in moment method, even though similar to the results of that in the usual method, does not help due to lack of an analogue of advanced composition theorem of the moments.</p>
<p><strong>Example 4 (Gaussian mechanism)</strong>. Applying the moment subsampling theorem to the Gaussian mechanism, we obtain <span class="math inline">\((\lambda, O(r \lambda / \sigma^2))\)</span>-rdp for a subsampled Gaussian mechanism with rate <span class="math inline">\(r\)</span>. Abadi-Chu-Goodfellow-McMahan-Mironov-Talwar-Zhang 2016 (ACGMMTZ16 in the following), however, gains an extra <span class="math inline">\(r\)</span> in the bound given certain assumptions.</p>
<h2 id="acgmmtz16">ACGMMTZ16</h2>
<p>What follows is my understanding of this result. I call it a conjecture because there is a gap which I am not able to reproduce their proof or prove it myself. This does not mean the result is false. On the contrary, I am inclined to believe it is true.</p>
<p><strong>Claim 26</strong>. Assuming Conjecture 1 (see below) is true. For a subsampled Gaussian mechanism with ratio <span class="math inline">\(r\)</span>, if <span class="math inline">\(r = O(\sigma^{-1})\)</span> and <span class="math inline">\(\lambda = O(\sigma^2)\)</span>, then we have <span class="math inline">\((\lambda, O(r^2 \lambda / \sigma^2))\)</span>-rdp.</p>
<p>Wait, why is there a conjecture? Well, I have tried but not been able to prove the following, which is a hidden assumption in the original proof:</p>
<p><strong>Conjecture 1</strong>. Let <span class="math inline">\(M\)</span> be the Gaussian mechanism with subsampling rate <span class="math inline">\(r\)</span>, and <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> be the laws of <span class="math inline">\(M(x)\)</span> and <span class="math inline">\(M(x')\)</span> respectively, where <span class="math inline">\(d(x, x') = 1\)</span>. Then</p>
<p><span class="math display">\[D_\lambda (p || q) \le D_\lambda (r \mu_1 + (1 - r) \mu_0 || \mu_0)\]</span></p>
<p>where <span class="math inline">\(\mu_i = N(i, \sigma^2)\)</span>.</p>
<p><strong>Remark</strong>. Conjecture 1 is heuristically reasonable. To see this, let us use the notations <span class="math inline">\(p_I\)</span> and <span class="math inline">\(q_I\)</span> to be <span class="math inline">\(q\)</span> and <span class="math inline">\(p\)</span> conditioned on the subsampling index <span class="math inline">\(I\)</span>, just like in the proof of the subsampling theorems (Claim 19 and 24). Then for <span class="math inline">\(I \in \mathcal I_\in\)</span>,</p>
<p><span class="math display">\[D_\lambda(p_I || q_I) \le D_\lambda(\mu_0 || \mu_1),\]</span></p>
<p>and for <span class="math inline">\(I \in \mathcal I_\notin\)</span>,</p>
<p><span class="math display">\[D_\lambda(p_I || q_I) = 0 = D_\lambda(\mu_0 || \mu_0).\]</span></p>
<p>Since we are taking an average over <span class="math inline">\(\mathcal I\)</span>, of which <span class="math inline">\(r |\mathcal I|\)</span> are in <span class="math inline">\(\mathcal I_\in\)</span> and <span class="math inline">\((1 - r) |\mathcal I|\)</span> are in <span class="math inline">\(\mathcal I_\notin\)</span>, (9.3) says "the inequalities carry over averaging".</p>
<p><a href="https://math.stackexchange.com/a/3152296/149540">A more general version of Conjecture 1 has been proven false</a>. The counter example for the general version does not apply here, so it is still possible Conjecture 1 is true.</p>
<p>Let <span class="math inline">\(p_\in\)</span> (resp. <span class="math inline">\(q_\in\)</span>) be the average of <span class="math inline">\(p_I\)</span> (resp. <span class="math inline">\(q_I\)</span>) over <span class="math inline">\(I \in \mathcal I_\in\)</span>, and <span class="math inline">\(p_\notin\)</span> (resp. <span class="math inline">\(q_\notin\)</span>) be the average of <span class="math inline">\(p_I\)</span> (resp. <span class="math inline">\(q_I\)</span>) over <span class="math inline">\(I \in \mathcal I_\notin\)</span>.</p>
<p>Immediately we have <span class="math inline">\(p_\notin = q_\notin\)</span>, hence</p>
<p><span class="math display">\[D_\lambda(p_\notin || q_\notin) = 0 = D_\lambda(\mu_0 || \mu_0). \qquad(9.7)\]</span></p>
<p>By Claim 25, we have</p>
<p><span class="math display">\[D_\lambda(p_\in || q_\in) \le D_\lambda (\mu_1 || \mu_0). \qquad(9.9) \]</span></p>
<p>So one way to prove Conjecture 1 is perhaps prove a more specialised comparison theorem than the false conjecture:</p>
<p>Given (9.7) and (9.9), show that</p>
<p><span class="math display">\[D_\lambda(r p_\in + (1 - r) p_\notin || r q_\in + (1 - r) q_\notin) \le D_\lambda(r \mu_1 + (1 - r) \mu_0 || \mu_0).\]</span></p>
<p>[End of Remark]</p>
<!---
**Conjecture 1** \[Probably [FALSE](https://math.stackexchange.com/a/3152296/149540), to be removed\]. Let $p_i$, $q_i$, $\mu_i$, $\nu_i$ be
probability densities on the same space for $i = 1 : n$. If
$D_\lambda(p_i || q_i) \le D_\lambda(\mu_i || \nu_i)$ for all $i$, then
$$D_\lambda(n^{-1} \sum_i p_i || n^{-1} \sum_i q_i) \le D_\lambda(n^{-1} \sum_i \mu_i || n^{-1} \sum_i \nu_i).$$
Basically, it is saying \"if for each $i$, $p_i$ and $q_i$ are closer to
each other than $\mu_i$ and $\nu_i$, then so are their averages over
$i$\".
So it is heuristically reasonable.
But it is probably [**FALSE**](https://math.stackexchange.com/a/3152296/149540).
This does not mean Claim 26 is false, as it might still be possible that Conjecture 2 (see below) is true.
This conjecture is equivalent to its special case when $n = 2$ by an induction argument
(replacing one pair of densities at a time).
-->
<p>Recall the definition of <span class="math inline">\(G_\lambda\)</span> under the definition of Rényi differential privacy. The following Claim will be useful.</p>
<p><strong>Claim 27</strong>. Let <span class="math inline">\(\lambda\)</span> be a positive integer, then</p>
<p><span class="math display">\[G_\lambda(r p + (1 - r) q || q) = \sum_{k = 1 : \lambda} {\lambda \choose k} r^k (1 - r)^{\lambda - k} G_k(p || q).\]</span></p>
<p><strong>Proof</strong>. Quite straightforward, by expanding the numerator <span class="math inline">\((r p + (1 - r) q)^\lambda\)</span> using binomial expansion. <span class="math inline">\(\square\)</span></p>
<p><strong>Proof of Claim 26</strong>. By Conjecture 1, it suffices to prove the following:</p>
<p>If <span class="math inline">\(r \le c_1 \sigma^{-1}\)</span> and <span class="math inline">\(\lambda \le c_2 \sigma^2\)</span> for some positive constant <span class="math inline">\(c_1\)</span> and <span class="math inline">\(c_2\)</span>, then there exists <span class="math inline">\(C = C(c_1, c_2)\)</span> such that <span class="math inline">\(G_\lambda (r \mu_1 + (1 - r) \mu_0 || \mu_0) \le C\)</span> (since <span class="math inline">\(O(r^2 \lambda^2 / \sigma^2) = O(1)\)</span>).</p>
<p><strong>Remark in the proof</strong>. Note that the choice of <span class="math inline">\(c_1\)</span>, <span class="math inline">\(c_2\)</span> and the function <span class="math inline">\(C(c_1, c_2)\)</span> are important to the practicality and usefulness of Claim 26.</p>
<!---
Part 1 can be derived using Conjecture 1, but since Conjecture 1 is probably false,
let us rename Part 1 itself _Conjecture 2_, which needs to be verified by other means.
We use the notations $p_I$ and $q_I$ to be $q$ and $p$ conditioned on
the subsampling index $I$, just like in the proof of the subsampling theorems (Claim 19 and 24).
Then
$$D_\lambda(q_I || p_I) = D_\lambda(p_I || q_I)
\begin{cases}
\le D_\lambda(\mu_0 || \mu_1) = D_\lambda(\mu_1 || \mu_0), & I \in \mathcal I_\in\\
= D_\lambda(\mu_0 || \mu_0) = D_\lambda(\mu_1 || \mu_1) = 0 & I \in \mathcal I_\notin
\end{cases}$$
Since $p = |\mathcal I|^{-1} \sum_{I \in \mathcal I} p_I$ and
$q = |\mathcal I|^{-1} \sum_{I \in \mathcal I} q_I$ and
$|\mathcal I_\in| = r |\mathcal I|$, by Conjecture 1, we have Part 1.
**Remark in the proof**. As we can see here, instead of trying to prove Conjecture 1,
it suffices to prove a weaker version of it, by specialising on mixture of Gaussians,
in order to have a Claim 26 without any conjectural assumptions.
I have in fact posted the Conjecture on [Stackexchange](https://math.stackexchange.com/questions/3147963/an-inequality-related-to-the-renyi-divergence).
Now let us verify Part 2.
-->
<p>Using Claim 27 and Example 1, we have</p>
<p><span class="math display">\[\begin{aligned}
G_\lambda(r \mu_1 + (1 - r) \mu_0 || \mu_0)) &= \sum_{j = 0 : \lambda} {\lambda \choose j} r^j (1 - r)^{\lambda - j} G_j(\mu_1 || \mu_0)\\
&=\sum_{j = 0 : \lambda} {\lambda \choose j} r^j (1 - r)^{\lambda - j} \exp(j (j - 1) / 2 \sigma^2). \qquad (9.5)
\end{aligned}\]</span></p>
<p>Denote by <span class="math inline">\(n = \lceil \sigma^2 \rceil\)</span>. It suffices to show</p>
<p><span class="math display">\[\sum_{j = 0 : n} {n \choose j} (c_1 n^{- 1 / 2})^j (1 - c_1 n^{- 1 / 2})^{n - j} \exp(c_2 j (j - 1) / 2 n) \le C\]</span></p>
<p>Note that we can discard the linear term <span class="math inline">\(- c_2 j / \sigma^2\)</span> in the exponential term since we want to bound the sum from above.</p>
<p>We examine the asymptotics of this sum when <span class="math inline">\(n\)</span> is large, and treat the sum as an approximation to an integration of a function <span class="math inline">\(\phi: [0, 1] \to \mathbb R\)</span>. For <span class="math inline">\(j = x n\)</span>, where <span class="math inline">\(x \in (0, 1)\)</span>, <span class="math inline">\(\phi\)</span> is thus defined as (note we multiply the summand with <span class="math inline">\(n\)</span> to compensate the uniform measure on <span class="math inline">\(1, ..., n\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
\phi_n(x) &:= n {n \choose j} (c_1 n^{- 1 / 2})^j (1 - c_1 n^{- 1 / 2})^{n - j} \exp(c_2 j^2 / 2 n) \\
&= n {n \choose x n} (c_1 n^{- 1 / 2})^{x n} (1 - c_1 n^{- 1 / 2})^{(1 - x) n} \exp(c_2 x^2 n / 2)
\end{aligned}\]</span></p>
<p>Using Stirling's approximation</p>
<p><span class="math display">\[n! \approx \sqrt{2 \pi n} n^n e^{- n},\]</span></p>
<p>we can approach the binomial coefficient:</p>
<p><span class="math display">\[{n \choose x n} \approx (\sqrt{2 \pi x (1 - x)} x^{x n} (1 - x)^{(1 - x) n})^{-1}.\]</span></p>
<p>We also approximate</p>
<p><span class="math display">\[(1 - c_1 n^{- 1 / 2})^{(1 - x) n} \approx \exp(- c_1 \sqrt{n} (1 - x)).\]</span></p>
<p>With these we have</p>
<p><span class="math display">\[\phi_n(x) \approx {1 \over \sqrt{2 \pi x (1 - x)}} \exp\left(- {1 \over 2} x n \log n + (x \log c_1 - x \log x - (1 - x) \log (1 - x) + {1 \over 2} c_2 x^2) n + {1 \over 2} \log n\right).\]</span></p>
<p>This vanishes as <span class="math inline">\(n \to \infty\)</span>, and since <span class="math inline">\(\phi_n(x)\)</span> is bounded above by the integrable function <span class="math inline">\({1 \over \sqrt{2 \pi x (1 - x)}}\)</span> (c.f. the arcsine law), and below by <span class="math inline">\(0\)</span>, we may invoke the dominant convergence theorem and exchange the integral with the limit and get</p>
<p><span class="math display">\[\begin{aligned}
\lim_{n \to \infty} &G_n (r \mu_1 + (1 - r) \mu_0 || \mu_0)) \\
&\le \lim_{n \to \infty} \int \phi_n(x) dx = \int \lim_{n \to \infty} \phi_n(x) dx = 0.
\end{aligned}\]</span></p>
<p>Thus we have that the generating function of the divergence variable <span class="math inline">\(L(r \mu_1 + (1 - r) \mu_0 || \mu_0)\)</span> is bounded.</p>
<p>Can this be true for better orders</p>
<p><span class="math display">\[r \le c_1 \sigma^{- d_r},\qquad \lambda \le c_2 \sigma^{d_\lambda}\]</span></p>
<p>for some <span class="math inline">\(d_r \in (0, 1]\)</span> and <span class="math inline">\(d_\lambda \in [2, \infty)\)</span>? If we follow the same approximation using these exponents, then letting <span class="math inline">\(n = c_2 \sigma^{d_\lambda}\)</span>,</p>
<p><span class="math display">\[\begin{aligned}
{n \choose j} &r^j (1 - r)^{n - j} G_j(\mu_0 || \mu_1) \le \phi_n(x) \\
&\approx {1 \over \sqrt{2 \pi x (1 - x)}} \exp\left({1 \over 2} c_2^{2 \over d_\lambda} x^2 n^{2 - {2 \over d_\lambda}} - {d_r \over 2} x n \log n + (x \log c_1 - x \log x - (1 - x) \log (1 - x)) n + {1 \over 2} \log n\right).
\end{aligned}\]</span></p>
<p>So we see that to keep the divergence moments bounded it is possible to have any <span class="math inline">\(r = O(\sigma^{- d_r})\)</span> for <span class="math inline">\(d_r \in (0, 1)\)</span>, but relaxing <span class="math inline">\(\lambda\)</span> may not be safe.</p>
<p>If we relax <span class="math inline">\(r\)</span>, then we get</p>
<p><span class="math display">\[G_\lambda(r \mu_1 + (1 - r) \mu_0 || \mu_0) = O(r^{2 / d_r} \lambda^2 \sigma^{-2}) = O(1).\]</span></p>
<p>Note that now the constant <span class="math inline">\(C\)</span> depends on <span class="math inline">\(d_r\)</span> as well. Numerical experiments seem to suggest that <span class="math inline">\(C\)</span> can increase quite rapidly as <span class="math inline">\(d_r\)</span> decreases from <span class="math inline">\(1\)</span>. <span class="math inline">\(\square\)</span></p>
<p>In the following for consistency we retain <span class="math inline">\(k\)</span> as the number of epochs, and use <span class="math inline">\(T := k / r\)</span> to denote the number of compositions / steps / minibatches. With Claim 26 we have:</p>
<p><strong>Claim 28</strong>. Assuming Conjecture 1 is true. Let <span class="math inline">\(\epsilon, c_1, c_2 > 0\)</span>, <span class="math inline">\(r \le c_1 \sigma^{-1}\)</span>, <span class="math inline">\(T = {c_2 \over 2 C(c_1, c_2)} \epsilon \sigma^2\)</span>. then the DP-SGD with subsampling rate <span class="math inline">\(r\)</span>, and <span class="math inline">\(T\)</span> steps is <span class="math inline">\((\epsilon, \delta)\)</span>-dp for</p>
<p><span class="math display">\[\delta = \exp(- {1 \over 2} c_2 \sigma^2 \epsilon).\]</span></p>
<p>In other words, for</p>
<p><span class="math display">\[\sigma \ge \sqrt{2 c_2^{-1}} \epsilon^{- {1 \over 2}} \sqrt{\log \delta^{-1}},\]</span></p>
<p>we can achieve <span class="math inline">\((\epsilon, \delta)\)</span>-dp.</p>
<p><strong>Proof</strong>. By Claim 26 and the Moment Composition Theorem (Claim 22), for <span class="math inline">\(\lambda = c_2 \sigma^2\)</span>, substituting <span class="math inline">\(T = {c_2 \over 2 C(c_1, c_2)} \epsilon \sigma^2\)</span>, we have</p>
<p><span class="math display">\[\mathbb P(L(p || q) \ge \epsilon) \le \exp(k C(c_1, c_2) - \lambda \epsilon) = \exp\left(- {1 \over 2} c_2 \sigma^2 \epsilon\right).\]</span></p>
<p><span class="math inline">\(\square\)</span></p>
<p><strong>Remark</strong>. Claim 28 is my understanding / version of Theorem 1 in [ACGMMTZ16], by using the same proof technique. Here I quote the original version of theorem with notions and notations altered for consistency with this post:</p>
<blockquote>
<p>There exists constants <span class="math inline">\(c_1', c_2' > 0\)</span> so that for any <span class="math inline">\(\epsilon < c_1' r^2 T\)</span>, DP-SGD is <span class="math inline">\((\epsilon, \delta)\)</span>-differentially private for any <span class="math inline">\(\delta > 0\)</span> if we choose</p>
</blockquote>
<p><span class="math display">\[\sigma \ge c_2' {r \sqrt{T \log (1 / \delta)} \over \epsilon}. \qquad (10)\]</span></p>
<p>I am however unable to reproduce this version, assuming Conjecture 1 is true, for the following reasons:</p>
<ol style="list-style-type: decimal">
<li><p>In the proof in the paper, we have <span class="math inline">\(\epsilon = c_1' r^2 T\)</span> instead of "less than" in the statement of the Theorem. If we change it to <span class="math inline">\(\epsilon < c_1' r^2 T\)</span> then the direction of the inequality becomes opposite to the direction we want to prove: <span class="math display">\[\exp(k C(c_1, c_2) - \lambda \epsilon) \ge ...\]</span></p></li>
<li><p>The condition <span class="math inline">\(r = O(\sigma^{-1})\)</span> of Claim 26 whose result is used in the proof of this theorem is not mentioned in the statement of the proof. The implication is that (10) becomes an ill-formed condition as the right hand side also depends on <span class="math inline">\(\sigma\)</span>.</p></li>
</ol>
<h2 id="tensorflow-implementation">Tensorflow implementation</h2>
<p>The DP-SGD is implemented in <a href="https://github.com/tensorflow/privacy">TensorFlow Privacy</a>. In the following I discuss the package in the current state (2019-03-11). It is divided into two parts: <a href="https://github.com/tensorflow/privacy/tree/master/privacy/optimizers"><code>optimizers</code></a> which implements the actual differentially private algorithms, and <a href="https://github.com/tensorflow/privacy/tree/master/privacy/analysis"><code>analysis</code></a> which computes the privacy guarantee.</p>
<p>The <code>analysis</code> part implements a privacy ledger that "keeps a record of all queries executed over a given dataset for the purpose of computing privacy guarantees". On the other hand, all the computation is done in <a href="https://github.com/tensorflow/privacy/blob/7e2d796bdee9b60dce21a82a397eefda35b0ac10/privacy/analysis/rdp_accountant.py"><code>rdp_accountant.py</code></a>. At this moment, <code>rdp_accountant.py</code> only implements the computation of the privacy guarantees for DP-SGD with Gaussian mechanism. In the following I will briefly explain the code in this file.</p>
<p>Some notational correspondences: their <code>alpha</code> is our <span class="math inline">\(\lambda\)</span>, their <code>q</code> is our <span class="math inline">\(r\)</span>, their <code>A_alpha</code> (in the comments) is our <span class="math inline">\(\kappa_{r N(1, \sigma^2) + (1 - r) N(0, \sigma^2)} (\lambda - 1)\)</span>, at least when <span class="math inline">\(\lambda\)</span> is an integer.</p>
<ul>
<li>The function <code>_compute_log_a</code> presumably computes the cumulants <span class="math inline">\(\kappa_{r N(1, \sigma^2) + (1 - r) N(0, \sigma^2), N(0, \sigma^2)}(\lambda - 1)\)</span>. It calls <code>_compute_log_a_int</code> or <code>_compute_log_a_frac</code> depending on whether <span class="math inline">\(\lambda\)</span> is an integer.</li>
<li>The function <code>_compute_log_a_int</code> computes the cumulant using (9.5).</li>
<li>When <span class="math inline">\(\lambda\)</span> is not an integer, we can't use (9.5). I have yet to decode how <code>_compute_log_a_frac</code> computes the cumulant (or an upper bound of it) in this case</li>
<li>The function <code>_compute_delta</code> computes <span class="math inline">\(\delta\)</span>s for a list of <span class="math inline">\(\lambda\)</span>s and <span class="math inline">\(\kappa\)</span>s using Item 1 of Claim 25 and return the smallest one, and the function <code>_compute_epsilon</code> computes epsilon uses Item 3 in Claim 25 in the same way.</li>
</ul>
<p>In <code>optimizers</code>, among other things, the DP-SGD with Gaussian mechanism is implemented in <code>dp_optimizer.py</code> and <code>gaussian_query.py</code>. See the definition of <code>DPGradientDescentGaussianOptimizer</code> in <code>dp_optimizer.py</code> and trace the calls therein.</p>
<p>At this moment, the privacy guarantee computation part and the optimizer part are separated, with <code>rdp_accountant.py</code> called in <code>compute_dp_sgd_privacy.py</code> with user-supplied parameters. I think this is due to the lack of implementation in <code>rdp_accountant.py</code> of any non-DPSGD-with-Gaussian privacy guarantee computation. There is already <a href="https://github.com/tensorflow/privacy/issues/23">an issue on this</a>, so hopefully it won't be long before the privacy guarantees can be automatically computed given a DP-SGD instance.</p>
<h2 id="comparison-among-different-methods">Comparison among different methods</h2>
<p>So far we have seen three routes to compute the privacy guarantees for DP-SGD with the Gaussian mechanism:</p>
<ol style="list-style-type: decimal">
<li>Claim 9 (single Gaussian mechanism privacy guarantee) -> Claim 19 (Subsampling theorem) -> Claim 18 (Advanced Adaptive Composition Theorem)</li>
<li>Example 1 (RDP for the Gaussian mechanism) -> Claim 22 (Moment Composition Theorem) -> Example 3 (Moment composition applied to the Gaussian mechanism)</li>
<li>Claim 26 (RDP for Gaussian mechanism with specific magnitudes for subsampling rate) -> Claim 28 (Moment Composition Theorem and translation to conventional DP)</li>
</ol>
<p>Which one is the best?</p>
<p>To make fair comparison, we may use one parameter as the metric and set all others to be the same. For example, we can</p>
<ol style="list-style-type: decimal">
<li>Given the same <span class="math inline">\(\epsilon\)</span>, <span class="math inline">\(r\)</span> (in Route 1 and 3), <span class="math inline">\(k\)</span>, <span class="math inline">\(\sigma\)</span>, compare the <span class="math inline">\(\delta\)</span>s</li>
<li>Given the same <span class="math inline">\(\epsilon\)</span>, <span class="math inline">\(r\)</span> (in Route 1 and 3), <span class="math inline">\(k\)</span>, <span class="math inline">\(\delta\)</span>, compare the <span class="math inline">\(\sigma\)</span>s</li>
<li>Given the same <span class="math inline">\(\delta\)</span>, <span class="math inline">\(r\)</span> (in Route 1 and 3), <span class="math inline">\(k\)</span>, <span class="math inline">\(\sigma\)</span>, compare the <span class="math inline">\(\epsilon\)</span>s.</li>
</ol>
<p>I find that the first one, where <span class="math inline">\(\delta\)</span> is used as a metric, the best. This is because we have the tightest bounds and the cleanest formula when comparing the <span class="math inline">\(\delta\)</span>. For example, the Azuma and Chernoff bounds are both expressed as a bound for <span class="math inline">\(\delta\)</span>. On the other hand, the inversion of these bounds either requires a cost in the tightness (Claim 9, bounds on <span class="math inline">\(\sigma\)</span>) or in the complexity of the formula (Claim 16 Advanced Adaptive Composition Theorem, bounds on <span class="math inline">\(\epsilon\)</span>).</p>
<p>So if we use <span class="math inline">\(\sigma\)</span> or <span class="math inline">\(\epsilon\)</span> as a metric, either we get a less fair comparison, or have to use a much more complicated formula as the bounds.</p>
<p>Let us first compare Route 1 and Route 2 without specialising to the Gaussian mechanism.</p>
<p><strong>Warning</strong>. What follows is a bit messy.</p>
<p>Suppose each mechanism <span class="math inline">\(N_i\)</span> satisfies <span class="math inline">\((\epsilon', \delta(\epsilon'))\)</span>-dp. Let <span class="math inline">\(\tilde \epsilon := \log (1 + r (e^{\epsilon'} - 1))\)</span>, then we have the subsampled mechanism <span class="math inline">\(M_i(x) = N_i(x_\gamma)\)</span> is <span class="math inline">\((\tilde \epsilon, r \tilde \delta(\tilde \epsilon))\)</span>-dp, where</p>
<p><span class="math display">\[\tilde \delta(\tilde \epsilon) = \delta(\log (r^{-1} (\exp(\tilde \epsilon) - 1) + 1))\]</span></p>
<p>Using the Azuma bound in the proof of Advanced Adaptive Composition Theorem (6.99):</p>
<p><span class="math display">\[\mathbb P(L(p^k || q^k) \ge \epsilon) \le \exp(- {(\epsilon - r^{-1} k a(\tilde\epsilon))^2 \over 2 r^{-1} k (\tilde\epsilon + a(\tilde\epsilon))^2}).\]</span></p>
<p>So we have the final bound for Route 1:</p>
<p><span class="math display">\[\delta_1(\epsilon) = \min_{\tilde \epsilon: \epsilon > r^{-1} k a(\tilde \epsilon)} \exp(- {(\epsilon - r^{-1} k a(\tilde\epsilon))^2 \over 2 r^{-1} k (\tilde\epsilon + a(\tilde\epsilon))^2}) + k \tilde \delta(\tilde \epsilon).\]</span></p>
<p>As for Route 2, since we do not gain anything from subsampling in RDP, we do not subsample at all.</p>
<p>By Claim 23, we have the bound for Route 2:</p>
<p><span class="math display">\[\delta_2(\epsilon) = \exp(- k \kappa^* (\epsilon / k)).\]</span></p>
<p>On one hand, one can compare <span class="math inline">\(\delta_1\)</span> and <span class="math inline">\(\delta_2\)</span> with numerical experiments. On the other hand, if we further specify <span class="math inline">\(\delta(\epsilon')\)</span> in Route 1 as the Chernoff bound for the cumulants of divergence variable, i.e.</p>
<p><span class="math display">\[\delta(\epsilon') = \exp(- \kappa^* (\epsilon')),\]</span></p>
<p>we have</p>
<p><span class="math display">\[\delta_1 (\epsilon) = \min_{\tilde \epsilon: a(\tilde \epsilon) < r k^{-1} \epsilon} \exp(- {(\epsilon - r^{-1} k a(\tilde\epsilon))^2 \over 2 r^{-1} k (\tilde\epsilon + a(\tilde\epsilon))^2}) + k \exp(- \kappa^* (b(\tilde\epsilon))),\]</span></p>
<p>where</p>
<p><span class="math display">\[b(\tilde \epsilon) := \log (r^{-1} (\exp(\tilde \epsilon) - 1) + 1) \le r^{-1} \tilde\epsilon.\]</span></p>
<p>We note that since <span class="math inline">\(a(\tilde \epsilon) = \tilde\epsilon(e^{\tilde \epsilon} - 1) 1_{\tilde\epsilon < \log 2} + \tilde\epsilon 1_{\tilde\epsilon \ge \log 2}\)</span>, we may compare the two cases separately.</p>
<p>Note that we have <span class="math inline">\(\kappa^*\)</span> is a monotonously increasing function, therefore</p>
<p><span class="math display">\[\kappa^* (b(\tilde\epsilon)) \le \kappa^*(r^{-1} \tilde\epsilon).\]</span></p>
<p>So for <span class="math inline">\(\tilde \epsilon \ge \log 2\)</span>, we have</p>
<p><span class="math display">\[k \exp(- \kappa^*(b(\tilde\epsilon))) \ge k \exp(- \kappa^*(r^{-1} \tilde \epsilon)) \ge k \exp(- \kappa^*(k^{-1} \epsilon)) \ge \delta_2(\epsilon).\]</span></p>
<p>For <span class="math inline">\(\tilde\epsilon < \log 2\)</span>, it is harder to compare, as now</p>
<p><span class="math display">\[k \exp(- \kappa^*(b(\tilde\epsilon))) \ge k \exp(- \kappa^*(\epsilon / \sqrt{r k})).\]</span></p>
<p>It is tempting to believe that this should also be greater than <span class="math inline">\(\delta_2(\epsilon)\)</span>. But I can not say for sure. At least in the special case of Gaussian, we have</p>
<p><span class="math display">\[k \exp(- \kappa^*(\epsilon / \sqrt{r k})) = k \exp(- (\sigma \sqrt{\epsilon / k r} - (2 \sigma)^{-1})^2) \ge \exp(- k ({\sigma \epsilon \over k} - (2 \sigma)^{-1})^2) = \delta_2(\epsilon)\]</span></p>
<p>when <span class="math inline">\(\epsilon\)</span> is sufficiently small. However we still need to consider the case where <span class="math inline">\(\epsilon\)</span> is not too small. But overall it seems most likely Route 2 is superior than Route 1.</p>
<p>So let us compare Route 2 with Route 3:</p>
<p>Given the condition to obtain the Chernoff bound</p>
<p><span class="math display">\[{\sigma \epsilon \over k} > (2 \sigma)^{-1}\]</span></p>
<p>we have</p>
<p><span class="math display">\[\delta_2(\epsilon) > \exp(- k (\sigma \epsilon / k)^2) = \exp(- \sigma^2 \epsilon^2 / k).\]</span></p>
<p>For this to achieve the same bound</p>
<p><span class="math display">\[\delta_3(\epsilon) = \exp\left(- {1 \over 2} c_2 \sigma^2 \epsilon\right)\]</span></p>
<p>we need <span class="math inline">\(k < {2 \epsilon \over c_2}\)</span>. This is only possible if <span class="math inline">\(c_2\)</span> is small or <span class="math inline">\(\epsilon\)</span> is large, since <span class="math inline">\(k\)</span> is a positive integer.</p>
<p>So taking at face value, Route 3 seems to achieve the best results. However, it also has some similar implicit conditions that need to be satisfied: First <span class="math inline">\(T\)</span> needs to be at least <span class="math inline">\(1\)</span>, meaning</p>
<p><span class="math display">\[{c_2 \over C(c_1, c_2)} \epsilon \sigma^2 \ge 1.\]</span></p>
<p>Second, <span class="math inline">\(k\)</span> needs to be at least <span class="math inline">\(1\)</span> as well, i.e.</p>
<p><span class="math display">\[k = r T \ge {c_1 c_2 \over C(c_1, c_2)} \epsilon \sigma \ge 1.\]</span></p>
<p>Both conditions rely on the magnitudes of <span class="math inline">\(\epsilon\)</span>, <span class="math inline">\(\sigma\)</span>, <span class="math inline">\(c_1\)</span>, <span class="math inline">\(c_2\)</span>, and the rate of growth of <span class="math inline">\(C(c_1, c_2)\)</span>. The biggest problem in this list is the last, because if we know how fast <span class="math inline">\(C\)</span> grows then we'll have a better idea what are the constraints for the parameters to achieve the result in Route 3.</p>
<h2 id="further-questions">Further questions</h2>
<p>Here is a list of what I think may be interesting topics or potential problems to look at, with no guarantee that they are all awesome untouched research problems:</p>
<ol style="list-style-type: decimal">
<li>Prove Conjecture 1</li>
<li>Find a theoretically definitive answer whether the methods in Part 1 or Part 2 yield better privacy guarantees.</li>
<li>Study the non-Gaussian cases, general or specific. Let <span class="math inline">\(p\)</span> be some probability density, what is the tail bound of <span class="math inline">\(L(p(y) || p(y + \alpha))\)</span> for <span class="math inline">\(|\alpha| \le 1\)</span>? Can you find anything better than Gaussian? For a start, perhaps the nice tables of Rényi divergence in Gil-Alajaji-Linder 2013 may be useful?</li>
<li>Find out how useful Claim 26 is. Perhaps start with computing the constant <span class="math inline">\(C\)</span> nemerically.</li>
<li>Help with <a href="https://github.com/tensorflow/privacy/issues/23">the aforementioned issue</a> in the Tensorflow privacy package.</li>
</ol>
<h2 id="references">References</h2>
<ul>
<li>Abadi, Martín, Andy Chu, Ian Goodfellow, H. Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. "Deep Learning with Differential Privacy." Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security - CCS'16, 2016, 308--18. <a href="https://doi.org/10.1145/2976749.2978318" class="uri">https://doi.org/10.1145/2976749.2978318</a>.</li>
<li>Erven, Tim van, and Peter Harremoës. "R\'enyi Divergence and Kullback-Leibler Divergence." IEEE Transactions on Information Theory 60, no. 7 (July 2014): 3797--3820. <a href="https://doi.org/10.1109/TIT.2014.2320500" class="uri">https://doi.org/10.1109/TIT.2014.2320500</a>.</li>
<li>Gil, M., F. Alajaji, and T. Linder. "Rényi Divergence Measures for Commonly Used Univariate Continuous Distributions." Information Sciences 249 (November 2013): 124--31. <a href="https://doi.org/10.1016/j.ins.2013.06.018" class="uri">https://doi.org/10.1016/j.ins.2013.06.018</a>.</li>
<li>Mironov, Ilya. "Renyi Differential Privacy." 2017 IEEE 30th Computer Security Foundations Symposium (CSF), August 2017, 263--75. <a href="https://doi.org/10.1109/CSF.2017.11" class="uri">https://doi.org/10.1109/CSF.2017.11</a>.</li>
</ul>
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A Tail of Two Densitiesposts/2019-03-13-a-tail-of-two-densities.html2019-03-13T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<ul>
<li><a href="#the-gist-of-differential-privacy">The gist of differential privacy</a></li>
<li><a href="#epsilon-dp"><span class="math inline">\(\epsilon\)</span>-dp</a></li>
<li><a href="#approximate-differential-privacy">Approximate differential privacy</a><ul>
<li><a href="#indistinguishability">Indistinguishability</a></li>
<li><a href="#back-to-approximate-differential-privacy">Back to approximate differential privacy</a></li>
</ul></li>
<li><a href="#composition-theorems">Composition theorems</a></li>
<li><a href="#subsampling">Subsampling</a></li>
<li><a href="#references">References</a></li>
</ul>
</div>
<p>This is Part 1 of a two-part post where I give an introduction to the mathematics of differential privacy.</p>
<p>Practically speaking, <a href="https://en.wikipedia.org/wiki/Differential_privacy">differential privacy</a> is a technique of perturbing database queries so that query results do not leak too much information while still being relatively accurate.</p>
<p>This post however focuses on the mathematical aspects of differential privacy, which is a study of <a href="https://en.wikipedia.org/wiki/Concentration_inequality">tail bounds</a> of the divergence between two probability measures, with the end goal of applying it to <a href="https://en.wikipedia.org/wiki/Stochastic_gradient_descent">stochastic gradient descent</a>. This post should be suitable for anyone familiar with probability theory.</p>
<p>I start with the definition of <span class="math inline">\(\epsilon\)</span>-differential privacy (corresponding to max divergence), followed by <span class="math inline">\((\epsilon, \delta)\)</span>-differential privacy (a.k.a. approximate differential privacy, corresponding to the <span class="math inline">\(\delta\)</span>-approximate max divergence). I show a characterisation of the <span class="math inline">\((\epsilon, \delta)\)</span>-differential privacy as conditioned <span class="math inline">\(\epsilon\)</span>-differential privacy. Also, as examples, I illustrate the <span class="math inline">\(\epsilon\)</span>-dp with Laplace mechanism and, using some common tail bounds, the approximate dp with the Gaussian mechanism.</p>
<p>Then I continue to show the effect of combinatorial and sequential compositions of randomised queries (called mechanisms) on privacy by stating and proving the composition theorems for differential privacy, as well as the effect of mixing mechanisms, by presenting the subsampling theorem (a.k.a. amplification theorem).</p>
<p>In <a href="/posts/2019-03-14-great-but-manageable-expectations.html">Part 2</a>, I discuss the Rényi differential privacy, corresponding to the Rényi divergence, a study of the <a href="https://en.wikipedia.org/wiki/Moment-generating_function">moment generating functions</a> of the divergence between probability measures to derive the tail bounds.</p>
<p>Like in Part 1, I prove a composition theorem and a subsampling theorem.</p>
<p>I also attempt to reproduce a seemingly better moment bound for the Gaussian mechanism with subsampling, with one intermediate step which I am not able to prove.</p>
<p>After that I explain the Tensorflow implementation of differential privacy in its <a href="https://github.com/tensorflow/privacy/tree/master/privacy">Privacy</a> module, which focuses on the differentially private stochastic gradient descent algorithm (DP-SGD).</p>
<p>Finally I use the results from both Part 1 and Part 2 to obtain some privacy guarantees for composed subsampling queries in general, and for DP-SGD in particular. I also compare these privacy guarantees.</p>
<p><strong>Acknowledgement</strong>. I would like to thank <a href="http://stockholm.ai">Stockholm AI</a> for introducing me to the subject of differential privacy. Thanks to Amir Hossein Rahnama for hosting the discussions at Stockholm AI. Thanks to (in chronological order) Reynaldo Boulogne, Martin Abedi, Ilya Mironov, Kurt Johansson, Mark Bun, Salil Vadhan, Jonathan Ullman, Yuanyuan Xu and Yiting Li for communication and discussions. Also thanks to the <a href="https://www.reddit.com/r/MachineLearning/">r/MachineLearning</a> community for comments and suggestions which result in improvement of readability of this post. The research was done while working at <a href="https://www.kth.se/en/sci/institutioner/math">KTH Department of Mathematics</a>.</p>
<p><em>If you are confused by any notations, ask me or try <a href="/notations.html">this</a>. This post (including both Part 1 and Part2) is licensed under <a href="https://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a> and <a href="https://www.gnu.org/licenses/fdl.html">GNU FDL</a>.</em></p>
<h2 id="the-gist-of-differential-privacy">The gist of differential privacy</h2>
<p>If you only have one minute, here is what differential privacy is about:</p>
<p>Let <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> be two probability densities, we define the <em>divergence variable</em><a href="#fn1" class="footnoteRef" id="fnref1"><sup>1</sup></a> of <span class="math inline">\((p, q)\)</span> to be</p>
<p><span class="math display">\[L(p || q) := \log {p(\xi) \over q(\xi)}\]</span></p>
<p>where <span class="math inline">\(\xi\)</span> is a random variable distributed according to <span class="math inline">\(p\)</span>.</p>
<p>Roughly speaking, differential privacy is the study of the tail bound of <span class="math inline">\(L(p || q)\)</span>: for certain <span class="math inline">\(p\)</span>s and <span class="math inline">\(q\)</span>s, and for <span class="math inline">\(\epsilon > 0\)</span>, find <span class="math inline">\(\delta(\epsilon)\)</span> such that</p>
<p><span class="math display">\[\mathbb P(L(p || q) > \epsilon) < \delta(\epsilon),\]</span></p>
<p>where <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are the laws of the outputs of a randomised functions on two very similar inputs. Moreover, to make matters even simpler, only three situations need to be considered:</p>
<ol style="list-style-type: decimal">
<li>(General case) <span class="math inline">\(q\)</span> is in the form of <span class="math inline">\(q(y) = p(y + \Delta)\)</span> for some bounded constant <span class="math inline">\(\Delta\)</span>.</li>
<li>(Compositions) <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are combinatorial or sequential compositions of some simpler <span class="math inline">\(p_i\)</span>'s and <span class="math inline">\(q_i\)</span>'s respectively</li>
<li>(Subsampling) <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are mixtures / averages of some simpler <span class="math inline">\(p_i\)</span>'s and <span class="math inline">\(q_i\)</span>'s respectively</li>
</ol>
<p>In applications, the inputs are databases and the randomised functions are queries with an added noise, and the tail bounds give privacy guarantees. When it comes to gradient descent, the input is the training dataset, and the query updates the parameters, and privacy is achieved by adding noise to the gradients.</p>
<p>Now if you have an hour...</p>
<h2 id="epsilon-dp"><span class="math inline">\(\epsilon\)</span>-dp</h2>
<p><strong>Definition (Mechanisms)</strong>. Let <span class="math inline">\(X\)</span> be a space with a metric <span class="math inline">\(d: X \times X \to \mathbb N\)</span>. A <em>mechanism</em> <span class="math inline">\(M\)</span> is a function that takes <span class="math inline">\(x \in X\)</span> as input and outputs a random variable on <span class="math inline">\(Y\)</span>.</p>
<p>In this post, <span class="math inline">\(X = Z^m\)</span> is the space of datasets of <span class="math inline">\(m\)</span> rows for some integer <span class="math inline">\(m\)</span>, where each item resides in some space <span class="math inline">\(Z\)</span>. In this case the distance <span class="math inline">\(d(x, x') := \#\{i: x_i \neq x'_i\}\)</span> is the number of rows that differ between <span class="math inline">\(x\)</span> and <span class="math inline">\(x'\)</span>.</p>
<p>Normally we have a query <span class="math inline">\(f: X \to Y\)</span>, and construct the mechanism <span class="math inline">\(M\)</span> from <span class="math inline">\(f\)</span> by adding a noise:</p>
<p><span class="math display">\[M(x) := f(x) + \text{noise}.\]</span></p>
<p>Later, we will also consider mechanisms constructed from composition or mixture of other mechanisms.</p>
<p>In this post <span class="math inline">\(Y = \mathbb R^d\)</span> for some <span class="math inline">\(d\)</span>.</p>
<p><strong>Definition (Sensitivity)</strong>. Let <span class="math inline">\(f: X \to \mathbb R^d\)</span> be a function. The <em>sensitivity</em> <span class="math inline">\(S_f\)</span> of <span class="math inline">\(f\)</span> is defined as</p>
<p><span class="math display">\[S_f := \sup_{x, x' \in X: d(x, x') = 1} \|f(x) - f(x')\|_2,\]</span></p>
<p>where <span class="math inline">\(\|y\|_2 = \sqrt{y_1^2 + ... + y_d^2}\)</span> is the <span class="math inline">\(\ell^2\)</span>-norm.</p>
<p><strong>Definition (Differential Privacy)</strong>. A mechanism <span class="math inline">\(M\)</span> is called <span class="math inline">\(\epsilon\)</span><em>-differential privacy</em> (<span class="math inline">\(\epsilon\)</span>-dp) if it satisfies the following condition: for all <span class="math inline">\(x, x' \in X\)</span> with <span class="math inline">\(d(x, x') = 1\)</span>, and for all measureable set <span class="math inline">\(S \subset \mathbb R^n\)</span>,</p>
<p><span class="math display">\[\mathbb P(M(x) \in S) \le e^\epsilon P(M(x') \in S). \qquad (1)\]</span></p>
<p>Practically speaking, this means given the results from perturbed query on two known databases that differs by one row, it is hard to determine which result is from which database.</p>
<p>An example of <span class="math inline">\(\epsilon\)</span>-dp mechanism is the Laplace mechanism.</p>
<p><strong>Definition</strong>. The <em>Laplace distribution</em> over <span class="math inline">\(\mathbb R\)</span> with parameter <span class="math inline">\(b > 0\)</span> has probability density function</p>
<p><span class="math display">\[f_{\text{Lap}(b)}(x) = {1 \over 2 b} e^{- {|x| \over b}}.\]</span></p>
<p><strong>Definition</strong>. Let <span class="math inline">\(d = 1\)</span>. The <em>Laplace mechanism</em> is defined by</p>
<p><span class="math display">\[M(x) = f(x) + \text{Lap}(b).\]</span></p>
<p><strong>Claim</strong>. The Laplace mechanism with</p>
<p><span class="math display">\[b \ge \epsilon^{-1} S_f \qquad (1.5)\]</span></p>
<p>is <span class="math inline">\(\epsilon\)</span>-dp.</p>
<p><strong>Proof</strong>. Quite straightforward. Let <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> be the laws of <span class="math inline">\(M(x)\)</span> and <span class="math inline">\(M(x')\)</span> respectively.</p>
<p><span class="math display">\[{p (y) \over q (y)} = {f_{\text{Lap}(b)} (y - f(x)) \over f_{\text{Lap}(b)} (y - f(x'))} = \exp(b^{-1} (|y - f(x')| - |y - f(x)|))\]</span></p>
<p>Using triangular inequality <span class="math inline">\(|A| - |B| \le |A - B|\)</span> on the right hand side, we have</p>
<p><span class="math display">\[{p (y) \over q (y)} \le \exp(b^{-1} (|f(x) - f(x')|)) \le \exp(\epsilon)\]</span></p>
<p>where in the last step we use the condition (1.5). <span class="math inline">\(\square\)</span></p>
<h2 id="approximate-differential-privacy">Approximate differential privacy</h2>
<p>Unfortunately, <span class="math inline">\(\epsilon\)</span>-dp does not apply to the most commonly used noise, the Gaussian noise. To fix this, we need to relax the definition a bit.</p>
<p><strong>Definition</strong>. A mechanism <span class="math inline">\(M\)</span> is said to be <span class="math inline">\((\epsilon, \delta)\)</span><em>-differentially private</em> if for all <span class="math inline">\(x, x' \in X\)</span> with <span class="math inline">\(d(x, x') = 1\)</span> and for all measureable <span class="math inline">\(S \subset \mathbb R^d\)</span></p>
<p><span class="math display">\[\mathbb P(M(x) \in S) \le e^\epsilon P(M(x') \in S) + \delta. \qquad (2)\]</span></p>
<p>Immediately we see that the <span class="math inline">\((\epsilon, \delta)\)</span>-dp is meaningful only if <span class="math inline">\(\delta < 1\)</span>.</p>
<h3 id="indistinguishability">Indistinguishability</h3>
<p>To understand <span class="math inline">\((\epsilon, \delta)\)</span>-dp, it is helpful to study <span class="math inline">\((\epsilon, \delta)\)</span>-indistinguishability.</p>
<p><strong>Definition</strong>. Two probability measures <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> on the same space are called <span class="math inline">\((\epsilon, \delta)\)</span><em>-ind(istinguishable)</em> if for all measureable sets <span class="math inline">\(S\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
p(S) \le e^\epsilon q(S) + \delta, \qquad (3) \\
q(S) \le e^\epsilon p(S) + \delta. \qquad (4)
\end{aligned}\]</span></p>
<p>As before, we also call random variables <span class="math inline">\(\xi\)</span> and <span class="math inline">\(\eta\)</span> to be <span class="math inline">\((\epsilon, \delta)\)</span>-ind if their laws are <span class="math inline">\((\epsilon, \delta)\)</span>-ind. When <span class="math inline">\(\delta = 0\)</span>, we call it <span class="math inline">\(\epsilon\)</span>-ind.</p>
<p>Immediately we have</p>
<p><strong>Claim 0</strong>. <span class="math inline">\(M\)</span> is <span class="math inline">\((\epsilon, \delta)\)</span>-dp (resp. <span class="math inline">\(\epsilon\)</span>-dp) iff <span class="math inline">\(M(x)\)</span> and <span class="math inline">\(M(x')\)</span> are <span class="math inline">\((\epsilon, \delta)\)</span>-ind (resp. <span class="math inline">\(\epsilon\)</span>-ind) for all <span class="math inline">\(x\)</span> and <span class="math inline">\(x'\)</span> with distance <span class="math inline">\(1\)</span>.</p>
<p><strong>Definition (Divergence Variable)</strong>. Let <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> be two probability measures. Let <span class="math inline">\(\xi\)</span> be a random variable distributed according to <span class="math inline">\(p\)</span>, we define a random variable <span class="math inline">\(L(p || q)\)</span> by</p>
<p><span class="math display">\[L(p || q) := \log {p(\xi) \over q(\xi)},\]</span></p>
<p>and call it the <em>divergence variable</em> of <span class="math inline">\((p, q)\)</span>.</p>
<p>One interesting and readily verifiable fact is</p>
<p><span class="math display">\[\mathbb E L(p || q) = D(p || q)\]</span></p>
<p>where <span class="math inline">\(D\)</span> is the <a href="https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence">KL-divergence</a>.</p>
<p><strong>Claim 1</strong>. If</p>
<p><span class="math display">\[\begin{aligned}
\mathbb P(L(p || q) \le \epsilon) &\ge 1 - \delta, \qquad(5) \\
\mathbb P(L(q || p) \le \epsilon) &\ge 1 - \delta
\end{aligned}\]</span></p>
<p>then <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\((\epsilon, \delta)\)</span>-ind.</p>
<p><strong>Proof</strong>. We verify (3), and (4) can be shown in the same way. Let <span class="math inline">\(A := \{y \in Y: \log {p(y) \over q(y)} > \epsilon\}\)</span>, then by (5) we have</p>
<p><span class="math display">\[p(A) < \delta.\]</span></p>
<p>So</p>
<p><span class="math display">\[p(S) = p(S \cap A) + p(S \setminus A) \le \delta + e^\epsilon q(S \setminus A) \le \delta + e^\epsilon q(S).\]</span></p>
<p><span class="math inline">\(\square\)</span></p>
<p>This Claim translates differential privacy to the tail bound of divergence variables, and for the rest of this post all dp results are obtained by estimating this tail bound.</p>
<p>In the following we discuss the converse of Claim 1. The discussions are rather technical, and readers can skip to the <a href="#back-to-approximate-differential-privacy">next subsection</a> on first reading.</p>
<p>The converse of Claim 1 is not true.</p>
<p><strong>Claim 2</strong>. There exists <span class="math inline">\(\epsilon, \delta > 0\)</span>, and <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> that are <span class="math inline">\((\epsilon, \delta)\)</span>-ind, such that</p>
<p><span class="math display">\[\begin{aligned}
\mathbb P(L(p || q) \le \epsilon) &< 1 - \delta, \\
\mathbb P(L(q || p) \le \epsilon) &< 1 - \delta
\end{aligned}\]</span></p>
<p><strong>Proof</strong>. Here's a example. Let <span class="math inline">\(Y = \{0, 1\}\)</span>, and <span class="math inline">\(p(0) = q(1) = 2 / 5\)</span> and <span class="math inline">\(p(1) = q(0) = 3 / 5\)</span>. Then it is not hard to verify that <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\((\log {4 \over 3}, {1 \over 3})\)</span>-ind: just check (3) for all four possible <span class="math inline">\(S \subset Y\)</span> and (4) holds by symmetry. On the other hand,</p>
<p><span class="math display">\[\mathbb P(L(p || q) \le \log {4 \over 3}) = \mathbb P(L(q || p) \le \log {4 \over 3}) = {2 \over 5} < {2 \over 3}.\]</span></p>
<p><span class="math inline">\(\square\)</span></p>
<p>A weaker version of the converse of Claim 1 is true (Kasiviswanathan-Smith 2015), though:</p>
<p><strong>Claim 3</strong>. Let <span class="math inline">\(\alpha > 1\)</span>. If <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\((\epsilon, \delta)\)</span>-ind, then</p>
<p><span class="math display">\[\mathbb P(L(p || q) > \alpha \epsilon) < {1 \over 1 - \exp((1 - \alpha) \epsilon)} \delta.\]</span></p>
<p><strong>Proof</strong>. Define</p>
<p><span class="math display">\[S = \{y: p(y) > e^{\alpha \epsilon} q(y)\}.\]</span></p>
<p>Then we have</p>
<p><span class="math display">\[e^{\alpha \epsilon} q(S) < p(S) \le e^\epsilon q(S) + \delta,\]</span></p>
<p>where the first inequality is due to the definition of <span class="math inline">\(S\)</span>, and the second due to the <span class="math inline">\((\epsilon, \delta)\)</span>-ind. Therefore</p>
<p><span class="math display">\[q(S) \le {\delta \over e^{\alpha \epsilon} - e^\epsilon}.\]</span></p>
<p>Using the <span class="math inline">\((\epsilon, \delta)\)</span>-ind again we have</p>
<p><span class="math display">\[p(S) \le e^\epsilon q(S) + \delta = {1 \over 1 - e^{(1 - \alpha) \epsilon}} \delta.\]</span></p>
<p><span class="math inline">\(\square\)</span></p>
<p>This can be quite bad if <span class="math inline">\(\epsilon\)</span> is small.</p>
<p>To prove the composition theorems in the next section, we need a condition better than that in Claim 1 so that we can go back and forth between indistinguishability and such condition. In other words, we need a <em>characterisation</em> of indistinguishability.</p>
<p>Let us take a careful look at the condition in Claim 1 and call it <strong>C1</strong>:</p>
<p><strong>C1</strong>. <span class="math inline">\(\mathbb P(L(p || q) \le \epsilon) \ge 1 - \delta\)</span> and <span class="math inline">\(\mathbb P(L(q || p) \le \epsilon) \ge 1 - \delta\)</span></p>
<p>It is equivalent to</p>
<p><strong>C2</strong>. there exist events <span class="math inline">\(A, B \subset Y\)</span> with probabilities <span class="math inline">\(p(A)\)</span> and <span class="math inline">\(q(B)\)</span> at least <span class="math inline">\(1 - \delta\)</span> such that <span class="math inline">\(\log p(y) - \log q(y) \le \epsilon\)</span> for all <span class="math inline">\(y \in A\)</span> and <span class="math inline">\(\log q(y) - \log p(y) \le \epsilon\)</span> for all <span class="math inline">\(y \in B\)</span>.</p>
<p>A similar-looking condition to <strong>C2</strong> is the following:</p>
<p><strong>C3</strong>. Let <span class="math inline">\(\Omega\)</span> be the <a href="https://en.wikipedia.org/wiki/Probability_space#Definition">underlying probability space</a>. There exist two events <span class="math inline">\(E, F \subset \Omega\)</span> with <span class="math inline">\(\mathbb P(E), \mathbb P(F) \ge 1 - \delta\)</span>, such that <span class="math inline">\(|\log p_{|E}(y) - \log q_{|F}(y)| \le \epsilon\)</span> for all <span class="math inline">\(y \in Y\)</span>.</p>
<p>Here <span class="math inline">\(p_{|E}\)</span> (resp. <span class="math inline">\(q_{|F}\)</span>) is <span class="math inline">\(p\)</span> (resp. <span class="math inline">\(q\)</span>) conditioned on event <span class="math inline">\(E\)</span> (resp. <span class="math inline">\(F\)</span>).</p>
<p><strong>Remark</strong>. Note that the events in <strong>C2</strong> and <strong>C3</strong> are in different spaces, and therefore we can not write <span class="math inline">\(p_{|E}(S)\)</span> as <span class="math inline">\(p(S | E)\)</span> or <span class="math inline">\(q_{|F}(S)\)</span> as <span class="math inline">\(q(S | F)\)</span>. In fact, if we let <span class="math inline">\(E\)</span> and <span class="math inline">\(F\)</span> in <strong>C3</strong> be subsets of <span class="math inline">\(Y\)</span> with <span class="math inline">\(p(E), q(F) \ge 1 - \delta\)</span> and assume <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> have the same supports, then <strong>C3</strong> degenerates to a stronger condition than <strong>C2</strong>. Indeed, in this case <span class="math inline">\(p_E(y) = p(y) 1_{y \in E}\)</span> and <span class="math inline">\(q_F(y) = q(y) 1_{y \in F}\)</span>, and so <span class="math inline">\(p_E(y) \le e^\epsilon q_F(y)\)</span> forces <span class="math inline">\(E \subset F\)</span>. We also obtain <span class="math inline">\(F \subset E\)</span> in the same way. This gives us <span class="math inline">\(E = F\)</span>, and <strong>C3</strong> becomes <strong>C2</strong> with <span class="math inline">\(A = B = E = F\)</span>.</p>
<p>As it turns out, <strong>C3</strong> is the condition we need.</p>
<p><strong>Claim 4</strong>. Two probability measures <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\((\epsilon, \delta)\)</span>-ind if and only if <strong>C3</strong> holds.</p>
<p><strong>Proof</strong>(Murtagh-Vadhan 2018). The "if" direction is proved in the same way as Claim 1. Without loss of generality we may assume <span class="math inline">\(\mathbb P(E) = \mathbb P(F) \ge 1 - \delta\)</span>. To see this, suppose <span class="math inline">\(F\)</span> has higher probability than <span class="math inline">\(E\)</span>, then we can substitute <span class="math inline">\(F\)</span> with a subset of <span class="math inline">\(F\)</span> that has the same probability as <span class="math inline">\(E\)</span> (with possible enlargement of the probability space).</p>
<p>Let <span class="math inline">\(\xi \sim p\)</span> and <span class="math inline">\(\eta \sim q\)</span> be two independent random variables, then</p>
<p><span class="math display">\[\begin{aligned}
p(S) &= \mathbb P(\xi \in S | E) \mathbb P(E) + \mathbb P(\xi \in S; E^c) \\
&\le e^\epsilon \mathbb P(\eta \in S | F) \mathbb P(E) + \delta \\
&= e^\epsilon \mathbb P(\eta \in S | F) \mathbb P(F) + \delta\\
&\le e^\epsilon q(S) + \delta.
\end{aligned}\]</span></p>
<p>The "only-if" direction is more involved.</p>
<p>We construct events <span class="math inline">\(E\)</span> and <span class="math inline">\(F\)</span> by constructing functions <span class="math inline">\(e, f: Y \to [0, \infty)\)</span> satisfying the following conditions:</p>
<ol style="list-style-type: decimal">
<li><span class="math inline">\(0 \le e(y) \le p(y)\)</span> and <span class="math inline">\(0 \le f(y) \le q(y)\)</span> for all <span class="math inline">\(y \in Y\)</span>.</li>
<li><span class="math inline">\(|\log e(y) - \log f(y)| \le \epsilon\)</span> for all <span class="math inline">\(y \in Y\)</span>.</li>
<li><span class="math inline">\(e(Y), f(Y) \ge 1 - \delta\)</span>.</li>
<li><span class="math inline">\(e(Y) = f(Y)\)</span>.</li>
</ol>
<p>Here for a set <span class="math inline">\(S \subset Y\)</span>, <span class="math inline">\(e(S) := \int_S e(y) dy\)</span>, and the same goes for <span class="math inline">\(f(S)\)</span>.</p>
<p>Let <span class="math inline">\(\xi \sim p\)</span> and <span class="math inline">\(\eta \sim q\)</span>. Then we define <span class="math inline">\(E\)</span> and <span class="math inline">\(F\)</span> by</p>
<p><span class="math display">\[\mathbb P(E | \xi = y) = e(y) / p(y) \\
\mathbb P(F | \eta = y) = f(y) / q(y).\]</span></p>
<p><strong>Remark inside proof</strong>. This can seem a bit confusing. Intuitively, we can think of it this way when <span class="math inline">\(Y\)</span> is finite: Recall a random variable on <span class="math inline">\(Y\)</span> is a function from the probability space <span class="math inline">\(\Omega\)</span> to <span class="math inline">\(Y\)</span>. Let event <span class="math inline">\(G_y \subset \Omega\)</span> be defined as <span class="math inline">\(G_y = \xi^{-1} (y)\)</span>. We cut <span class="math inline">\(G_y\)</span> into the disjoint union of <span class="math inline">\(E_y\)</span> and <span class="math inline">\(G_y \setminus E_y\)</span> such that <span class="math inline">\(\mathbb P(E_y) = e(y)\)</span>. Then <span class="math inline">\(E = \bigcup_{y \in Y} E_y\)</span>. So <span class="math inline">\(e(y)\)</span> can be seen as the "density" of <span class="math inline">\(E\)</span>.</p>
<p>Indeed, given <span class="math inline">\(E\)</span> and <span class="math inline">\(F\)</span> defined this way, we have</p>
<p><span class="math display">\[p_E(y) = {e(y) \over e(Y)} \le {\exp(\epsilon) f(y) \over e(Y)} = {\exp(\epsilon) f(y) \over f(Y)} = \exp(\epsilon) q_F(y).\]</span></p>
<p>and</p>
<p><span class="math display">\[\mathbb P(E) = \int \mathbb P(E | \xi = y) p(y) dy = e(Y) \ge 1 - \delta,\]</span></p>
<p>and the same goes for <span class="math inline">\(\mathbb P(F)\)</span>.</p>
<p>What remains is to construct <span class="math inline">\(e(y)\)</span> and <span class="math inline">\(f(y)\)</span> satisfying the four conditions.</p>
<p>Like in the proof of Claim 1, let <span class="math inline">\(S, T \subset Y\)</span> be defined as</p>
<p><span class="math display">\[\begin{aligned}
S := \{y: p(y) > \exp(\epsilon) q(y)\},\\
T := \{y: q(y) > \exp(\epsilon) p(y)\}.
\end{aligned}\]</span></p>
<p>Let</p>
<p><span class="math display">\[\begin{aligned}
e(y) &:= \exp(\epsilon) q(y) 1_{y \in S} + p(y) 1_{y \notin S}\\
f(y) &:= \exp(\epsilon) p(y) 1_{y \in T} + q(y) 1_{y \notin T}. \qquad (6)
\end{aligned}\]</span></p>
<p>By checking them on the three disjoint subsets <span class="math inline">\(S\)</span>, <span class="math inline">\(T\)</span>, <span class="math inline">\((S \cup T)^c\)</span>, it is not hard to verify that the <span class="math inline">\(e(y)\)</span> and <span class="math inline">\(f(y)\)</span> constructed this way satisfy the first two conditions. They also satisfy the third condition:</p>
<p><span class="math display">\[\begin{aligned}
e(Y) &= 1 - (p(S) - \exp(\epsilon) q(S)) \ge 1 - \delta, \\
f(Y) &= 1 - (q(T) - \exp(\epsilon) p(T)) \ge 1 - \delta.
\end{aligned}\]</span></p>
<p>If <span class="math inline">\(e(Y) = f(Y)\)</span> then we are done. Otherwise, without loss of generality, assume <span class="math inline">\(e(Y) < f(Y)\)</span>, then all it remains to do is to reduce the value of <span class="math inline">\(f(y)\)</span> while preserving Condition 1, 2 and 3, until <span class="math inline">\(f(Y) = e(Y)\)</span>.</p>
<p>As it turns out, this can be achieved by reducing <span class="math inline">\(f(y)\)</span> on the set <span class="math inline">\(\{y \in Y: q(y) > p(y)\}\)</span>. To see this, let us rename the <span class="math inline">\(f(y)\)</span> defined in (6) <span class="math inline">\(f_+(y)\)</span>, and construct <span class="math inline">\(f_-(y)\)</span> by</p>
<p><span class="math display">\[f_-(y) := p(y) 1_{y \in T} + (q(y) \wedge p(y)) 1_{y \notin T}.\]</span></p>
<p>It is not hard to show that not only <span class="math inline">\(e(y)\)</span> and <span class="math inline">\(f_-(y)\)</span> also satisfy conditions 1-3, but</p>
<p><span class="math display">\[e(y) \ge f_-(y), \forall y \in Y,\]</span></p>
<p>and thus <span class="math inline">\(e(Y) \ge f_-(Y)\)</span>. Therefore there exists an <span class="math inline">\(f\)</span> that interpolates between <span class="math inline">\(f_-\)</span> and <span class="math inline">\(f_+\)</span> with <span class="math inline">\(f(Y) = e(Y)\)</span>. <span class="math inline">\(\square\)</span></p>
<p>To prove the adaptive composition theorem for approximate differential privacy, we need a similar claim (We use index shorthand <span class="math inline">\(\xi_{< i} = \xi_{1 : i - 1}\)</span> and similarly for other notations):</p>
<p><strong>Claim 5</strong>. Let <span class="math inline">\(\xi_{1 : i}\)</span> and <span class="math inline">\(\eta_{1 : i}\)</span> be random variables. Let</p>
<p><span class="math display">\[\begin{aligned}
p_i(S | y_{1 : i - 1}) := \mathbb P(\xi_i \in S | \xi_{1 : i - 1} = y_{1 : i - 1})\\
q_i(S | y_{1 : i - 1}) := \mathbb P(\eta_i \in S | \eta_{1 : i - 1} = y_{1 : i - 1})
\end{aligned}\]</span></p>
<p>be the conditional laws of <span class="math inline">\(\xi_i | \xi_{< i}\)</span> and <span class="math inline">\(\eta_i | \eta_{< i}\)</span> respectively. Then the following are equivalent:</p>
<ol style="list-style-type: decimal">
<li>For any <span class="math inline">\(y_{< i} \in Y^{i - 1}\)</span>, <span class="math inline">\(p_i(\cdot | y_{< i})\)</span> and <span class="math inline">\(q_i(\cdot | y_{< i})\)</span> are <span class="math inline">\((\epsilon, \delta)\)</span>-ind</li>
<li><p>There exists events <span class="math inline">\(E_i, F_i \subset \Omega\)</span> with <span class="math inline">\(\mathbb P(E_i | \xi_{<i} = y_{<i}) = \mathbb P(F_i | \eta_{<i} = y_{< i}) \ge 1 - \delta\)</span> for any <span class="math inline">\(y_{< i}\)</span>, such that <span class="math inline">\(p_{i | E_i}(\cdot | y_{< i})\)</span> and <span class="math inline">\(q_{i | E_i} (\cdot | y_{< i})\)</span> are <span class="math inline">\(\epsilon\)</span>-ind for any <span class="math inline">\(y_{< i}\)</span>, where <span class="math display">\[\begin{aligned}
p_{i | E_i}(S | y_{1 : i - 1}) := \mathbb P(\xi_i \in S | E_i, \xi_{1 : i - 1} = y_{1 : i - 1})\\
q_{i | F_i}(S | y_{1 : i - 1}) := \mathbb P(\eta_i \in S | F_i, \eta_{1 : i - 1} = y_{1 : i - 1})
\end{aligned}\]</span></p>
<p>are <span class="math inline">\(p_i\)</span> and <span class="math inline">\(q_i\)</span> conditioned on <span class="math inline">\(E_i\)</span> and <span class="math inline">\(F_i\)</span> respectively.</p></li>
</ol>
<p><strong>Proof</strong>. Item 2 => Item 1: as in the Proof of Claim 4,</p>
<p><span class="math display">\[\begin{aligned}
p_i(S | y_{< i}) &= p_{i | E_i} (S | y_{< i}) \mathbb P(E_i | \xi_{< i} = y_{< i}) + p_{i | E_i^c}(S | y_{< i}) \mathbb P(E_i^c | \xi_{< i} = y_{< i}) \\
&\le p_{i | E_i} (S | y_{< i}) \mathbb P(E_i | \xi_{< i} = y_{< i}) + \delta \\
&= p_{i | E_i} (S | y_{< i}) \mathbb P(F_i | \xi_{< i} = y_{< i}) + \delta \\
&\le e^\epsilon q_{i | F_i} (S | y_{< i}) \mathbb P(F_i | \xi_{< i} = y_{< i}) + \delta \\
&= e^\epsilon q_i (S | y_{< i}) + \delta.
\end{aligned}\]</span></p>
<p>The direction from <span class="math inline">\(q_i(S | y_{< i}) \le e^\epsilon p_i(S | y_{< i}) + \delta\)</span> can be shown in the same way.</p>
<p>Item 1 => Item 2: as in the Proof of Claim 4 we construct <span class="math inline">\(e(y_{1 : i})\)</span> and <span class="math inline">\(f(y_{1 : i})\)</span> as "densities" of events <span class="math inline">\(E_i\)</span> and <span class="math inline">\(F_i\)</span>.</p>
<p>Let</p>
<p><span class="math display">\[\begin{aligned}
e(y_{1 : i}) &:= e^\epsilon q_i(y_i | y_{< i}) 1_{y_i \in S_i(y_{< i})} + p_i(y_i | y_{< i}) 1_{y_i \notin S_i(y_{< i})}\\
f(y_{1 : i}) &:= e^\epsilon p_i(y_i | y_{< i}) 1_{y_i \in T_i(y_{< i})} + q_i(y_i | y_{< i}) 1_{y_i \notin T_i(y_{< i})}\\
\end{aligned}\]</span></p>
<p>where</p>
<p><span class="math display">\[\begin{aligned}
S_i(y_{< i}) = \{y_i \in Y: p_i(y_i | y_{< i}) > e^\epsilon q_i(y_i | y_{< i})\}\\
T_i(y_{< i}) = \{y_i \in Y: q_i(y_i | y_{< i}) > e^\epsilon p_i(y_i | y_{< i})\}.
\end{aligned}\]</span></p>
<p>Then <span class="math inline">\(E_i\)</span> and <span class="math inline">\(F_i\)</span> are defined as</p>
<p><span class="math display">\[\begin{aligned}
\mathbb P(E_i | \xi_{\le i} = y_{\le i}) &= {e(y_{\le i}) \over p_i(y_{\le i})},\\
\mathbb P(F_i | \xi_{\le i} = y_{\le i}) &= {f(y_{\le i}) \over q_i(y_{\le i})}.
\end{aligned}\]</span></p>
<p>The rest of the proof is almost the same as the proof of Claim 4. <span class="math inline">\(\square\)</span></p>
<h3 id="back-to-approximate-differential-privacy">Back to approximate differential privacy</h3>
<p>By Claim 0 and 1 we have</p>
<p><strong>Claim 6</strong>. If for all <span class="math inline">\(x, x' \in X\)</span> with distance <span class="math inline">\(1\)</span></p>
<p><span class="math display">\[\mathbb P(L(M(x) || M(x')) \le \epsilon) \ge 1 - \delta,\]</span></p>
<p>then <span class="math inline">\(M\)</span> is <span class="math inline">\((\epsilon, \delta)\)</span>-dp.</p>
<p>Note that in the literature the divergence variable <span class="math inline">\(L(M(x) || M(x'))\)</span> is also called the <em>privacy loss</em>.</p>
<p>By Claim 0 and Claim 4 we have</p>
<p><strong>Claim 7</strong>. <span class="math inline">\(M\)</span> is <span class="math inline">\((\epsilon, \delta)\)</span>-dp if and only if for every <span class="math inline">\(x, x' \in X\)</span> with distance <span class="math inline">\(1\)</span>, there exist events <span class="math inline">\(E, F \subset \Omega\)</span> with <span class="math inline">\(\mathbb P(E) = \mathbb P(F) \ge 1 - \delta\)</span>, <span class="math inline">\(M(x) | E\)</span> and <span class="math inline">\(M(x') | F\)</span> are <span class="math inline">\(\epsilon\)</span>-ind.</p>
<p>We can further simplify the privacy loss <span class="math inline">\(L(M(x) || M(x'))\)</span>, by observing the translational and scaling invariance of <span class="math inline">\(L(\cdot||\cdot)\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
L(\xi || \eta) &\overset{d}{=} L(\alpha \xi + \beta || \alpha \eta + \beta), \qquad \alpha \neq 0. \qquad (6.1)
\end{aligned}\]</span></p>
<p>With this and the definition</p>
<p><span class="math display">\[M(x) = f(x) + \zeta\]</span></p>
<p>for some random variable <span class="math inline">\(\zeta\)</span>, we have</p>
<p><span class="math display">\[L(M(x) || M(x')) \overset{d}{=} L(\zeta || \zeta + f(x') - f(x)).\]</span></p>
<p>Without loss of generality, we can consider <span class="math inline">\(f\)</span> with sensitivity <span class="math inline">\(1\)</span>, for</p>
<p><span class="math display">\[L(f(x) + S_f \zeta || f(x') + S_f \zeta) \overset{d}{=} L(S_f^{-1} f(x) + \zeta || S_f^{-1} f(x') + \zeta)\]</span></p>
<p>so for any noise <span class="math inline">\(\zeta\)</span> that achieves <span class="math inline">\((\epsilon, \delta)\)</span>-dp for a function with sensitivity <span class="math inline">\(1\)</span>, we have the same privacy guarantee by for an arbitrary function with sensitivity <span class="math inline">\(S_f\)</span> by adding a noise <span class="math inline">\(S_f \zeta\)</span>.</p>
<p>With Claim 6 we can show that the Gaussian mechanism is approximately differentially private. But first we need to define it.</p>
<p><strong>Definition (Gaussian mechanism)</strong>. Given a query <span class="math inline">\(f: X \to Y\)</span>, the <em>Gaussian mechanism</em> <span class="math inline">\(M\)</span> adds a Gaussian noise to the query:</p>
<p><span class="math display">\[M(x) = f(x) + N(0, \sigma^2 I).\]</span></p>
<p>Some tail bounds for the Gaussian distribution will be useful.</p>
<p><strong>Claim 8 (Gaussian tail bounds)</strong>. Let <span class="math inline">\(\xi \sim N(0, 1)\)</span> be a standard normal distribution. Then for <span class="math inline">\(t > 0\)</span></p>
<p><span class="math display">\[\mathbb P(\xi > t) < {1 \over \sqrt{2 \pi} t} e^{- {t^2 \over 2}}, \qquad (6.3)\]</span></p>
<p>and</p>
<p><span class="math display">\[\mathbb P(\xi > t) < e^{- {t^2 \over 2}}. \qquad (6.5)\]</span></p>
<p><strong>Proof</strong>. Both bounds are well known. The first can be proved using</p>
<p><span class="math display">\[\int_t^\infty e^{- {y^2 \over 2}} dy < \int_t^\infty {y \over t} e^{- {y^2 \over 2}} dy.\]</span></p>
<p>The second is shown using <a href="https://en.wikipedia.org/wiki/Chernoff_bound">Chernoff bound</a>. For any random variable <span class="math inline">\(\xi\)</span>,</p>
<p><span class="math display">\[\mathbb P(\xi > t) < {\mathbb E \exp(\lambda \xi) \over \exp(\lambda t)} = \exp(\kappa_\xi(\lambda) - \lambda t), \qquad (6.7)\]</span></p>
<p>where <span class="math inline">\(\kappa_\xi(\lambda) = \log \mathbb E \exp(\lambda \xi)\)</span> is the cumulant of <span class="math inline">\(\xi\)</span>. Since (6.7) holds for any <span class="math inline">\(\lambda\)</span>, we can get the best bound by minimising <span class="math inline">\(\kappa_\xi(\lambda) - \lambda t\)</span> (a.k.a. the <a href="https://en.wikipedia.org/wiki/Legendre_transformation">Legendre transformation</a>). When <span class="math inline">\(\xi\)</span> is standard normal, we get (6.5). <span class="math inline">\(\square\)</span></p>
<p><strong>Remark</strong>. We will use the Chernoff bound extensively in the second part of this post when considering Rényi differential privacy.</p>
<p><strong>Claim 9</strong>. The Gaussian mechanism on a query <span class="math inline">\(f\)</span> is <span class="math inline">\((\epsilon, \delta)\)</span>-dp, where</p>
<p><span class="math display">\[\delta = \exp(- (\epsilon \sigma / S_f - (2 \sigma / S_f)^{-1})^2 / 2). \qquad (6.8)\]</span></p>
<p>Conversely, to achieve give <span class="math inline">\((\epsilon, \delta)\)</span>-dp, we may set</p>
<p><span class="math display">\[\sigma > \left(\epsilon^{-1} \sqrt{2 \log \delta^{-1}} + (2 \epsilon)^{- {1 \over 2}}\right) S_f \qquad (6.81)\]</span></p>
<p>or</p>
<p><span class="math display">\[\sigma > (\epsilon^{-1} (1 \vee \sqrt{(\log (2 \pi)^{-1} \delta^{-2})_+}) + (2 \epsilon)^{- {1 \over 2}}) S_f \qquad (6.82)\]</span></p>
<p>or</p>
<p><span class="math display">\[\sigma > \epsilon^{-1} \sqrt{\log e^\epsilon \delta^{-2}} S_f \qquad (6.83)\]</span></p>
<p>or</p>
<p><span class="math display">\[\sigma > \epsilon^{-1} (\sqrt{1 + \epsilon} \vee \sqrt{(\log e^\epsilon (2 \pi)^{-1} \delta^{-2})_+}) S_f. \qquad (6.84)\]</span></p>
<p><strong>Proof</strong>. As discussed before we only need to consider the case where <span class="math inline">\(S_f = 1\)</span>. Fix arbitrary <span class="math inline">\(x, x' \in X\)</span> with <span class="math inline">\(d(x, x') = 1\)</span>. Let <span class="math inline">\(\zeta = (\zeta_1, ..., \zeta_d) \sim N(0, I_d)\)</span>.</p>
<p>By Claim 6 it suffices to bound</p>
<p><span class="math display">\[\mathbb P(L(M(x) || M(x')) > \epsilon)\]</span></p>
<p>We have by the linear invariance of <span class="math inline">\(L\)</span>,</p>
<p><span class="math display">\[L(M(x) || M(x')) = L(f(x) + \sigma \zeta || f(x') + \sigma \zeta) \overset{d}{=} L(\zeta|| \zeta + \Delta / \sigma),\]</span></p>
<p>where <span class="math inline">\(\Delta := f(x') - f(x)\)</span>.</p>
<p>Plugging in the Gaussian density, we have</p>
<p><span class="math display">\[L(M(x) || M(x')) \overset{d}{=} \sum_i {\Delta_i \over \sigma} \zeta_i + \sum_i {\Delta_i^2 \over 2 \sigma^2} \overset{d}{=} {\|\Delta\|_2 \over \sigma} \xi + {\|\Delta\|_2^2 \over 2 \sigma^2}.\]</span></p>
<p>where <span class="math inline">\(\xi \sim N(0, 1)\)</span>.</p>
<p>Hence</p>
<p><span class="math display">\[\mathbb P(L(M(x) || M(x')) > \epsilon) = \mathbb P(\zeta > {\sigma \over \|\Delta\|_2} \epsilon - {\|\Delta\|_2 \over 2 \sigma}).\]</span></p>
<p>Since <span class="math inline">\(\|\Delta\|_2 \le S_f = 1\)</span>, we have</p>
<p><span class="math display">\[\mathbb P(L(M(x) || M(x')) > \epsilon) \le \mathbb P(\xi > \sigma \epsilon - (2 \sigma)^{-1}).\]</span></p>
<p>Thus the problem is reduced to the tail bound of a standard normal distribution, so we can use Claim 8. Note that we implicitly require <span class="math inline">\(\sigma > (2 \epsilon)^{- 1 / 2}\)</span> here so that <span class="math inline">\(\sigma \epsilon - (2 \sigma)^{-1} > 0\)</span> and we can use the tail bounds.</p>
<p>Using (6.3) we have</p>
<p><span class="math display">\[\mathbb P(L(M(x) || M(x')) > \epsilon) < \exp(- (\epsilon \sigma - (2 \sigma)^{-1})^2 / 2).\]</span></p>
<p>This gives us (6.8).</p>
<p>To bound the right hand by <span class="math inline">\(\delta\)</span>, we require</p>
<p><span class="math display">\[\epsilon \sigma - {1 \over 2 \sigma} > \sqrt{2 \log \delta^{-1}}. \qquad (6.91)\]</span></p>
<p>Solving this inequality we have</p>
<p><span class="math display">\[\sigma > {\sqrt{2 \log \delta^{-1}} + \sqrt{2 \log \delta^{-1} + 2 \epsilon} \over 2 \epsilon}.\]</span></p>
<p>Using <span class="math inline">\(\sqrt{2 \log \delta^{-1} + 2 \epsilon} \le \sqrt{2 \log \delta^{-1}} + \sqrt{2 \epsilon}\)</span>, we can achieve the above inequality by having</p>
<p><span class="math display">\[\sigma > \epsilon^{-1} \sqrt{2 \log \delta^{-1}} + (2 \epsilon)^{-{1 \over 2}}.\]</span></p>
<p>This gives us (6.81).</p>
<p>Alternatively, we can use the concavity of <span class="math inline">\(\sqrt{\cdot}\)</span>:</p>
<p><span class="math display">\[(2 \epsilon)^{-1} (\sqrt{2 \log \delta^{-1}} + \sqrt{2 \log \delta^{-1} + 2 \epsilon}) \le \epsilon^{-1} \sqrt{\log e^\epsilon \delta^{-2}},\]</span></p>
<p>which gives us (6.83)</p>
<p>Back to (6.9), if we use (6.5) instead, we need</p>
<p><span class="math display">\[\log t + {t^2 \over 2} > \log {(2 \pi)^{- 1 / 2} \delta^{-1}}\]</span></p>
<p>where <span class="math inline">\(t = \epsilon \sigma - (2 \sigma)^{-1}\)</span>. This can be satisfied if</p>
<p><span class="math display">\[\begin{aligned}
t &> 1 \qquad (6.93)\\
t &> \sqrt{\log (2 \pi)^{-1} \delta^{-2}}. \qquad (6.95)
\end{aligned}\]</span></p>
<p>We can solve both inequalities as before and obtain</p>
<p><span class="math display">\[\sigma > \epsilon^{-1} (1 \vee \sqrt{(\log (2 \pi)^{-1} \delta^{-2})_+}) + (2 \epsilon)^{- {1 \over 2}},\]</span></p>
<p>or</p>
<p><span class="math display">\[\sigma > \epsilon^{-1}(\sqrt{1 + \epsilon} \vee \sqrt{(\log e^\epsilon (2 \pi)^{-1} \delta^{-2})_+}).\]</span></p>
<p>This gives us (6.82)(6.84). <span class="math inline">\(\square\)</span></p>
<p>When <span class="math inline">\(\epsilon \le \alpha\)</span> is bounded, by (6.83) (6.84) we can require either</p>
<p><span class="math display">\[\sigma > \epsilon^{-1} (\sqrt{\log e^\alpha \delta^{-2}}) S_f\]</span></p>
<p>or</p>
<p><span class="math display">\[\sigma > \epsilon^{-1} (\sqrt{1 + \alpha} \vee \sqrt{(\log (2 \pi)^{-1} e^\alpha \delta^{-2})_+}) S_f.\]</span></p>
<p>The second bound is similar to and slightly better than the one in Theorem A.1 of Dwork-Roth 2013, where <span class="math inline">\(\alpha = 1\)</span>:</p>
<p><span class="math display">\[\sigma > \epsilon^{-1} \left({3 \over 2} \vee \sqrt{(2 \log {5 \over 4} \delta^{-1})_+}\right) S_f.\]</span></p>
<p>Note that the lower bound of <span class="math inline">\({3 \over 2}\)</span> is implicitly required in the proof of Theorem A.1.</p>
<h2 id="composition-theorems">Composition theorems</h2>
<p>So far we have seen how a mechanism made of a single query plus a noise can be proved to be differentially private. But we need to understand the privacy when composing several mechanisms, combinatorially or sequentially. Let us first define the combinatorial case:</p>
<p><strong>Definition (Independent composition)</strong>. Let <span class="math inline">\(M_1, ..., M_k\)</span> be <span class="math inline">\(k\)</span> mechanisms with independent noises. The mechanism <span class="math inline">\(M = (M_1, ..., M_k)\)</span> is called the <em>independent composition</em> of <span class="math inline">\(M_{1 : k}\)</span>.</p>
<p>To define the adaptive composition, let us motivate it with an example of gradient descent. Consider the loss function <span class="math inline">\(\ell(x; \theta)\)</span> of a neural network, where <span class="math inline">\(\theta\)</span> is the parameter and <span class="math inline">\(x\)</span> the input, gradient descent updates its parameter <span class="math inline">\(\theta\)</span> at each time <span class="math inline">\(t\)</span>:</p>
<p><span class="math display">\[\theta_{t} = \theta_{t - 1} - \alpha m^{-1} \sum_{i = 1 : m} \nabla_\theta \ell(x_i; \theta) |_{\theta = \theta_{t - 1}}.\]</span></p>
<p>We may add privacy by adding noise <span class="math inline">\(\zeta_t\)</span> at each step:</p>
<p><span class="math display">\[\theta_{t} = \theta_{t - 1} - \alpha m^{-1} \sum_{i = 1 : m} \nabla_\theta \ell(x_i; \theta) |_{\theta = \theta_{t - 1}} + \zeta_t. \qquad (6.97)\]</span></p>
<p>Viewed as a sequence of mechanism, we have that at each time <span class="math inline">\(t\)</span>, the mechanism <span class="math inline">\(M_t\)</span> takes input <span class="math inline">\(x\)</span>, and outputs <span class="math inline">\(\theta_t\)</span>. But <span class="math inline">\(M_t\)</span> also depends on the output of the previous mechanism <span class="math inline">\(M_{t - 1}\)</span>. To this end, we define the adaptive composition.</p>
<p><strong>Definition (Adaptive composition)</strong>. Let <span class="math inline">\(({M_i(y_{1 : i - 1})})_{i = 1 : k}\)</span> be <span class="math inline">\(k\)</span> mechanisms with independent noises, where <span class="math inline">\(M_1\)</span> has no parameter, <span class="math inline">\(M_2\)</span> has one parameter in <span class="math inline">\(Y\)</span>, <span class="math inline">\(M_3\)</span> has two parameters in <span class="math inline">\(Y\)</span> and so on. For <span class="math inline">\(x \in X\)</span>, define <span class="math inline">\(\xi_i\)</span> recursively by</p>
<p><span class="math display">\[\begin{aligned}
\xi_1 &:= M_1(x)\\
\xi_i &:= M_i(\xi_1, \xi_2, ..., \xi_{i - 1}) (x).
\end{aligned}\]</span></p>
<p>The <em>adaptive composition</em> of <span class="math inline">\(M_{1 : k}\)</span> is defined by <span class="math inline">\(M(x) := (\xi_1, \xi_2, ..., \xi_k)\)</span>.</p>
<p>The definition of adaptive composition may look a bit complicated, but the point is to describe <span class="math inline">\(k\)</span> mechanisms such that for each <span class="math inline">\(i\)</span>, the output of the first, second, ..., <span class="math inline">\(i - 1\)</span>th mechanisms determine the <span class="math inline">\(i\)</span>th mechanism, like in the case of gradient descent.</p>
<p>It is not hard to write down the differentially private gradient descent as a sequential composition:</p>
<p><span class="math display">\[M_t(\theta_{1 : t - 1})(x) = \theta_{t - 1} - \alpha m^{-1} \sum_{i = 1 : m} \nabla_\theta \ell(x_i; \theta) |_{\theta = \theta_{t - 1}} + \zeta_t.\]</span></p>
<p>In Dwork-Rothblum-Vadhan 2010 (see also Dwork-Roth 2013) the adaptive composition is defined in a more general way, but the definition is based on the same principle, and proofs in this post on adaptive compositions carry over.</p>
<p>It is not hard to see that the adaptive composition degenerates to independent composition when each <span class="math inline">\(M_i(y_{1 : i})\)</span> evaluates to the same mechanism regardless of <span class="math inline">\(y_{1 : i}\)</span>, in which case the <span class="math inline">\(\xi_i\)</span>s are independent.</p>
<p>In the following when discussing adaptive compositions we sometimes omit the parameters for convenience without risk of ambiguity, and write <span class="math inline">\(M_i(y_{1 : i})\)</span> as <span class="math inline">\(M_i\)</span>, but keep in mind of the dependence on the parameters.</p>
<p>It is time to state and prove the composition theorems. In this section we consider <span class="math inline">\(2 \times 2 \times 2 = 8\)</span> cases, i.e. situations of three dimensions, where there are two choices in each dimension:</p>
<ol style="list-style-type: decimal">
<li>Composition of <span class="math inline">\(\epsilon\)</span>-dp or more generally <span class="math inline">\((\epsilon, \delta)\)</span>-dp mechanisms</li>
<li>Composition of independent or more generally adaptive mechanisms</li>
<li>Basic or advanced compositions</li>
</ol>
<p>Note that in the first two dimensions the second choice is more general than the first.</p>
<p>The proofs of these composition theorems will be laid out as follows:</p>
<ol style="list-style-type: decimal">
<li>Claim 10 - Basic composition theorem for <span class="math inline">\((\epsilon, \delta)\)</span>-dp with adaptive mechanisms: by a direct proof with an induction argument</li>
<li>Claim 14 - Advanced composition theorem for <span class="math inline">\(\epsilon\)</span>-dp with independent mechanisms: by factorising privacy loss and using <a href="https://en.wikipedia.org/wiki/Hoeffding%27s_inequality">Hoeffding's Inequality</a></li>
<li>Claim 16 - Advanced composition theorem for <span class="math inline">\(\epsilon\)</span>-dp with adaptive mechanisms: by factorising privacy loss and using <a href="https://en.wikipedia.org/wiki/Azuma%27s_inequality">Azuma's Inequality</a></li>
<li>Claims 17 and 18 - Advanced composition theorem for <span class="math inline">\((\epsilon, \delta)\)</span>-dp with independent / adaptive mechanisms: by using characterisations of <span class="math inline">\((\epsilon, \delta)\)</span>-dp in Claims 4 and 5 as an approximation of <span class="math inline">\(\epsilon\)</span>-dp and then using Proofs in Item 2 or 3.</li>
</ol>
<p><strong>Claim 10 (Basic composition theorem).</strong> Let <span class="math inline">\(M_{1 : k}\)</span> be <span class="math inline">\(k\)</span> mechanisms with independent noises such that for each <span class="math inline">\(i\)</span> and <span class="math inline">\(y_{1 : i - 1}\)</span>, <span class="math inline">\(M_i(y_{1 : i - 1})\)</span> is <span class="math inline">\((\epsilon_i, \delta_i)\)</span>-dp. Then the adpative composition of <span class="math inline">\(M_{1 : k}\)</span> is <span class="math inline">\((\sum_i \epsilon_i, \sum_i \delta_i)\)</span>-dp.</p>
<p><strong>Proof (Dwork-Lei 2009, see also Dwork-Roth 2013 Appendix B.1)</strong>. Let <span class="math inline">\(x\)</span> and <span class="math inline">\(x'\)</span> be neighbouring points in <span class="math inline">\(X\)</span>. Let <span class="math inline">\(M\)</span> be the adaptive composition of <span class="math inline">\(M_{1 : k}\)</span>. Define</p>
<p><span class="math display">\[\xi_{1 : k} := M(x), \qquad \eta_{1 : k} := M(x').\]</span></p>
<p>Let <span class="math inline">\(p^i\)</span> and <span class="math inline">\(q^i\)</span> be the laws of <span class="math inline">\((\xi_{1 : i})\)</span> and <span class="math inline">\((\eta_{1 : i})\)</span> respectively.</p>
<p>Let <span class="math inline">\(S_1, ..., S_k \subset Y\)</span> and <span class="math inline">\(T_i := \prod_{j = 1 : i} S_j\)</span>. We use two tricks.</p>
<ol style="list-style-type: decimal">
<li><p>Since <span class="math inline">\(\xi_i | \xi_{< i} = y_{< i}\)</span> and <span class="math inline">\(\eta_i | \eta_{< i} = y_{< i}\)</span> are <span class="math inline">\((\epsilon_i, \delta_i)\)</span>-ind, and a probability is no greater than <span class="math inline">\(1\)</span>, <span class="math display">\[\begin{aligned}
\mathbb P(\xi_i \in S_i | \xi_{< i} = y_{< i}) &\le (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) + \delta_i) \wedge 1 \\
&\le (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) + \delta_i) \wedge (1 + \delta_i) \\
&= (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) \wedge 1) + \delta_i
\end{aligned}\]</span></p></li>
<li><p>Given <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> that are <span class="math inline">\((\epsilon, \delta)\)</span>-ind, define <span class="math display">\[\mu(x) = (p(x) - e^\epsilon q(x))_+.\]</span></p>
<p>We have <span class="math display">\[\mu(S) \le \delta, \forall S\]</span></p>
<p>In the following we define <span class="math inline">\(\mu^{i - 1} = (p^{i - 1} - e^\epsilon q^{i - 1})_+\)</span> for the same purpose.</p></li>
</ol>
<p>We use an inductive argument to prove the theorem:</p>
<p><span class="math display">\[\begin{aligned}
\mathbb P(\xi_{\le i} \in T_i) &= \int_{T_{i - 1}} \mathbb P(\xi_i \in S_i | \xi_{< i} = y_{< i}) p^{i - 1} (y_{< i}) dy_{< i} \\
&\le \int_{T_{i - 1}} (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) \wedge 1) p^{i - 1}(y_{< i}) dy_{< i} + \delta_i\\
&\le \int_{T_{i - 1}} (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) \wedge 1) (e^{\epsilon_1 + ... + \epsilon_{i - 1}} q^{i - 1}(y_{< i}) + \mu^{i - 1} (y_{< i})) dy_{< i} + \delta_i\\
&\le \int_{T_{i - 1}} e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) e^{\epsilon_1 + ... + \epsilon_{i - 1}} q^{i - 1}(y_{< i}) dy_{< i} + \mu_{i - 1}(T_{i - 1}) + \delta_i\\
&\le e^{\epsilon_1 + ... + \epsilon_i} \mathbb P(\eta_{\le i} \in T_i) + \delta_1 + ... + \delta_{i - 1} + \delta_i.\\
\end{aligned}\]</span></p>
<p>In the second line we use Trick 1; in the third line we use the induction assumption; in the fourth line we multiply the first term in the first braket with first term in the second braket, and the second term (i.e. <span class="math inline">\(1\)</span>) in the first braket with the second term in the second braket (i.e. the <span class="math inline">\(\mu\)</span> term); in the last line we use Trick 2.</p>
<p>The base case <span class="math inline">\(i = 1\)</span> is true since <span class="math inline">\(M_1\)</span> is <span class="math inline">\((\epsilon_1, \delta_1)\)</span>-dp. <span class="math inline">\(\square\)</span></p>
<p>To prove the advanced composition theorem, we start with some lemmas.</p>
<p><strong>Claim 11</strong>. If <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\(\epsilon\)</span>-ind, then</p>
<p><span class="math display">\[D(p || q) + D(q || p) \le \epsilon(e^\epsilon - 1).\]</span></p>
<p><strong>Proof</strong>. Since <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\(\epsilon\)</span>-ind, we have <span class="math inline">\(|\log p(x) - \log q(x)| \le \epsilon\)</span> for all <span class="math inline">\(x\)</span>. Let <span class="math inline">\(S := \{x: p(x) > q(x)\}\)</span>. Then we have on</p>
<p><span class="math display">\[\begin{aligned}
D(p || q) + D(q || p) &= \int (p(x) - q(x)) (\log p(x) - \log q(x)) dx\\
&= \int_S (p(x) - q(x)) (\log p(x) - \log q(x)) dx + \int_{S^c} (q(x) - p(x)) (\log q(x) - \log p(x)) dx\\
&\le \epsilon(\int_S p(x) - q(x) dx + \int_{S^c} q(x) - p(x) dx)
\end{aligned}\]</span></p>
<p>Since on <span class="math inline">\(S\)</span> we have <span class="math inline">\(q(x) \le p(x) \le e^\epsilon q(x)\)</span>, and on <span class="math inline">\(S^c\)</span> we have <span class="math inline">\(p(x) \le q(x) \le e^\epsilon p(x)\)</span>, we obtain</p>
<p><span class="math display">\[D(p || q) + D(q || p) \le \epsilon \int_S (1 - e^{-\epsilon}) p(x) dx + \epsilon \int_{S^c} (e^{\epsilon} - 1) p(x) dx \le \epsilon (e^{\epsilon} - 1),\]</span></p>
<p>where in the last step we use <span class="math inline">\(e^\epsilon - 1 \ge 1 - e^{- \epsilon}\)</span> and <span class="math inline">\(p(S) + p(S^c) = 1\)</span>. <span class="math inline">\(\square\)</span></p>
<p><strong>Claim 12</strong>. If <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\(\epsilon\)</span>-ind, then</p>
<p><span class="math display">\[D(p || q) \le a(\epsilon) \ge D(q || p),\]</span></p>
<p>where</p>
<p><span class="math display">\[a(\epsilon) = \epsilon (e^\epsilon - 1) 1_{\epsilon \le \log 2} + \epsilon 1_{\epsilon > \log 2} \le (\log 2)^{-1} \epsilon^2 1_{\epsilon \le \log 2} + \epsilon 1_{\epsilon > \log 2}. \qquad (6.98)\]</span></p>
<p><strong>Proof</strong>. Since <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are <span class="math inline">\(\epsilon\)</span>-ind, we have</p>
<p><span class="math display">\[D(p || q) = \mathbb E_{\xi \sim p} \log {p(\xi) \over q(\xi)} \le \max_y {\log p(y) \over \log q(y)} \le \epsilon.\]</span></p>
<p>Comparing the quantity in Claim 11 (<span class="math inline">\(\epsilon(e^\epsilon - 1)\)</span>) with the quantity above (<span class="math inline">\(\epsilon\)</span>), we arrive at the conclusion. <span class="math inline">\(\square\)</span></p>
<p><strong>Claim 13 (<a href="https://en.wikipedia.org/wiki/Hoeffding%27s_inequality">Hoeffding's Inequality</a>)</strong>. Let <span class="math inline">\(L_i\)</span> be independent random variables with <span class="math inline">\(|L_i| \le b\)</span>, and let <span class="math inline">\(L = L_1 + ... + L_k\)</span>, then for <span class="math inline">\(t > 0\)</span>,</p>
<p><span class="math display">\[\mathbb P(L - \mathbb E L \ge t) \le \exp(- {t^2 \over 2 k b^2}).\]</span></p>
<p><strong>Claim 14 (Advanced Independent Composition Theorem)</strong> (<span class="math inline">\(\delta = 0\)</span>). Fix <span class="math inline">\(0 < \beta < 1\)</span>. Let <span class="math inline">\(M_1, ..., M_k\)</span> be <span class="math inline">\(\epsilon\)</span>-dp, then the independent composition <span class="math inline">\(M\)</span> of <span class="math inline">\(M_{1 : k}\)</span> is <span class="math inline">\((k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} \epsilon, \beta)\)</span>-dp.</p>
<p><strong>Remark</strong>. By (6.98) we know that <span class="math inline">\(k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} \epsilon = \sqrt{2 k \log \beta^{-1}} \epsilon + k O(\epsilon^2)\)</span> when <span class="math inline">\(\epsilon\)</span> is sufficiently small, in which case the leading term is of order <span class="math inline">\(O(\sqrt k \epsilon)\)</span> and we save a <span class="math inline">\(\sqrt k\)</span> in the <span class="math inline">\(\epsilon\)</span>-part compared to the Basic Composition Theorem (Claim 10).</p>
<p><strong>Remark</strong>. In practice one can try different choices of <span class="math inline">\(\beta\)</span> and settle with the one that gives the best privacy guarantee. See the discussions at the end of <a href="/posts/2019-03-14-great-but-manageable-expectations.html">Part 2 of this post</a>.</p>
<p><strong>Proof</strong>. Let <span class="math inline">\(p_i\)</span>, <span class="math inline">\(q_i\)</span>, <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> be the laws of <span class="math inline">\(M_i(x)\)</span>, <span class="math inline">\(M_i(x')\)</span>, <span class="math inline">\(M(x)\)</span> and <span class="math inline">\(M(x')\)</span> respectively.</p>
<p><span class="math display">\[\mathbb E L_i = D(p_i || q_i) \le a(\epsilon),\]</span></p>
<p>where <span class="math inline">\(L_i := L(p_i || q_i)\)</span>. Due to <span class="math inline">\(\epsilon\)</span>-ind also have</p>
<p><span class="math display">\[|L_i| \le \epsilon.\]</span></p>
<p>Therefore, by Hoeffding's Inequality,</p>
<p><span class="math display">\[\mathbb P(L - k a(\epsilon) \ge t) \le \mathbb P(L - \mathbb E L \ge t) \le \exp(- t^2 / 2 k \epsilon^2),\]</span></p>
<p>where <span class="math inline">\(L := \sum_i L_i = L(p || q)\)</span>.</p>
<p>Plugging in <span class="math inline">\(t = \sqrt{2 k \epsilon^2 \log \beta^{-1}}\)</span>, we have</p>
<p><span class="math display">\[\mathbb P(L(p || q) \le k a(\epsilon) + \sqrt{2 k \epsilon^2 \log \beta^{-1}}) \ge 1 - \beta.\]</span></p>
<p>Similarly we also have</p>
<p><span class="math display">\[\mathbb P(L(q || p) \le k a(\epsilon) + \sqrt{2 k \epsilon^2 \log \beta^{-1}}) \ge 1 - \beta.\]</span></p>
<p>By Claim 1 we arrive at the conclusion. <span class="math inline">\(\square\)</span></p>
<p><strong>Claim 15 (<a href="https://en.wikipedia.org/wiki/Azuma%27s_inequality">Azuma's Inequality</a>)</strong>. Let <span class="math inline">\(X_{0 : k}\)</span> be a <a href="https://en.wikipedia.org/wiki/Martingale_(probability_theory)">supermartingale</a>. If <span class="math inline">\(|X_i - X_{i - 1}| \le b\)</span>, then</p>
<p><span class="math display">\[\mathbb P(X_k - X_0 \ge t) \le \exp(- {t^2 \over 2 k b^2}).\]</span></p>
<p>Azuma's Inequality implies a slightly weaker version of Hoeffding's Inequality. To see this, let <span class="math inline">\(L_{1 : k}\)</span> be independent variables with <span class="math inline">\(|L_i| \le b\)</span>. Let <span class="math inline">\(X_i = \sum_{j = 1 : i} L_j - \mathbb E L_j\)</span>. Then <span class="math inline">\(X_{0 : k}\)</span> is a martingale, and</p>
<p><span class="math display">\[| X_i - X_{i - 1} | = | L_i - \mathbb E L_i | \le 2 b,\]</span></p>
<p>since <span class="math inline">\(\|L_i\|_1 \le \|L_i\|_\infty\)</span>. Hence by Azuma's Inequality,</p>
<p><span class="math display">\[\mathbb P(L - \mathbb E L \ge t) \le \exp(- {t^2 \over 8 k b^2}).\]</span></p>
<p>Of course here we have made no assumption on <span class="math inline">\(\mathbb E L_i\)</span>. If instead we have some bound for the expectation, say <span class="math inline">\(|\mathbb E L_i| \le a\)</span>, then by the same derivation we have</p>
<p><span class="math display">\[\mathbb P(L - \mathbb E L \ge t) \le \exp(- {t^2 \over 2 k (a + b)^2}).\]</span></p>
<p>It is not hard to see what Azuma is to Hoeffding is like adaptive composition to independent composition. Indeed, we can use Azuma's Inequality to prove the Advanced Adaptive Composition Theorem for <span class="math inline">\(\delta = 0\)</span>.</p>
<p><strong>Claim 16 (Advanced Adaptive Composition Theorem)</strong> (<span class="math inline">\(\delta = 0\)</span>). Let <span class="math inline">\(\beta > 0\)</span>. Let <span class="math inline">\(M_{1 : k}\)</span> be <span class="math inline">\(k\)</span> mechanisms with independent noises such that for each <span class="math inline">\(i\)</span> and <span class="math inline">\(y_{1 : i}\)</span>, <span class="math inline">\(M_i(y_{1 : i})\)</span> is <span class="math inline">\((\epsilon, 0)\)</span>-dp. Then the adpative composition of <span class="math inline">\(M_{1 : k}\)</span> is <span class="math inline">\((k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} (\epsilon + a(\epsilon)), \beta)\)</span>-dp.</p>
<p><strong>Proof</strong>. As before, let <span class="math inline">\(\xi_{1 : k} \overset{d}{=} M(x)\)</span> and <span class="math inline">\(\eta_{1 : k} \overset{d}{=} M(x')\)</span>, where <span class="math inline">\(M\)</span> is the adaptive composition of <span class="math inline">\(M_{1 : k}\)</span>. Let <span class="math inline">\(p_i\)</span> (resp. <span class="math inline">\(q_i\)</span>) be the law of <span class="math inline">\(\xi_i | \xi_{< i}\)</span> (resp. <span class="math inline">\(\eta_i | \eta_{< i}\)</span>). Let <span class="math inline">\(p^i\)</span> (resp. <span class="math inline">\(q^i\)</span>) be the law of <span class="math inline">\(\xi_{\le i}\)</span> (resp. <span class="math inline">\(\eta_{\le i}\)</span>). We want to construct supermartingale <span class="math inline">\(X\)</span>. To this end, let</p>
<p><span class="math display">\[X_i = \log {p^i(\xi_{\le i}) \over q^i(\xi_{\le i})} - i a(\epsilon) \]</span></p>
<p>We show that <span class="math inline">\((X_i)\)</span> is a supermartingale:</p>
<p><span class="math display">\[\begin{aligned}
\mathbb E(X_i - X_{i - 1} | X_{i - 1}) &= \mathbb E \left(\log {p_i (\xi_i | \xi_{< i}) \over q_i (\xi_i | \xi_{< i})} - a(\epsilon) | \log {p^{i - 1} (\xi_{< i}) \over q^{i - 1} (\xi_{< i})}\right) \\
&= \mathbb E \left( \mathbb E \left(\log {p_i (\xi_i | \xi_{< i}) \over q_i (\xi_i | \xi_{< i})} | \xi_{< i}\right) | \log {p^{i - 1} (\xi_{< i}) \over q^{i - 1} (\xi_{< i})}\right) - a(\epsilon) \\
&= \mathbb E \left( D(p_i (\cdot | \xi_{< i}) || q_i (\cdot | \xi_{< i})) | \log {p^{i - 1} (\xi_{< i}) \over q^{i - 1} (\xi_{< i})}\right) - a(\epsilon) \\
&\le 0,
\end{aligned}\]</span></p>
<p>since by Claim 12 <span class="math inline">\(D(p_i(\cdot | y_{< i}) || q_i(\cdot | y_{< i})) \le a(\epsilon)\)</span> for all <span class="math inline">\(y_{< i}\)</span>.</p>
<p>Since</p>
<p><span class="math display">\[| X_i - X_{i - 1} | = | \log {p_i(\xi_i | \xi_{< i}) \over q_i(\xi_i | \xi_{< i})} - a(\epsilon) | \le \epsilon + a(\epsilon),\]</span></p>
<p>by Azuma's Inequality,</p>
<p><span class="math display">\[\mathbb P(\log {p^k(\xi_{1 : k}) \over q^k(\xi_{1 : k})} \ge k a(\epsilon) + t) \le \exp(- {t^2 \over 2 k (\epsilon + a(\epsilon))^2}). \qquad(6.99)\]</span></p>
<p>Let <span class="math inline">\(t = \sqrt{2 k \log \beta^{-1}} (\epsilon + a(\epsilon))\)</span> we are done. <span class="math inline">\(\square\)</span></p>
<p><strong>Claim 17 (Advanced Independent Composition Theorem)</strong>. Fix <span class="math inline">\(0 < \beta < 1\)</span>. Let <span class="math inline">\(M_1, ..., M_k\)</span> be <span class="math inline">\((\epsilon, \delta)\)</span>-dp, then the independent composition <span class="math inline">\(M\)</span> of <span class="math inline">\(M_{1 : k}\)</span> is <span class="math inline">\((k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} \epsilon, k \delta + \beta)\)</span>-dp.</p>
<p><strong>Proof</strong>. By Claim 4, there exist events <span class="math inline">\(E_{1 : k}\)</span> and <span class="math inline">\(F_{1 : k}\)</span> such that</p>
<ol style="list-style-type: decimal">
<li>The laws <span class="math inline">\(p_{i | E_i}\)</span> and <span class="math inline">\(q_{i | F_i}\)</span> are <span class="math inline">\(\epsilon\)</span>-ind.</li>
<li><span class="math inline">\(\mathbb P(E_i), \mathbb P(F_i) \ge 1 - \delta\)</span>.</li>
</ol>
<p>Let <span class="math inline">\(E := \bigcap E_i\)</span> and <span class="math inline">\(F := \bigcap F_i\)</span>, then they both have probability at least <span class="math inline">\(1 - k \delta\)</span>, and <span class="math inline">\(p_{i | E}\)</span> and <span class="math inline">\(q_{i | F}\)</span> are <span class="math inline">\(\epsilon\)</span>-ind.</p>
<p>By Claim 14, <span class="math inline">\(p_{|E}\)</span> and <span class="math inline">\(q_{|F}\)</span> are <span class="math inline">\((\epsilon' := k a(\epsilon) + \sqrt{2 k \epsilon^2 \log \beta^{-1}}, \beta)\)</span>-ind. Let us shrink the bigger event between <span class="math inline">\(E\)</span> and <span class="math inline">\(F\)</span> so that they have equal probabilities. Then</p>
<p><span class="math display">\[\begin{aligned}
p (S) &\le p_{|E}(S) \mathbb P(E) + \mathbb P(E^c) \\
&\le (e^{\epsilon'} q_{|F}(S) + \beta) \mathbb P(F) + k \delta\\
&\le e^{\epsilon'} q(S) + \beta + k \delta.
\end{aligned}\]</span></p>
<p><span class="math inline">\(\square\)</span></p>
<p><strong>Claim 18 (Advanced Adaptive Composition Theorem)</strong>. Fix <span class="math inline">\(0 < \beta < 1\)</span>. Let <span class="math inline">\(M_{1 : k}\)</span> be <span class="math inline">\(k\)</span> mechanisms with independent noises such that for each <span class="math inline">\(i\)</span> and <span class="math inline">\(y_{1 : i}\)</span>, <span class="math inline">\(M_i(y_{1 : i})\)</span> is <span class="math inline">\((\epsilon, \delta)\)</span>-dp. Then the adpative composition of <span class="math inline">\(M_{1 : k}\)</span> is <span class="math inline">\((k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} (\epsilon + a(\epsilon)), \beta + k \delta)\)</span>-dp.</p>
<p><strong>Remark</strong>. This theorem appeared in Dwork-Rothblum-Vadhan 2010, but I could not find a proof there. A proof can be found in Dwork-Roth 2013 (See Theorem 3.20 there). Here I prove it in a similar way, except that instead of the use of an intermediate random variable there, I use the conditional probability results from Claim 5, the approach mentioned in Vadhan 2017.</p>
<p><strong>Proof</strong>. By Claim 5, there exist events <span class="math inline">\(E_{1 : k}\)</span> and <span class="math inline">\(F_{1 : k}\)</span> such that</p>
<ol style="list-style-type: decimal">
<li>The laws <span class="math inline">\(p_{i | E_i}(\cdot | y_{< i})\)</span> and <span class="math inline">\(q_{i | F_i}(\cdot | y_{< i})\)</span> are <span class="math inline">\(\epsilon\)</span>-ind for all <span class="math inline">\(y_{< i}\)</span>.</li>
<li><span class="math inline">\(\mathbb P(E_i | y_{< i}), \mathbb P(F_i | y_{< i}) \ge 1 - \delta\)</span> for all <span class="math inline">\(y_{< i}\)</span>.</li>
</ol>
<p>Let <span class="math inline">\(E := \bigcap E_i\)</span> and <span class="math inline">\(F := \bigcap F_i\)</span>, then they both have probability at least <span class="math inline">\(1 - k \delta\)</span>, and <span class="math inline">\(p_{i | E}(\cdot | y_{< i}\)</span> and <span class="math inline">\(q_{i | F}(\cdot | y_{< i})\)</span> are <span class="math inline">\(\epsilon\)</span>-ind.</p>
<p>By Advanced Adaptive Composition Theorem (<span class="math inline">\(\delta = 0\)</span>), <span class="math inline">\(p_{|E}\)</span> and <span class="math inline">\(q_{|F}\)</span> are <span class="math inline">\((\epsilon' := k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} (\epsilon + a(\epsilon)), \beta)\)</span>-ind.</p>
<p>The rest is the same as in the proof of Claim 17. <span class="math inline">\(\square\)</span></p>
<h2 id="subsampling">Subsampling</h2>
<p>Stochastic gradient descent is like gradient descent, but with random subsampling.</p>
<p>Recall we have been considering databases in the space <span class="math inline">\(Z^m\)</span>. Let <span class="math inline">\(n < m\)</span> be a positive integer, <span class="math inline">\(\mathcal I := \{I \subset [m]: |I| = n\}\)</span> be the set of subsets of <span class="math inline">\([m]\)</span> of size <span class="math inline">\(n\)</span>, and <span class="math inline">\(\gamma\)</span> a random subset sampled uniformly from <span class="math inline">\(\mathcal I\)</span>. Let <span class="math inline">\(r = {n \over m}\)</span> which we call the subsampling rate. Then we may add a subsampling module to the noisy gradient descent algorithm (6.97) considered before</p>
<p><span class="math display">\[\theta_{t} = \theta_{t - 1} - \alpha n^{-1} \sum_{i \in \gamma} \nabla_\theta h_\theta(x_i) |_{\theta = \theta_{t - 1}} + \zeta_t. \qquad (7)\]</span></p>
<p>It turns out subsampling has an amplification effect on privacy.</p>
<p><strong>Claim 19 (Ullman 2017)</strong>. Fix <span class="math inline">\(r \in [0, 1]\)</span>. Let <span class="math inline">\(n \le m\)</span> be two nonnegative integers with <span class="math inline">\(n = r m\)</span>. Let <span class="math inline">\(N\)</span> be an <span class="math inline">\((\epsilon, \delta)\)</span>-dp machanism on <span class="math inline">\(Z^n\)</span>. Define mechanism <span class="math inline">\(M\)</span> on <span class="math inline">\(Z^m\)</span> by</p>
<p><span class="math display">\[M(x) = N(x_\gamma)\]</span></p>
<p>Then <span class="math inline">\(M\)</span> is <span class="math inline">\((\log (1 + r(e^\epsilon - 1)), r \delta)\)</span>-dp.</p>
<p><strong>Remark</strong>. Some seem to cite Kasiviswanathan-Lee-Nissim-Raskhodnikova-Smith 2005 for this result, but it is not clear to me how it appears there.</p>
<p><strong>Proof</strong>. Let <span class="math inline">\(x, x' \in Z^n\)</span> such that they differ by one row <span class="math inline">\(x_i \neq x_i'\)</span>. Naturally we would like to consider the cases where the index <span class="math inline">\(i\)</span> is picked and the ones where it is not separately. Let <span class="math inline">\(\mathcal I_\in\)</span> and <span class="math inline">\(\mathcal I_\notin\)</span> be these two cases:</p>
<p><span class="math display">\[\begin{aligned}
\mathcal I_\in = \{J \subset \mathcal I: i \in J\}\\
\mathcal I_\notin = \{J \subset \mathcal I: i \notin J\}\\
\end{aligned}\]</span></p>
<p>We will use these notations later. Let <span class="math inline">\(A\)</span> be the event <span class="math inline">\(\{\gamma \ni i\}\)</span>.</p>
<p>Let <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> be the laws of <span class="math inline">\(M(x)\)</span> and <span class="math inline">\(M(x')\)</span> respectively. We collect some useful facts about them. First due to <span class="math inline">\(N\)</span> being <span class="math inline">\((\epsilon, \delta)\)</span>-dp,</p>
<p><span class="math display">\[p_{|A}(S) \le e^\epsilon q_{|A}(S) + \delta.\]</span></p>
<p>Also,</p>
<p><span class="math display">\[p_{|A}(S) \le e^\epsilon p_{|A^c}(S) + \delta.\]</span></p>
<p>To see this, note that being conditional laws, <span class="math inline">\(p_A\)</span> and <span class="math inline">\(p_{A^c}\)</span> are averages of laws over <span class="math inline">\(\mathcal I_\in\)</span> and <span class="math inline">\(\mathcal I_\notin\)</span> respectively:</p>
<p><span class="math display">\[\begin{aligned}
p_{|A}(S) = |\mathcal I_\in|^{-1} \sum_{I \in \mathcal I_\in} \mathbb P(N(x_I) \in S)\\
p_{|A^c}(S) = |\mathcal I_\notin|^{-1} \sum_{J \in \mathcal I_\notin} \mathbb P(N(x_J) \in S).
\end{aligned}\]</span></p>
<p>Now we want to pair the <span class="math inline">\(I\)</span>'s in <span class="math inline">\(\mathcal I_\in\)</span> and <span class="math inline">\(J\)</span>'s in <span class="math inline">\(\mathcal I_\notin\)</span> so that they differ by one index only, which means <span class="math inline">\(d(x_I, x_J) = 1\)</span>. Formally, this means we want to consider the set:</p>
<p><span class="math display">\[\mathcal D := \{(I, J) \in \mathcal I_\in \times \mathcal I_\notin: |I \cap J| = n - 1\}.\]</span></p>
<p>We may observe by trying out some simple cases that every <span class="math inline">\(I \in \mathcal I_\in\)</span> is paired with <span class="math inline">\(n\)</span> elements in <span class="math inline">\(\mathcal I_\notin\)</span>, and every <span class="math inline">\(J \in \mathcal I_\notin\)</span> is paired with <span class="math inline">\(m - n\)</span> elements in <span class="math inline">\(\mathcal I_\in\)</span>. Therefore</p>
<p><span class="math display">\[p_{|A}(S) = |\mathcal D|^{-1} \sum_{(I, J) \in \mathcal D} \mathbb P(N(x_I \in S)) \le |\mathcal D|^{-1} \sum_{(I, J) \in \mathcal D} (e^\epsilon \mathbb P(N(x_J \in S)) + \delta) = e^\epsilon p_{|A^c} (S) + \delta.\]</span></p>
<p>Since each of the <span class="math inline">\(m\)</span> indices is picked independently with probability <span class="math inline">\(r\)</span>, we have</p>
<p><span class="math display">\[\mathbb P(A) = r.\]</span></p>
<p>Let <span class="math inline">\(t \in [0, 1]\)</span> to be determined. We may write</p>
<p><span class="math display">\[\begin{aligned}
p(S) &= r p_{|A} (S) + (1 - r) p_{|A^c} (S)\\
&\le r(t e^\epsilon q_{|A}(S) + (1 - t) e^\epsilon q_{|A^c}(S) + \delta) + (1 - r) q_{|A^c} (S)\\
&= rte^\epsilon q_{|A}(S) + (r(1 - t) e^\epsilon + (1 - r)) q_{|A^c} (S) + r \delta\\
&= te^\epsilon r q_{|A}(S) + \left({r \over 1 - r}(1 - t) e^\epsilon + 1\right) (1 - r) q_{|A^c} (S) + r \delta \\
&\le \left(t e^\epsilon \wedge \left({r \over 1 - r} (1 - t) e^\epsilon + 1\right)\right) q(S) + r \delta. \qquad (7.5)
\end{aligned}\]</span></p>
<p>We can see from the last line that the best bound we can get is when</p>
<p><span class="math display">\[t e^\epsilon = {r \over 1 - r} (1 - t) e^\epsilon + 1.\]</span></p>
<p>Solving this equation we obtain</p>
<p><span class="math display">\[t = r + e^{- \epsilon} - r e^{- \epsilon}\]</span></p>
<p>and plugging this in (7.5) we have</p>
<p><span class="math display">\[p(S) \le (1 + r(e^\epsilon - 1)) q(S) + r \delta.\]</span></p>
<p><span class="math inline">\(\square\)</span></p>
<p>Since <span class="math inline">\(\log (1 + x) < x\)</span> for <span class="math inline">\(x > 0\)</span>, we can rewrite the conclusion of the Claim to <span class="math inline">\((r(e^\epsilon - 1), r \delta)\)</span>-dp. Further more, if <span class="math inline">\(\epsilon < \alpha\)</span> for some <span class="math inline">\(\alpha\)</span>, we can rewrite it as <span class="math inline">\((r \alpha^{-1} (e^\alpha - 1) \epsilon, r \delta)\)</span>-dp or <span class="math inline">\((O(r \epsilon), r \delta)\)</span>-dp.</p>
<p>Let <span class="math inline">\(\epsilon < 1\)</span>. We see that if the mechanism <span class="math inline">\(N\)</span> is <span class="math inline">\((\epsilon, \delta)\)</span>-dp on <span class="math inline">\(Z^n\)</span>, then <span class="math inline">\(M\)</span> is <span class="math inline">\((2 r \epsilon, r \delta)\)</span>-dp, and if we run it over <span class="math inline">\(k / r\)</span> minibatches, by Advanced Adaptive Composition theorem, we have <span class="math inline">\((\sqrt{2 k r \log \beta^{-1}} \epsilon + 2 k r \epsilon^2, k \delta + \beta)\)</span>-dp.</p>
<p>This is better than the privacy guarantee without subsampling, where we run over <span class="math inline">\(k\)</span> iterations and obtain <span class="math inline">\((\sqrt{2 k \log \beta^{-1}} \epsilon + 2 k \epsilon^2, k \delta + \beta)\)</span>-dp. So with subsampling we gain an extra <span class="math inline">\(\sqrt r\)</span> in the <span class="math inline">\(\epsilon\)</span>-part of the privacy guarantee. But, smaller subsampling rate means smaller minibatch size, which would result in bigger variance, so there is a trade-off here.</p>
<p>Finally we define the differentially private stochastic gradient descent (DP-SGD) with the Gaussian mechanism (Abadi-Chu-Goodfellow-McMahan-Mironov-Talwar-Zhang 2016), which is (7) with the noise specialised to Gaussian and an added clipping operation to bound to sensitivity of the query to a chosen <span class="math inline">\(C\)</span>:</p>
<p><span class="math display">\[\theta_{t} = \theta_{t - 1} - \alpha \left(n^{-1} \sum_{i \in \gamma} \nabla_\theta \ell(x_i; \theta) |_{\theta = \theta_{t - 1}}\right)_{\text{Clipped at }C / 2} + N(0, \sigma^2 C^2 I),\]</span></p>
<p>where</p>
<p><span class="math display">\[y_{\text{Clipped at } \alpha} := y / (1 \vee {\|y\|_2 \over \alpha})\]</span></p>
<p>is <span class="math inline">\(y\)</span> clipped to have norm at most <span class="math inline">\(\alpha\)</span>.</p>
<p>Note that the clipping in DP-SGD is much stronger than making the query have sensitivity <span class="math inline">\(C\)</span>. It makes the difference between the query results of two <em>arbitrary</em> inputs bounded by <span class="math inline">\(C\)</span>, rather than <em>neighbouring</em> inputs.</p>
<p>In <a href="/posts/2019-03-14-great-but-manageable-expectations.html">Part 2 of this post</a> we will use the tools developed above to discuss the privacy guarantee for DP-SGD, among other things.</p>
<h2 id="references">References</h2>
<ul>
<li>Abadi, Martín, Andy Chu, Ian Goodfellow, H. Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. "Deep Learning with Differential Privacy." Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security - CCS'16, 2016, 308--18. <a href="https://doi.org/10.1145/2976749.2978318" class="uri">https://doi.org/10.1145/2976749.2978318</a>.</li>
<li>Dwork, Cynthia, and Aaron Roth. "The Algorithmic Foundations of Differential Privacy." Foundations and Trends® in Theoretical Computer Science 9, no. 3--4 (2013): 211--407. <a href="https://doi.org/10.1561/0400000042" class="uri">https://doi.org/10.1561/0400000042</a>.</li>
<li>Dwork, Cynthia, Guy N. Rothblum, and Salil Vadhan. "Boosting and Differential Privacy." In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 51--60. Las Vegas, NV, USA: IEEE, 2010. <a href="https://doi.org/10.1109/FOCS.2010.12" class="uri">https://doi.org/10.1109/FOCS.2010.12</a>.</li>
<li>Shiva Prasad Kasiviswanathan, Homin K. Lee, Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. "What Can We Learn Privately?" In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05). Pittsburgh, PA, USA: IEEE, 2005. <a href="https://doi.org/10.1109/SFCS.2005.1" class="uri">https://doi.org/10.1109/SFCS.2005.1</a>.</li>
<li>Murtagh, Jack, and Salil Vadhan. "The Complexity of Computing the Optimal Composition of Differential Privacy." In Theory of Cryptography, edited by Eyal Kushilevitz and Tal Malkin, 9562:157--75. Berlin, Heidelberg: Springer Berlin Heidelberg, 2016. <a href="https://doi.org/10.1007/978-3-662-49096-9_7" class="uri">https://doi.org/10.1007/978-3-662-49096-9_7</a>.</li>
<li>Ullman, Jonathan. "Solution to CS7880 Homework 1.", 2017. <a href="http://www.ccs.neu.edu/home/jullman/cs7880s17/HW1sol.pdf" class="uri">http://www.ccs.neu.edu/home/jullman/cs7880s17/HW1sol.pdf</a></li>
<li>Vadhan, Salil. "The Complexity of Differential Privacy." In Tutorials on the Foundations of Cryptography, edited by Yehuda Lindell, 347--450. Cham: Springer International Publishing, 2017. <a href="https://doi.org/10.1007/978-3-319-57048-8_7" class="uri">https://doi.org/10.1007/978-3-319-57048-8_7</a>.</li>
</ul>
<div class="footnotes">
<hr />
<ol>
<li id="fn1"><p>For those who have read about differential privacy and never heard of the term "divergence variable", it is closely related to the notion of "privacy loss", see the paragraph under Claim 6 in <a href="#back-to-approximate-differential-privacy">Back to approximate differential privacy</a>. I defined the term this way so that we can focus on the more general stuff: compared to the privacy loss <span class="math inline">\(L(M(x) || M(x'))\)</span>, the term <span class="math inline">\(L(p || q)\)</span> removes the "distracting information" that <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are related to databases, queries, mechanisms etc., but merely probability laws. By removing the distraction, we simplify the analysis. And once we are done with the analysis of <span class="math inline">\(L(p || q)\)</span>, we can apply the results obtained in the general setting to the special case where <span class="math inline">\(p\)</span> is the law of <span class="math inline">\(M(x)\)</span> and <span class="math inline">\(q\)</span> is the law of <span class="math inline">\(M(x')\)</span>.<a href="#fnref1">↩</a></p></li>
</ol>
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Raise your ELBOposts/2019-02-14-raise-your-elbo.html2019-02-14T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<ul>
<li><a href="#kl-divergence-and-elbo">KL divergence and ELBO</a></li>
<li><a href="#em">EM</a><ul>
<li><a href="#gmm">GMM</a></li>
<li><a href="#smm">SMM</a></li>
<li><a href="#plsa">pLSA</a></li>
<li><a href="#hmm">HMM</a></li>
</ul></li>
<li><a href="#fully-bayesian-em-mfa">Fully Bayesian EM / MFA</a><ul>
<li><a href="#application-to-mixture-models">Application to mixture models</a></li>
<li><a href="#fully-bayesian-gmm">Fully Bayesian GMM</a></li>
<li><a href="#lda">LDA</a></li>
<li><a href="#dpmm">DPMM</a></li>
</ul></li>
<li><a href="#svi">SVI</a></li>
<li><a href="#aevb">AEVB</a><ul>
<li><a href="#vae">VAE</a></li>
<li><a href="#fully-bayesian-aevb">Fully Bayesian AEVB</a></li>
</ul></li>
<li><a href="#references">References</a></li>
</ul>
</div>
<p>In this post I give an introduction to variational inference, which is about maximising the evidence lower bound (ELBO).</p>
<p>I use a top-down approach, starting with the KL divergence and the ELBO, to lay the mathematical framework of all the models in this post.</p>
<p>Then I define mixture models and the EM algorithm, with Gaussian mixture model (GMM), probabilistic latent semantic analysis (pLSA) and the hidden markov model (HMM) as examples.</p>
<p>After that I present the fully Bayesian version of EM, also known as mean field approximation (MFA), and apply it to fully Bayesian mixture models, with fully Bayesian GMM (also known as variational GMM), latent Dirichlet allocation (LDA) and Dirichlet process mixture model (DPMM) as examples.</p>
<p>Then I explain stochastic variational inference, a modification of EM and MFA to improve efficiency.</p>
<p>Finally I talk about autoencoding variational Bayes (AEVB), a Monte-Carlo + neural network approach to raising the ELBO, exemplified by the variational autoencoder (VAE). I also show its fully Bayesian version.</p>
<p><strong>Acknowledgement</strong>. The following texts and resources were illuminating during the writing of this post: the Stanford CS228 notes (<a href="https://ermongroup.github.io/cs228-notes/inference/variational/">1</a>,<a href="https://ermongroup.github.io/cs228-notes/learning/latent/">2</a>), the <a href="https://www.cs.tau.ac.il/~rshamir/algmb/presentations/EM-BW-Ron-16%20.pdf">Tel Aviv Algorithms in Molecular Biology slides</a> (clear explanations of the connection between EM and Baum-Welch), Chapter 10 of <a href="https://www.springer.com/us/book/9780387310732">Bishop's book</a> (brilliant introduction to variational GMM), Section 2.5 of <a href="http://cs.brown.edu/~sudderth/papers/sudderthPhD.pdf">Sudderth's thesis</a> and <a href="https://metacademy.org">metacademy</a>. Also thanks to Josef Lindman Hörnlund for discussions. The research was done while working at KTH mathematics department.</p>
<p><em>If you are reading on a mobile device, you may need to "request desktop site" for the equations to be properly displayed. This post is licensed under CC BY-SA and GNU FDL.</em></p>
<h2 id="kl-divergence-and-elbo">KL divergence and ELBO</h2>
<p>Let <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> be two probability measures. The Kullback-Leibler (KL) divergence is defined as</p>
<p><span class="math display">\[D(q||p) = E_q \log{q \over p}.\]</span></p>
<p>It achieves minimum <span class="math inline">\(0\)</span> when <span class="math inline">\(p = q\)</span>.</p>
<p>If <span class="math inline">\(p\)</span> can be further written as</p>
<p><span class="math display">\[p(x) = {w(x) \over Z}, \qquad (0)\]</span></p>
<p>where <span class="math inline">\(Z\)</span> is a normaliser, then</p>
<p><span class="math display">\[\log Z = D(q||p) + L(w, q), \qquad(1)\]</span></p>
<p>where <span class="math inline">\(L(w, q)\)</span> is called the evidence lower bound (ELBO), defined by</p>
<p><span class="math display">\[L(w, q) = E_q \log{w \over q}. \qquad (1.25)\]</span></p>
<p>From (1), we see that to minimise the nonnegative term <span class="math inline">\(D(q || p)\)</span>, one can maximise the ELBO.</p>
<p>To this end, we can simply discard <span class="math inline">\(D(q || p)\)</span> in (1) and obtain:</p>
<p><span class="math display">\[\log Z \ge L(w, q) \qquad (1.3)\]</span></p>
<p>and keep in mind that the inequality becomes an equality when <span class="math inline">\(q = {w \over Z}\)</span>.</p>
<p>It is time to define the task of variational inference (VI), also known as variational Bayes (VB).</p>
<p><strong>Definition</strong>. Variational inference is concerned with maximising the ELBO <span class="math inline">\(L(w, q)\)</span>.</p>
<p>There are mainly two versions of VI, the half Bayesian and the fully Bayesian cases. Half Bayesian VI, to which expectation-maximisation algorithms (EM) apply, instantiates (1.3) with</p>
<p><span class="math display">\[\begin{aligned}
Z &= p(x; \theta)\\
w &= p(x, z; \theta)\\
q &= q(z)
\end{aligned}\]</span></p>
<p>and the dummy variable <span class="math inline">\(x\)</span> in Equation (0) is substituted with <span class="math inline">\(z\)</span>.</p>
<p>Fully Bayesian VI, often just called VI, has the following instantiations:</p>
<p><span class="math display">\[\begin{aligned}
Z &= p(x) \\
w &= p(x, z, \theta) \\
q &= q(z, \theta)
\end{aligned}\]</span></p>
<p>and <span class="math inline">\(x\)</span> in Equation (0) is substituted with <span class="math inline">\((z, \theta)\)</span>.</p>
<p>In both cases <span class="math inline">\(\theta\)</span> are parameters and <span class="math inline">\(z\)</span> are latent variables.</p>
<p><strong>Remark on the naming of things</strong>. The term "variational" comes from the fact that we perform calculus of variations: maximise some functional (<span class="math inline">\(L(w, q)\)</span>) over a set of functions (<span class="math inline">\(q\)</span>). Note however, most of the VI / VB algorithms do not concern any techniques in calculus of variations, but only uses Jensen's inequality / the fact the <span class="math inline">\(D(q||p)\)</span> reaches minimum when <span class="math inline">\(p = q\)</span>. Due to this reasoning of the naming, EM is also a kind of VI, even though in the literature VI often referes to its fully Bayesian version.</p>
<h2 id="em">EM</h2>
<p>To illustrate the EM algorithms, we first define the mixture model.</p>
<p><strong>Definition (mixture model)</strong>. Given dataset <span class="math inline">\(x_{1 : m}\)</span>, we assume the data has some underlying latent variable <span class="math inline">\(z_{1 : m}\)</span> that may take a value from a finite set <span class="math inline">\(\{1, 2, ..., n_z\}\)</span>. Let <span class="math inline">\(p(z_{i}; \pi)\)</span> be categorically distributed according to the probability vector <span class="math inline">\(\pi\)</span>. That is, <span class="math inline">\(p(z_{i} = k; \pi) = \pi_k\)</span>. Also assume <span class="math inline">\(p(x_{i} | z_{i} = k; \eta) = p(x_{i}; \eta_k)\)</span>. Find <span class="math inline">\(\theta = (\pi, \eta)\)</span> that maximises the likelihood <span class="math inline">\(p(x_{1 : m}; \theta)\)</span>.</p>
<p>Represented as a DAG (a.k.a the plate notations), the model looks like this:</p>
<div class="figure">
<img src="/assets/resources/mixture-model.png" style="width:250px" />
</div>
<p>where the boxes with <span class="math inline">\(m\)</span> mean repitition for <span class="math inline">\(m\)</span> times, since there <span class="math inline">\(m\)</span> indepdent pairs of <span class="math inline">\((x, z)\)</span>, and the same goes for <span class="math inline">\(\eta\)</span>.</p>
<p>The direct maximisation</p>
<p><span class="math display">\[\max_\theta \sum_i \log p(x_{i}; \theta) = \max_\theta \sum_i \log \int p(x_{i} | z_i; \theta) p(z_i; \theta) dz_i\]</span></p>
<p>is hard because of the integral in the log.</p>
<p>We can fit this problem in (1.3) by having <span class="math inline">\(Z = p(x_{1 : m}; \theta)\)</span> and <span class="math inline">\(w = p(z_{1 : m}, x_{1 : m}; \theta)\)</span>. The plan is to update <span class="math inline">\(\theta\)</span> repeatedly so that <span class="math inline">\(L(p(z, x; \theta_t), q(z))\)</span> is non decreasing over time <span class="math inline">\(t\)</span>.</p>
<p>Equation (1.3) at time <span class="math inline">\(t\)</span> for the <span class="math inline">\(i\)</span>th datapoint is</p>
<p><span class="math display">\[\log p(x_{i}; \theta_t) \ge L(p(z_i, x_{i}; \theta_t), q(z_i)) \qquad (2)\]</span></p>
<p>Each timestep consists of two steps, the E-step and the M-step.</p>
<p>At E-step, we set</p>
<p><span class="math display">\[q(z_{i}) = p(z_{i}|x_{i}; \theta_t), \]</span></p>
<p>to turn the inequality into equality. We denote <span class="math inline">\(r_{ik} = q(z_i = k)\)</span> and call them responsibilities, so the posterior <span class="math inline">\(q(z_i)\)</span> is categorical distribution with parameter <span class="math inline">\(r_i = r_{i, 1 : n_z}\)</span>.</p>
<p>At M-step, we maximise <span class="math inline">\(\sum_i L(p(x_{i}, z_{i}; \theta), q(z_{i}))\)</span> over <span class="math inline">\(\theta\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
\theta_{t + 1} &= \text{argmax}_\theta \sum_i L(p(x_{i}, z_{i}; \theta), p(z_{i} | x_{i}; \theta_t)) \\
&= \text{argmax}_\theta \sum_i \mathbb E_{p(z_{i} | x_{i}; \theta_t)} \log p(x_{i}, z_{i}; \theta) \qquad (2.3)
\end{aligned}\]</span></p>
<p>So <span class="math inline">\(\sum_i L(p(x_{i}, z_{i}; \theta), q(z_i))\)</span> is non-decreasing at both the E-step and the M-step.</p>
<p>We can see from this derivation that EM is half-Bayesian. The E-step is Bayesian it computes the posterior of the latent variables and the M-step is frequentist because it performs maximum likelihood estimate of <span class="math inline">\(\theta\)</span>.</p>
<p>It is clear that the ELBO sum coverges as it is nondecreasing with an upper bound, but it is not clear whether the sum converges to the correct value, i.e. <span class="math inline">\(\max_\theta p(x_{1 : m}; \theta)\)</span>. In fact it is said that the EM does get stuck in local maximum sometimes.</p>
<p>A different way of describing EM, which will be useful in hidden Markov model is:</p>
<ul>
<li><p>At E-step, one writes down the formula <span class="math display">\[\sum_i \mathbb E_{p(z_i | x_{i}; \theta_t)} \log p(x_{i}, z_i; \theta). \qquad (2.5)\]</span></p></li>
<li><p>At M-setp, one finds <span class="math inline">\(\theta_{t + 1}\)</span> to be the <span class="math inline">\(\theta\)</span> that maximises the above formula.</p></li>
</ul>
<h3 id="gmm">GMM</h3>
<p>Gaussian mixture model (GMM) is an example of mixture models.</p>
<p>The space of the data is <span class="math inline">\(\mathbb R^n\)</span>. We use the hypothesis that the data is Gaussian conditioned on the latent variable:</p>
<p><span class="math display">\[(x_i; \eta_k) \sim N(\mu_k, \Sigma_k),\]</span></p>
<p>so we write <span class="math inline">\(\eta_k = (\mu_k, \Sigma_k)\)</span>,</p>
<p>During E-step, the <span class="math inline">\(q(z_i)\)</span> can be directly computed using Bayes' theorem:</p>
<p><span class="math display">\[r_{ik} = q(z_i = k) = \mathbb P(z_i = k | x_{i}; \theta_t)
= {g_{\mu_{t, k}, \Sigma_{t, k}} (x_{i}) \pi_{t, k} \over \sum_{j = 1 : n_z} g_{\mu_{t, j}, \Sigma_{t, j}} (x_{i}) \pi_{t, j}},\]</span></p>
<p>where <span class="math inline">\(g_{\mu, \Sigma} (x) = (2 \pi)^{- n / 2} \det \Sigma^{-1 / 2} \exp(- {1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu))\)</span> is the pdf of the Gaussian distribution <span class="math inline">\(N(\mu, \Sigma)\)</span>.</p>
<p>During M-step, we need to compute</p>
<p><span class="math display">\[\text{argmax}_{\Sigma, \mu, \pi} \sum_{i = 1 : m} \sum_{k = 1 : n_z} r_{ik} \log (g_{\mu_k, \Sigma_k}(x_{i}) \pi_k).\]</span></p>
<p>This is similar to the quadratic discriminant analysis, and the solution is</p>
<p><span class="math display">\[\begin{aligned}
\pi_{k} &= {1 \over m} \sum_{i = 1 : m} r_{ik}, \\
\mu_{k} &= {\sum_i r_{ik} x_{i} \over \sum_i r_{ik}}, \\
\Sigma_{k} &= {\sum_i r_{ik} (x_{i} - \mu_{t, k}) (x_{i} - \mu_{t, k})^T \over \sum_i r_{ik}}.
\end{aligned}\]</span></p>
<p><strong>Remark</strong>. The k-means algorithm is the <span class="math inline">\(\epsilon \to 0\)</span> limit of the GMM with constraints <span class="math inline">\(\Sigma_k = \epsilon I\)</span>. See Section 9.3.2 of Bishop 2006 for derivation. It is also briefly mentioned there that a variant in this setting where the covariance matrix is not restricted to <span class="math inline">\(\epsilon I\)</span> is called elliptical k-means algorithm.</p>
<h3 id="smm">SMM</h3>
<p>As a transition to the next models to study, let us consider a simpler mixture model obtained by making one modification to GMM: change <span class="math inline">\((x; \eta_k) \sim N(\mu_k, \Sigma_k)\)</span> to <span class="math inline">\(\mathbb P(x = w; \eta_k) = \eta_{kw}\)</span> where <span class="math inline">\(\eta\)</span> is a stochastic matrix and <span class="math inline">\(w\)</span> is an arbitrary element of the space for <span class="math inline">\(x\)</span>. So now the space for both <span class="math inline">\(x\)</span> and <span class="math inline">\(z\)</span> are finite. We call this model the simple mixture model (SMM).</p>
<p>As in GMM, at E-step <span class="math inline">\(r_{ik}\)</span> can be explicitly computed using Bayes' theorem.</p>
<p>It is not hard to write down the solution to the M-step in this case:</p>
<p><span class="math display">\[\begin{aligned}
\pi_{k} &= {1 \over m} \sum_i r_{ik}, \qquad (2.7)\\
\eta_{k, w} &= {\sum_i r_{ik} 1_{x_i = w} \over \sum_i r_{ik}}. \qquad (2.8)
\end{aligned}\]</span></p>
<p>where <span class="math inline">\(1_{x_i = w}\)</span> is the <a href="https://en.wikipedia.org/wiki/Indicator_function">indicator function</a>, and evaluates to <span class="math inline">\(1\)</span> if <span class="math inline">\(x_i = w\)</span> and <span class="math inline">\(0\)</span> otherwise.</p>
<p>Two trivial variants of the SMM are the two versions of probabilistic latent semantic analysis (pLSA), which we call pLSA1 and pLSA2.</p>
<p>The model pLSA1 is a probabilistic version of latent semantic analysis, which is basically a simple matrix factorisation model in collaborative filtering, whereas pLSA2 has a fully Bayesian version called latent Dirichlet allocation (LDA), not to be confused with the other LDA (linear discriminant analysis).</p>
<h3 id="plsa">pLSA</h3>
<p>The pLSA model (Hoffman 2000) is a mixture model, where the dataset is now pairs <span class="math inline">\((d_i, x_i)_{i = 1 : m}\)</span>. In natural language processing, <span class="math inline">\(x\)</span> are words and <span class="math inline">\(d\)</span> are documents, and a pair <span class="math inline">\((d, x)\)</span> represent an ocurrance of word <span class="math inline">\(x\)</span> in document <span class="math inline">\(d\)</span>.</p>
<p>For each datapoint <span class="math inline">\((d_{i}, x_{i})\)</span>,</p>
<p><span class="math display">\[\begin{aligned}
p(d_i, x_i; \theta) &= \sum_{z_i} p(z_i; \theta) p(d_i | z_i; \theta) p(x_i | z_i; \theta) \qquad (2.91)\\
&= p(d_i; \theta) \sum_z p(x_i | z_i; \theta) p (z_i | d_i; \theta) \qquad (2.92).
\end{aligned}\]</span></p>
<p>Of the two formulations, (2.91) corresponds to pLSA type 1, and (2.92) corresponds to type 2.</p>
<h4 id="plsa1">pLSA1</h4>
<p>The pLSA1 model (Hoffman 2000) is basically SMM with <span class="math inline">\(x_i\)</span> substituted with <span class="math inline">\((d_i, x_i)\)</span>, which conditioned on <span class="math inline">\(z_i\)</span> are independently categorically distributed:</p>
<p><span class="math display">\[p(d_i = u, x_i = w | z_i = k; \theta) = p(d_i ; \xi_k) p(x_i; \eta_k) = \xi_{ku} \eta_{kw}.\]</span></p>
<p>The model can be illustrated in the plate notations:</p>
<div class="figure">
<img src="/assets/resources/plsa1.png" style="width:350px" />
</div>
<p>So the solution of the M-step is</p>
<p><span class="math display">\[\begin{aligned}
\pi_{k} &= {1 \over m} \sum_i r_{ik} \\
\xi_{k, u} &= {\sum_i r_{ik} 1_{d_{i} = u} \over \sum_i r_{ik}} \\
\eta_{k, w} &= {\sum_i r_{ik} 1_{x_{i} = w} \over \sum_i r_{ik}}.
\end{aligned}\]</span></p>
<p><strong>Remark</strong>. pLSA1 is the probabilistic version of LSA, also known as matrix factorisation.</p>
<p>Let <span class="math inline">\(n_d\)</span> and <span class="math inline">\(n_x\)</span> be the number of values <span class="math inline">\(d_i\)</span> and <span class="math inline">\(x_i\)</span> can take.</p>
<p><strong>Problem</strong> (LSA). Let <span class="math inline">\(R\)</span> be a <span class="math inline">\(n_d \times n_x\)</span> matrix, fix <span class="math inline">\(s \le \min\{n_d, n_x\}\)</span>. Find <span class="math inline">\(n_d \times s\)</span> matrix <span class="math inline">\(D\)</span> and <span class="math inline">\(n_x \times s\)</span> matrix <span class="math inline">\(X\)</span> that minimises</p>
<p><span class="math display">\[J(D, X) = \|R - D X^T\|_F.\]</span></p>
<p>where <span class="math inline">\(\|\cdot\|_F\)</span> is the Frobenius norm.</p>
<p><strong>Claim</strong>. Let <span class="math inline">\(R = U \Sigma V^T\)</span> be the SVD of <span class="math inline">\(R\)</span>, then the solution to the above problem is <span class="math inline">\(D = U_s \Sigma_s^{{1 \over 2}}\)</span> and <span class="math inline">\(X = V_s \Sigma_s^{{1 \over 2}}\)</span>, where <span class="math inline">\(U_s\)</span> (resp. <span class="math inline">\(V_s\)</span>) is the matrix of the first <span class="math inline">\(s\)</span> columns of <span class="math inline">\(U\)</span> (resp. <span class="math inline">\(V\)</span>) and <span class="math inline">\(\Sigma_s\)</span> is the <span class="math inline">\(s \times s\)</span> submatrix of <span class="math inline">\(\Sigma\)</span>.</p>
<p>One can compare pLSA1 with LSA. Both procedures produce embeddings of <span class="math inline">\(d\)</span> and <span class="math inline">\(x\)</span>: in pLSA we obtain <span class="math inline">\(n_z\)</span> dimensional embeddings <span class="math inline">\(\xi_{\cdot, u}\)</span> and <span class="math inline">\(\eta_{\cdot, w}\)</span>, whereas in LSA we obtain <span class="math inline">\(s\)</span> dimensional embeddings <span class="math inline">\(D_{u, \cdot}\)</span> and <span class="math inline">\(X_{w, \cdot}\)</span>.</p>
<h4 id="plsa2">pLSA2</h4>
<p>Let us turn to pLSA2 (Hoffman 2004), corresponding to (2.92). We rewrite it as</p>
<p><span class="math display">\[p(x_i | d_i; \theta) = \sum_{z_i} p(x_i | z_i; \theta) p(z_i | d_i; \theta).\]</span></p>
<p>To simplify notations, we collect all the <span class="math inline">\(x_i\)</span>s with the corresponding <span class="math inline">\(d_i\)</span> equal to 1 (suppose there are <span class="math inline">\(m_1\)</span> of them), and write them as <span class="math inline">\((x_{1, j})_{j = 1 : m_1}\)</span>. In the same fashion we construct <span class="math inline">\(x_{2, 1 : m_2}, x_{3, 1 : m_3}, ... x_{n_d, 1 : m_{n_d}}\)</span>. Similarly, we relabel the corresponding <span class="math inline">\(d_i\)</span> and <span class="math inline">\(z_i\)</span> accordingly.</p>
<p>With almost no loss of generality, we assume all <span class="math inline">\(m_\ell\)</span>s are equal and write them as <span class="math inline">\(m\)</span>.</p>
<p>Now the model becomes</p>
<p><span class="math display">\[p(x_{\ell, i} | d_{\ell, i} = \ell; \theta) = \sum_k p(x_{\ell, i} | z_{\ell, i} = k; \theta) p(z_{\ell, i} = k | d_{\ell, i} = \ell; \theta).\]</span></p>
<p>Since we have regrouped the <span class="math inline">\(x\)</span>'s and <span class="math inline">\(z\)</span>'s whose indices record the values of the <span class="math inline">\(d\)</span>'s, we can remove the <span class="math inline">\(d\)</span>'s from the equation altogether:</p>
<p><span class="math display">\[p(x_{\ell, i}; \theta) = \sum_k p(x_{\ell, i} | z_{\ell, i} = k; \theta) p(z_{\ell, i} = k; \theta).\]</span></p>
<p>It is effectively a modification of SMM by making <span class="math inline">\(n_d\)</span> copies of <span class="math inline">\(\pi\)</span>. More specifically the parameters are <span class="math inline">\(\theta = (\pi_{1 : n_d, 1 : n_z}, \eta_{1 : n_z, 1 : n_x})\)</span>, where we model <span class="math inline">\((z | d = \ell) \sim \text{Cat}(\pi_{\ell, \cdot})\)</span> and, as in pLSA1, <span class="math inline">\((x | z = k) \sim \text{Cat}(\eta_{k, \cdot})\)</span>.</p>
<p>Illustrated in the plate notations, pLSA2 is:</p>
<div class="figure">
<img src="/assets/resources/plsa2.png" style="width:350px" />
</div>
<p>The computation is basically adding an index <span class="math inline">\(\ell\)</span> to the computation of SMM wherever applicable.</p>
<p>The updates at the E-step is</p>
<p><span class="math display">\[r_{\ell i k} = p(z_{\ell i} = k | x_{\ell i}; \theta) \propto \pi_{\ell k} \eta_{k, x_{\ell i}}.\]</span></p>
<p>And at the M-step</p>
<p><span class="math display">\[\begin{aligned}
\pi_{\ell k} &= {1 \over m} \sum_i r_{\ell i k} \\
\eta_{k w} &= {\sum_{\ell, i} r_{\ell i k} 1_{x_{\ell i} = w} \over \sum_{\ell, i} r_{\ell i k}}.
\end{aligned}\]</span></p>
<h3 id="hmm">HMM</h3>
<p>The hidden markov model (HMM) is a sequential version of SMM, in the same sense that recurrent neural networks are sequential versions of feed-forward neural networks.</p>
<p>HMM is an example where the posterior <span class="math inline">\(p(z_i | x_i; \theta)\)</span> is not easy to compute, and one has to utilise properties of the underlying Bayesian network to go around it.</p>
<p>Now each sample is a sequence <span class="math inline">\(x_i = (x_{ij})_{j = 1 : T}\)</span>, and so are the latent variables <span class="math inline">\(z_i = (z_{ij})_{j = 1 : T}\)</span>.</p>
<p>The latent variables are assumed to form a Markov chain with transition matrix <span class="math inline">\((\xi_{k \ell})_{k \ell}\)</span>, and <span class="math inline">\(x_{ij}\)</span> is completely dependent on <span class="math inline">\(z_{ij}\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
p(z_{ij} | z_{i, j - 1}) &= \xi_{z_{i, j - 1}, z_{ij}},\\
p(x_{ij} | z_{ij}) &= \eta_{z_{ij}, x_{ij}}.
\end{aligned}\]</span></p>
<p>Also, the distribution of <span class="math inline">\(z_{i1}\)</span> is again categorical with parameter <span class="math inline">\(\pi\)</span>:</p>
<p><span class="math display">\[p(z_{i1}) = \pi_{z_{i1}}\]</span></p>
<p>So the parameters are <span class="math inline">\(\theta = (\pi, \xi, \eta)\)</span>. And HMM can be shown in plate notations as:</p>
<div class="figure">
<img src="/assets/resources/hmm.png" style="width:350px" />
</div>
<p>Now we apply EM to HMM, which is called the <a href="https://en.wikipedia.org/wiki/Baum%E2%80%93Welch_algorithm">Baum-Welch algorithm</a>. Unlike the previous examples, it is too messy to compute <span class="math inline">\(p(z_i | x_{i}; \theta)\)</span>, so during the E-step we instead write down formula (2.5) directly in hope of simplifying it:</p>
<p><span class="math display">\[\begin{aligned}
\mathbb E_{p(z_i | x_i; \theta_t)} \log p(x_i, z_i; \theta_t) &=\mathbb E_{p(z_i | x_i; \theta_t)} \left(\log \pi_{z_{i1}} + \sum_{j = 2 : T} \log \xi_{z_{i, j - 1}, z_{ij}} + \sum_{j = 1 : T} \log \eta_{z_{ij}, x_{ij}}\right). \qquad (3)
\end{aligned}\]</span></p>
<p>Let us compute the summand in second term:</p>
<p><span class="math display">\[\begin{aligned}
\mathbb E_{p(z_i | x_{i}; \theta_t)} \log \xi_{z_{i, j - 1}, z_{ij}} &= \sum_{k, \ell} (\log \xi_{k, \ell}) \mathbb E_{p(z_{i} | x_{i}; \theta_t)} 1_{z_{i, j - 1} = k, z_{i, j} = \ell} \\
&= \sum_{k, \ell} p(z_{i, j - 1} = k, z_{ij} = \ell | x_{i}; \theta_t) \log \xi_{k, \ell}. \qquad (4)
\end{aligned}\]</span></p>
<p>Similarly, one can write down the first term and the summand in the third term to obtain</p>
<p><span class="math display">\[\begin{aligned}
\mathbb E_{p(z_i | x_{i}; \theta_t)} \log \pi_{z_{i1}} &= \sum_k p(z_{i1} = k | x_{i}; \theta_t), \qquad (5) \\
\mathbb E_{p(z_i | x_{i}; \theta_t)} \log \eta_{z_{i, j}, x_{i, j}} &= \sum_{k, w} 1_{x_{ij} = w} p(z_{i, j} = k | x_i; \theta_t) \log \eta_{k, w}. \qquad (6)
\end{aligned}\]</span></p>
<p>plugging (4)(5)(6) back into (3) and summing over <span class="math inline">\(j\)</span>, we obtain the formula to maximise over <span class="math inline">\(\theta\)</span> on:</p>
<p><span class="math display">\[\sum_k \sum_i r_{i1k} \log \pi_k + \sum_{k, \ell} \sum_{j = 2 : T, i} s_{ijk\ell} \log \xi_{k, \ell} + \sum_{k, w} \sum_{j = 1 : T, i} r_{ijk} 1_{x_{ij} = w} \log \eta_{k, w},\]</span></p>
<p>where</p>
<p><span class="math display">\[\begin{aligned}
r_{ijk} &:= p(z_{ij} = k | x_{i}; \theta_t), \\
s_{ijk\ell} &:= p(z_{i, j - 1} = k, z_{ij} = \ell | x_{i}; \theta_t).
\end{aligned}\]</span></p>
<p>Now we proceed to the M-step. Since each of the <span class="math inline">\(\pi_k, \xi_{k, \ell}, \eta_{k, w}\)</span> is nicely confined in the inner sum of each term, together with the constraint <span class="math inline">\(\sum_k \pi_k = \sum_\ell \xi_{k, \ell} = \sum_w \eta_{k, w} = 1\)</span> it is not hard to find the argmax at time <span class="math inline">\(t + 1\)</span> (the same way one finds the MLE for any categorical distribution):</p>
<p><span class="math display">\[\begin{aligned}
\pi_{k} &= {1 \over m} \sum_i r_{i1k}, \qquad (6.1) \\
\xi_{k, \ell} &= {\sum_{j = 2 : T, i} s_{ijk\ell} \over \sum_{j = 1 : T - 1, i} r_{ijk}}, \qquad(6.2) \\
\eta_{k, w} &= {\sum_{ij} 1_{x_{ij} = w} r_{ijk} \over \sum_{ij} r_{ijk}}. \qquad(6.3)
\end{aligned}\]</span></p>
<p>Note that (6.1)(6.3) are almost identical to (2.7)(2.8). This makes sense as the only modification HMM makes over SMM is the added dependencies between the latent variables.</p>
<p>What remains is to compute <span class="math inline">\(r\)</span> and <span class="math inline">\(s\)</span>.</p>
<p>This is done by using the forward and backward procedures which takes advantage of the conditioned independence / topology of the underlying Bayesian network. It is out of scope of this post, but for the sake of completeness I include it here.</p>
<p>Let</p>
<p><span class="math display">\[\begin{aligned}
\alpha_k(i, j) &:= p(x_{i, 1 : j}, z_{ij} = k; \theta_t), \\
\beta_k(i, j) &:= p(x_{i, j + 1 : T} | z_{ij} = k; \theta_t).
\end{aligned}\]</span></p>
<p>They can be computed recursively as</p>
<p><span class="math display">\[\begin{aligned}
\alpha_k(i, j) &= \begin{cases}
\eta_{k, x_{1j}} \pi_k, & j = 1; \\
\eta_{k, x_{ij}} \sum_\ell \alpha_\ell(j - 1, i) \xi_{k\ell}, & j \ge 2.
\end{cases}\\
\beta_k(i, j) &= \begin{cases}
1, & j = T;\\
\sum_\ell \xi_{k\ell} \beta_\ell(j + 1, i) \eta_{\ell, x_{i, j + 1}}, & j < T.
\end{cases}
\end{aligned}\]</span></p>
<p>Then</p>
<p><span class="math display">\[\begin{aligned}
p(z_{ij} = k, x_{i}; \theta_t) &= \alpha_k(i, j) \beta_k(i, j), \qquad (7)\\
p(x_{i}; \theta_t) &= \sum_k \alpha_k(i, j) \beta_k(i, j),\forall j = 1 : T \qquad (8)\\
p(z_{i, j - 1} = k, z_{i, j} = \ell, x_{i}; \theta_t) &= \alpha_k(i, j) \xi_{k\ell} \beta_\ell(i, j + 1) \eta_{\ell, x_{j + 1, i}}. \qquad (9)
\end{aligned}\]</span></p>
<p>And this yields <span class="math inline">\(r_{ijk}\)</span> and <span class="math inline">\(s_{ijk\ell}\)</span> since they can be computed as <span class="math inline">\({(7) \over (8)}\)</span> and <span class="math inline">\({(9) \over (8)}\)</span> respectively.</p>
<h2 id="fully-bayesian-em-mfa">Fully Bayesian EM / MFA</h2>
<p>Let us now venture into the realm of full Bayesian.</p>
<p>In EM we aim to maximise the ELBO</p>
<p><span class="math display">\[\int q(z) \log {p(x, z; \theta) \over q(z)} dz d\theta\]</span></p>
<p>alternately over <span class="math inline">\(q\)</span> and <span class="math inline">\(\theta\)</span>. As mentioned before, the E-step of maximising over <span class="math inline">\(q\)</span> is Bayesian, in that it computes the posterior of <span class="math inline">\(z\)</span>, whereas the M-step of maximising over <span class="math inline">\(\theta\)</span> is maximum likelihood and frequentist.</p>
<p>The fully Bayesian EM makes the M-step Bayesian by making <span class="math inline">\(\theta\)</span> a random variable, so the ELBO becomes</p>
<p><span class="math display">\[L(p(x, z, \theta), q(z, \theta)) = \int q(z, \theta) \log {p(x, z, \theta) \over q(z, \theta)} dz d\theta\]</span></p>
<p>We further assume <span class="math inline">\(q\)</span> can be factorised into distributions on <span class="math inline">\(z\)</span> and <span class="math inline">\(\theta\)</span>: <span class="math inline">\(q(z, \theta) = q_1(z) q_2(\theta)\)</span>. So the above formula is rewritten as</p>
<p><span class="math display">\[L(p(x, z, \theta), q(z, \theta)) = \int q_1(z) q_2(\theta) \log {p(x, z, \theta) \over q_1(z) q_2(\theta)} dz d\theta\]</span></p>
<p>To find argmax over <span class="math inline">\(q_1\)</span>, we rewrite</p>
<p><span class="math display">\[\begin{aligned}
L(p(x, z, \theta), q(z, \theta)) &= \int q_1(z) \left(\int q_2(\theta) \log p(x, z, \theta) d\theta\right) dz - \int q_1(z) \log q_1(z) dz - \int q_2(\theta) \log q_2(\theta) \\&= - D(q_1(z) || p_x(z)) + C,
\end{aligned}\]</span></p>
<p>where <span class="math inline">\(p_x\)</span> is a density in <span class="math inline">\(z\)</span> with</p>
<p><span class="math display">\[\log p_x(z) = \mathbb E_{q_2(\theta)} \log p(x, z, \theta) + C.\]</span></p>
<p>So the <span class="math inline">\(q_1\)</span> that maximises the ELBO is <span class="math inline">\(q_1^* = p_x\)</span>.</p>
<p>Similarly, the optimal <span class="math inline">\(q_2\)</span> is such that</p>
<p><span class="math display">\[\log q_2^*(\theta) = \mathbb E_{q_1(z)} \log p(x, z, \theta) + C.\]</span></p>
<p>The fully Bayesian EM thus alternately evaluates <span class="math inline">\(q_1^*\)</span> (E-step) and <span class="math inline">\(q_2^*\)</span> (M-step).</p>
<p>It is also called mean field approximation (MFA), and can be easily generalised to models with more than two groups of latent variables, see e.g. Section 10.1 of Bishop 2006.</p>
<h3 id="application-to-mixture-models">Application to mixture models</h3>
<p><strong>Definition (Fully Bayesian mixture model)</strong>. The relations between <span class="math inline">\(\pi\)</span>, <span class="math inline">\(\eta\)</span>, <span class="math inline">\(x\)</span>, <span class="math inline">\(z\)</span> are the same as in the definition of mixture models. Furthermore, we assume the distribution of <span class="math inline">\((x | \eta_k)\)</span> belongs to the <a href="https://en.wikipedia.org/wiki/Exponential_family">exponential family</a> (the definition of the exponential family is briefly touched at the end of this section). But now both <span class="math inline">\(\pi\)</span> and <span class="math inline">\(\eta\)</span> are random variables. Let the prior distribution <span class="math inline">\(p(\pi)\)</span> is Dirichlet with parameter <span class="math inline">\((\alpha, \alpha, ..., \alpha)\)</span>. Let the prior <span class="math inline">\(p(\eta_k)\)</span> be the conjugate prior of <span class="math inline">\((x | \eta_k)\)</span>, with parameter <span class="math inline">\(\beta\)</span>, we will see later in this section that the posterior <span class="math inline">\(q(\eta_k)\)</span> belongs to the same family as <span class="math inline">\(p(\eta_k)\)</span>. Represented in a plate notations, a fully Bayesian mixture model looks like:</p>
<div class="figure">
<img src="/assets/resources/fully-bayesian-mm.png" style="width:450px" />
</div>
<p>Given this structure we can write down the mean-field approximation:</p>
<p><span class="math display">\[\log q(z) = \mathbb E_{q(\eta)q(\pi)} (\log(x | z, \eta) + \log(z | \pi)) + C.\]</span></p>
<p>Both sides can be factored into per-sample expressions, giving us</p>
<p><span class="math display">\[\log q(z_i) = \mathbb E_{q(\eta)} \log p(x_i | z_i, \eta) + \mathbb E_{q(\pi)} \log p(z_i | \pi) + C\]</span></p>
<p>Therefore</p>
<p><span class="math display">\[\log r_{ik} = \log q(z_i = k) = \mathbb E_{q(\eta_k)} \log p(x_i | \eta_k) + \mathbb E_{q(\pi)} \log \pi_k + C. \qquad (9.1)\]</span></p>
<p>So the posterior of each <span class="math inline">\(z_i\)</span> is categorical regardless of the <span class="math inline">\(p\)</span>s and <span class="math inline">\(q\)</span>s.</p>
<p>Computing the posterior of <span class="math inline">\(\pi\)</span> and <span class="math inline">\(\eta\)</span>:</p>
<p><span class="math display">\[\log q(\pi) + \log q(\eta) = \log p(\pi) + \log p(\eta) + \sum_i \mathbb E_{q(z_i)} p(x_i | z_i, \eta) + \sum_i \mathbb E_{q(z_i)} p(z_i | \pi) + C.\]</span></p>
<p>So we can separate the terms involving <span class="math inline">\(\pi\)</span> and those involving <span class="math inline">\(\eta\)</span>. First compute the posterior of <span class="math inline">\(\pi\)</span>:</p>
<p><span class="math display">\[\log q(\pi) = \log p(\pi) + \sum_i \mathbb E_{q(z_i)} \log p(z_i | \pi) = \log p(\pi) + \sum_i \sum_k r_{ik} \log \pi_k + C.\]</span></p>
<p>The right hand side is the log of a Dirichlet modulus the constant <span class="math inline">\(C\)</span>, from which we can update the posterior parameter <span class="math inline">\(\phi^\pi\)</span>:</p>
<p><span class="math display">\[\phi^\pi_k = \alpha + \sum_i r_{ik}. \qquad (9.3)\]</span></p>
<p>Similarly we can obtain the posterior of <span class="math inline">\(\eta\)</span>:</p>
<p><span class="math display">\[\log q(\eta) = \log p(\eta) + \sum_i \sum_k r_{ik} \log p(x_i | \eta_k) + C.\]</span></p>
<p>Again we can factor the terms with respect to <span class="math inline">\(k\)</span> and get:</p>
<p><span class="math display">\[\log q(\eta_k) = \log p(\eta_k) + \sum_i r_{ik} \log p(x_i | \eta_k) + C. \qquad (9.5)\]</span></p>
<p>Here we can see why conjugate prior works. Mathematically, given a probability distribution <span class="math inline">\(p(x | \theta)\)</span>, the distribution <span class="math inline">\(p(\theta)\)</span> is called conjugate prior of <span class="math inline">\(p(x | \theta)\)</span> if <span class="math inline">\(\log p(\theta) + \log p(x | \theta)\)</span> has the same form as <span class="math inline">\(\log p(\theta)\)</span>.</p>
<p>For example, the conjugate prior for the exponential family <span class="math inline">\(p(x | \theta) = h(x) \exp(\theta \cdot T(x) - A(\theta))\)</span> where <span class="math inline">\(T\)</span>, <span class="math inline">\(A\)</span> and <span class="math inline">\(h\)</span> are some functions is <span class="math inline">\(p(\theta; \chi, \nu) \propto \exp(\chi \cdot \theta - \nu A(\theta))\)</span>.</p>
<p>Here what we want is a bit different from conjugate priors because of the coefficients <span class="math inline">\(r_{ik}\)</span>. But the computation carries over to the conjugate priors of the exponential family (try it yourself and you'll see). That is, if <span class="math inline">\(p(x_i | \eta_k)\)</span> belongs to the exponential family</p>
<p><span class="math display">\[p(x_i | \eta_k) = h(x) \exp(\eta_k \cdot T(x) - A(\eta_k))\]</span></p>
<p>and if <span class="math inline">\(p(\eta_k)\)</span> is the conjugate prior of <span class="math inline">\(p(x_i | \eta_k)\)</span></p>
<p><span class="math display">\[p(\eta_k) \propto \exp(\chi \cdot \eta_k - \nu A(\eta_k))\]</span></p>
<p>then <span class="math inline">\(q(\eta_k)\)</span> has the same form as <span class="math inline">\(p(\eta_k)\)</span>, and from (9.5) we can compute the updates of <span class="math inline">\(\phi^{\eta_k}\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
\phi^{\eta_k}_1 &= \chi + \sum_i r_{ik} T(x_i), \qquad (9.7) \\
\phi^{\eta_k}_2 &= \nu + \sum_i r_{ik}. \qquad (9.9)
\end{aligned}\]</span></p>
<p>So the mean field approximation for the fully Bayesian mixture model is the alternate iteration of (9.1) (E-step) and (9.3)(9.7)(9.9) (M-step) until convergence.</p>
<h3 id="fully-bayesian-gmm">Fully Bayesian GMM</h3>
<p>A typical example of fully Bayesian mixture models is the fully Bayesian Gaussian mixture model (Attias 2000, also called variational GMM in the literature). It is defined by applying the same modification to GMM as the difference between Fully Bayesian mixture model and the vanilla mixture model.</p>
<p>More specifically:</p>
<ul>
<li><span class="math inline">\(p(z_{i}) = \text{Cat}(\pi)\)</span> as in vanilla GMM</li>
<li><span class="math inline">\(p(\pi) = \text{Dir}(\alpha, \alpha, ..., \alpha)\)</span> has Dirichlet distribution, the conjugate prior to the parameters of the categorical distribution.</li>
<li><span class="math inline">\(p(x_i | z_i = k) = p(x_i | \eta_k) = N(\mu_{k}, \Sigma_{k})\)</span> as in vanilla GMM</li>
<li><span class="math inline">\(p(\mu_k, \Sigma_k) = \text{NIW} (\mu_0, \lambda, \Psi, \nu)\)</span> is the normal-inverse-Wishart distribution, the conjugate prior to the mean and covariance matrix of the Gaussian distribution.</li>
</ul>
<p>The E-step and M-step can be computed using (9.1) and (9.3)(9.7)(9.9) in the previous section. The details of the computation can be found in Chapter 10.2 of Bishop 2006 or Attias 2000.</p>
<h3 id="lda">LDA</h3>
<p>As the second example of fully Bayesian mixture models, Latent Dirichlet allocation (LDA) (Blei-Ng-Jordan 2003) is the fully Bayesian version of pLSA2, with the following plate notations:</p>
<div class="figure">
<img src="/assets/resources/lda.png" style="width:450px" />
</div>
<p>It is the smoothed version in the paper.</p>
<p>More specifically, on the basis of pLSA2, we add prior distributions to <span class="math inline">\(\eta\)</span> and <span class="math inline">\(\pi\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
p(\eta_k) &= \text{Dir} (\beta, ..., \beta), \qquad k = 1 : n_z \\
p(\pi_\ell) &= \text{Dir} (\alpha, ..., \alpha), \qquad \ell = 1 : n_d \\
\end{aligned}\]</span></p>
<p>And as before, the prior of <span class="math inline">\(z\)</span> is</p>
<p><span class="math display">\[p(z_{\ell, i}) = \text{Cat} (\pi_\ell), \qquad \ell = 1 : n_d, i = 1 : m\]</span></p>
<p>We also denote posterior distribution</p>
<p><span class="math display">\[\begin{aligned}
q(\eta_k) &= \text{Dir} (\phi^{\eta_k}), \qquad k = 1 : n_z \\
q(\pi_\ell) &= \text{Dir} (\phi^{\pi_\ell}), \qquad \ell = 1 : n_d \\
p(z_{\ell, i}) &= \text{Cat} (r_{\ell, i}), \qquad \ell = 1 : n_d, i = 1 : m
\end{aligned}\]</span></p>
<p>As before, in E-step we update <span class="math inline">\(r\)</span>, and in M-step we update <span class="math inline">\(\lambda\)</span> and <span class="math inline">\(\gamma\)</span>.</p>
<p>But in the LDA paper, one treats optimisation over <span class="math inline">\(r\)</span>, <span class="math inline">\(\lambda\)</span> and <span class="math inline">\(\gamma\)</span> as a E-step, and treats <span class="math inline">\(\alpha\)</span> and <span class="math inline">\(\beta\)</span> as parameters, which is optmised over at M-step. This makes it more akin to the classical EM where the E-step is Bayesian and M-step MLE. This is more complicated, and we do not consider it this way here.</p>
<p>Plugging in (9.1) we obtain the updates at E-step</p>
<p><span class="math display">\[r_{\ell i k} \propto \exp(\psi(\phi^{\pi_\ell}_k) + \psi(\phi^{\eta_k}_{x_{\ell i}}) - \psi(\sum_w \phi^{\eta_k}_w)), \qquad (10)\]</span></p>
<p>where <span class="math inline">\(\psi\)</span> is the digamma function. Similarly, plugging in (9.3)(9.7)(9.9), at M-step, we update the posterior of <span class="math inline">\(\pi\)</span> and <span class="math inline">\(\eta\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
\phi^{\pi_\ell}_k &= \alpha + \sum_i r_{\ell i k}. \qquad (11)\\
%%}}$
%%As for $\eta$, we have
%%{{$%align%
%%\log q(\eta) &= \sum_k \log p(\eta_k) + \sum_{\ell, i} \mathbb E_{q(z_{\ell i})} \log p(x_{\ell i} | z_{\ell i}, \eta) \\
%%&= \sum_{k, j} (\sum_{\ell, i} r_{\ell i k} 1_{x_{\ell i} = j} + \beta - 1) \log \eta_{k j}
%%}}$
%%which gives us
%%{{$
\phi^{\eta_k}_w &= \beta + \sum_{\ell, i} r_{\ell i k} 1_{x_{\ell i} = w}. \qquad (12)
\end{aligned}\]</span></p>
<p>So the algorithm iterates over (10) and (11)(12) until convergence.</p>
<h3 id="dpmm">DPMM</h3>
<p>The Dirichlet process mixture model (DPMM) is like the fully Bayesian mixture model except <span class="math inline">\(n_z = \infty\)</span>, i.e. <span class="math inline">\(z\)</span> can take any positive integer value.</p>
<p>The probability of <span class="math inline">\(z_i = k\)</span> is defined using the so called stick-breaking process: let <span class="math inline">\(v_i \sim \text{Beta} (\alpha, \beta)\)</span> be i.i.d. random variables with Beta distributions, then</p>
<p><span class="math display">\[\mathbb P(z_i = k | v_{1:\infty}) = (1 - v_1) (1 - v_2) ... (1 - v_{k - 1}) v_k.\]</span></p>
<p>So <span class="math inline">\(v\)</span> plays a similar role to <span class="math inline">\(\pi\)</span> in the previous models.</p>
<p>As before, we have that the distribution of <span class="math inline">\(x\)</span> belongs to the exponential family:</p>
<p><span class="math display">\[p(x | z = k, \eta) = p(x | \eta_k) = h(x) \exp(\eta_k \cdot T(x) - A(\eta_k))\]</span></p>
<p>so the prior of <span class="math inline">\(\eta_k\)</span> is</p>
<p><span class="math display">\[p(\eta_k) \propto \exp(\chi \cdot \eta_k - \nu A(\eta_k)).\]</span></p>
<p>Because of the infinities we can't directly apply the formulas in the general fully Bayesian mixture models. So let us carefully derive the whole thing again.</p>
<p>As before, we can write down the ELBO:</p>
<p><span class="math display">\[L(p(x, z, \theta), q(z, \theta)) = \mathbb E_{q(\theta)} \log {p(\theta) \over q(\theta)} + \mathbb E_{q(\theta) q(z)} \log {p(x, z | \theta) \over q(z)}.\]</span></p>
<p>Both terms are infinite series:</p>
<p><span class="math display">\[L(p, q) = \sum_{k = 1 : \infty} \mathbb E_{q(\theta_k)} \log {p(\theta_k) \over q(\theta_k)} + \sum_{i = 1 : m} \sum_{k = 1 : \infty} q(z_i = k) \mathbb E_{q(\theta)} \log {p(x_i, z_i = k | \theta) \over q(z_i = k)}.\]</span></p>
<p>There are several ways to deal with the infinities. One is to fix some level <span class="math inline">\(T > 0\)</span> and set <span class="math inline">\(v_T = 1\)</span> almost surely (Blei-Jordan 2006). This effectively turns the model into a finite one, and both terms become finite sums over <span class="math inline">\(k = 1 : T\)</span>.</p>
<p>Another walkaround (Kurihara-Welling-Vlassis 2007) is also a kind of truncation, but less heavy-handed: setting the posterior <span class="math inline">\(q(\theta) = q(\eta) q(v)\)</span> to be:</p>
<p><span class="math display">\[q(\theta) = q(\theta_{1 : T}) p(\theta_{T + 1 : \infty}) =: q(\theta_{\le T}) p(\theta_{> T}).\]</span></p>
<p>That is, tie the posterior after <span class="math inline">\(T\)</span> to the prior. This effectively turns the first term in the ELBO to a finite sum over <span class="math inline">\(k = 1 : T\)</span>, while keeping the second sum an infinite series:</p>
<p><span class="math display">\[L(p, q) = \sum_{k = 1 : T} \mathbb E_{q(\theta_k)} \log {p(\theta_k) \over q(\theta_k)} + \sum_i \sum_{k = 1 : \infty} q(z_i = k) \mathbb E_{q(\theta)} \log {p(x_i, z_i = k | \theta) \over q(z_i = k)}. \qquad (13)\]</span></p>
<p>The plate notation of this model is:</p>
<div class="figure">
<img src="/assets/resources/dpmm.png" style="width:450px" />
</div>
<p>As it turns out, the infinities can be tamed in this case.</p>
<p>As before, the optimal <span class="math inline">\(q(z_i)\)</span> is computed as</p>
<p><span class="math display">\[r_{ik} = q(z_i = k) = s_{ik} / S_i\]</span></p>
<p>where</p>
<p><span class="math display">\[\begin{aligned}
s_{ik} &= \exp(\mathbb E_{q(\theta)} \log p(x_i, z_i = k | \theta)) \\
S_i &= \sum_{k = 1 : \infty} s_{ik}.
\end{aligned}\]</span></p>
<p>Plugging this back to (13) we have</p>
<p><span class="math display">\[\begin{aligned}
\sum_{k = 1 : \infty} r_{ik} &\mathbb E_{q(\theta)} \log {p(x_i, z_i = k | \theta) \over r_{ik}} \\
&= \sum_{k = 1 : \infty} r_{ik} \mathbb E_{q(\theta)} (\log p(x_i, z_i = k | \theta) - \mathbb E_{q(\theta)} \log p(x_i, z_i = k | \theta) + \log S_i) = \log S_i.
\end{aligned}\]</span></p>
<p>So it all rests upon <span class="math inline">\(S_i\)</span> being finite.</p>
<p>For <span class="math inline">\(k \le T + 1\)</span>, we compute the quantity <span class="math inline">\(s_{ik}\)</span> directly. For <span class="math inline">\(k > T\)</span>, it is not hard to show that</p>
<p><span class="math display">\[s_{ik} = s_{i, T + 1} \exp((k - T - 1) \mathbb E_{p(w)} \log (1 - w)),\]</span></p>
<p>where <span class="math inline">\(w\)</span> is a random variable with same distribution as <span class="math inline">\(p(v_k)\)</span>, i.e. <span class="math inline">\(\text{Beta}(\alpha, \beta)\)</span>.</p>
<p>Hence</p>
<p><span class="math display">\[S_i = \sum_{k = 1 : T} s_{ik} + {s_{i, T + 1} \over 1 - \exp(\psi(\beta) - \psi(\alpha + \beta))}\]</span></p>
<p>is indeed finite. Similarly we also obtain</p>
<p><span class="math display">\[q(z_i > k) = S^{-1} \left(\sum_{\ell = k + 1 : T} s_\ell + {s_{i, T + 1} \over 1 - \exp(\psi(\beta) - \psi(\alpha + \beta))}\right), k \le T \qquad (14)\]</span></p>
<p>Now let us compute the posterior of <span class="math inline">\(\theta_{\le T}\)</span>. In the following we exchange the integrals without justifying them (c.f. Fubini's Theorem). Equation (13) can be rewritten as</p>
<p><span class="math display">\[L(p, q) = \mathbb E_{q(\theta_{\le T})} \left(\log p(\theta_{\le T}) + \sum_i \mathbb E_{q(z_i) p(\theta_{> T})} \log {p(x_i, z_i | \theta) \over q(z_i)} - \log q(\theta_{\le T})\right).\]</span></p>
<p>Note that unlike the derivation of the mean-field approximation, we keep the <span class="math inline">\(- \mathbb E_{q(z)} \log q(z)\)</span> term even though we are only interested in <span class="math inline">\(\theta\)</span> at this stage. This is again due to the problem of infinities: as in the computation of <span class="math inline">\(S\)</span>, we would like to cancel out some undesirable unbounded terms using <span class="math inline">\(q(z)\)</span>. We now have</p>
<p><span class="math display">\[\log q(\theta_{\le T}) = \log p(\theta_{\le T}) + \sum_i \mathbb E_{q(z_i) p(\theta_{> T})} \log {p(x_i, z_i | \theta) \over q(z_i)} + C.\]</span></p>
<p>By plugging in <span class="math inline">\(q(z = k)\)</span> we obtain</p>
<p><span class="math display">\[\log q(\theta_{\le T}) = \log p(\theta_{\le T}) + \sum_{k = 1 : \infty} q(z_i = k) \left(\mathbb E_{p(\theta_{> T})} \log {p(x_i, z_i = k | \theta) \over q(z_i = k)} - \mathbb E_{q(\theta)} \log {p(x_i, z_i = k | \theta) \over q(z_i = k)}\right) + C.\]</span></p>
<p>Again, we separate the <span class="math inline">\(v_k\)</span>'s and the <span class="math inline">\(\eta_k\)</span>'s to obtain</p>
<p><span class="math display">\[q(v_{\le T}) = \log p(v_{\le T}) + \sum_i \sum_k q(z_i = k) \left(\mathbb E_{p(v_{> T})} \log p(z_i = k | v) - \mathbb E_{q(v)} \log p (z_i = k | v)\right).\]</span></p>
<p>Denote by <span class="math inline">\(D_k\)</span> the difference between the two expetations on the right hand side. It is easy to show that</p>
<p><span class="math display">\[D_k = \begin{cases}
\log(1 - v_1) ... (1 - v_{k - 1}) v_k - \mathbb E_{q(v)} \log (1 - v_1) ... (1 - v_{k - 1}) v_k & k \le T\\
\log(1 - v_1) ... (1 - v_T) - \mathbb E_{q(v)} \log (1 - v_1) ... (1 - v_T) & k > T
\end{cases}\]</span></p>
<p>so <span class="math inline">\(D_k\)</span> is bounded. With this we can derive the update for <span class="math inline">\(\phi^{v, 1}\)</span> and <span class="math inline">\(\phi^{v, 2}\)</span>:</p>
<p><span class="math display">\[\begin{aligned}
\phi^{v, 1}_k &= \alpha + \sum_i q(z_i = k) \\
\phi^{v, 2}_k &= \beta + \sum_i q(z_i > k),
\end{aligned}\]</span></p>
<p>where <span class="math inline">\(q(z_i > k)\)</span> can be computed as in (14).</p>
<p>When it comes to <span class="math inline">\(\eta\)</span>, we have</p>
<p><span class="math display">\[\log q(\eta_{\le T}) = \log p(\eta_{\le T}) + \sum_i \sum_{k = 1 : \infty} q(z_i = k) (\mathbb E_{p(\eta_k)} \log p(x_i | \eta_k) - \mathbb E_{q(\eta_k)} \log p(x_i | \eta_k)).\]</span></p>
<p>Since <span class="math inline">\(q(\eta_k) = p(\eta_k)\)</span> for <span class="math inline">\(k > T\)</span>, the inner sum on the right hand side is a finite sum over <span class="math inline">\(k = 1 : T\)</span>. By factorising <span class="math inline">\(q(\eta_{\le T})\)</span> and <span class="math inline">\(p(\eta_{\le T})\)</span>, we have</p>
<p><span class="math display">\[\log q(\eta_k) = \log p(\eta_k) + \sum_i q(z_i = k) \log (x_i | \eta_k) + C,\]</span></p>
<p>which gives us</p>
<p><span class="math display">\[\begin{aligned}
\phi^{\eta, 1}_k &= \xi + \sum_i q(z_i = k) T(x_i) \\
\phi^{\eta, 2}_k &= \nu + \sum_i q(z_i = k).
\end{aligned}\]</span></p>
<h2 id="svi">SVI</h2>
<p>In variational inference, the computation of some parameters are more expensive than others.</p>
<p>For example, the computation of M-step is often much more expensive than that of E-step:</p>
<ul>
<li>In the vanilla mixture models with the EM algorithm, the update of <span class="math inline">\(\theta\)</span> requires the computation of <span class="math inline">\(r_{ik}\)</span> for all <span class="math inline">\(i = 1 : m\)</span>, see Eq (2.3).</li>
<li>In the fully Bayesian mixture model with mean field approximation, the updates of <span class="math inline">\(\phi^\pi\)</span> and <span class="math inline">\(\phi^\eta\)</span> require the computation of a sum over all samples (see Eq (9.3)(9.7)(9.9)).</li>
</ul>
<p>Similarly, in pLSA2 (resp. LDA), the updates of <span class="math inline">\(\eta_k\)</span> (resp. <span class="math inline">\(\phi^{\eta_k}\)</span>) requires a sum over <span class="math inline">\(\ell = 1 : n_d\)</span>, whereas the updates of other parameters do not.</p>
<p>In these cases, the parameter that requires more computations are called global and the other ones local.</p>
<p>Stochastic variational inference (SVI, Hoffman-Blei-Wang-Paisley 2012) addresses this problem in the same way as stochastic gradient descent improves efficiency of gradient descent.</p>
<p>Each time SVI picks a sample, updates the corresponding local parameters, and computes the update of the global parameters as if all the <span class="math inline">\(m\)</span> samples are identical to the picked sample. Finally it incorporates this global parameter value into previous computations of the global parameters, by means of an exponential moving average.</p>
<p>As an example, here's SVI applied to LDA:</p>
<ol style="list-style-type: decimal">
<li>Set <span class="math inline">\(t = 1\)</span>.</li>
<li>Pick <span class="math inline">\(\ell\)</span> uniformly from <span class="math inline">\(\{1, 2, ..., n_d\}\)</span>.</li>
<li>Repeat until convergence:
<ol style="list-style-type: decimal">
<li>Compute <span class="math inline">\((r_{\ell i k})_{i = 1 : m, k = 1 : n_z}\)</span> using (10).</li>
<li>Compute <span class="math inline">\((\phi^{\pi_\ell}_k)_{k = 1 : n_z}\)</span> using (11).</li>
</ol></li>
<li><p>Compute <span class="math inline">\((\tilde \phi^{\eta_k}_w)_{k = 1 : n_z, w = 1 : n_x}\)</span> using the following formula (compare with (12)) <span class="math display">\[\tilde \phi^{\eta_k}_w = \beta + n_d \sum_{i} r_{\ell i k} 1_{x_{\ell i} = w}\]</span></p></li>
<li><p>Update the exponential moving average <span class="math inline">\((\phi^{\eta_k}_w)_{k = 1 : n_z, w = 1 : n_x}\)</span>: <span class="math display">\[\phi^{\eta_k}_w = (1 - \rho_t) \phi^{\eta_k}_w + \rho_t \tilde \phi^{\eta_k}_w\]</span></p></li>
<li><p>Increment <span class="math inline">\(t\)</span> and go back to Step 2.</p></li>
</ol>
<p>In the original paper, <span class="math inline">\(\rho_t\)</span> needs to satisfy some conditions that guarantees convergence of the global parameters:</p>
<p><span class="math display">\[\begin{aligned}
\sum_t \rho_t = \infty \\
\sum_t \rho_t^2 < \infty
\end{aligned}\]</span></p>
<p>and the choice made there is</p>
<p><span class="math display">\[\rho_t = (t + \tau)^{-\kappa}\]</span></p>
<p>for some <span class="math inline">\(\kappa \in (.5, 1]\)</span> and <span class="math inline">\(\tau \ge 0\)</span>.</p>
<h2 id="aevb">AEVB</h2>
<p>SVI adds to variational inference stochastic updates similar to stochastic gradient descent. Why not just use neural networks with stochastic gradient descent while we are at it? Autoencoding variational Bayes (AEVB) (Kingma-Welling 2013) is such an algorithm.</p>
<p>Let's look back to the original problem of maximising the ELBO:</p>
<p><span class="math display">\[\max_{\theta, q} \sum_{i = 1 : m} L(p(x_i | z_i; \theta) p(z_i; \theta), q(z_i))\]</span></p>
<p>Since for any given <span class="math inline">\(\theta\)</span>, the optimal <span class="math inline">\(q(z_i)\)</span> is the posterior <span class="math inline">\(p(z_i | x_i; \theta)\)</span>, the problem reduces to</p>
<p><span class="math display">\[\max_{\theta} \sum_i L(p(x_i | z_i; \theta) p(z_i; \theta), p(z_i | x_i; \theta))\]</span></p>
<p>Let us assume <span class="math inline">\(p(z_i; \theta) = p(z_i)\)</span> is independent of <span class="math inline">\(\theta\)</span> to simplify the problem. In the old mixture models, we have <span class="math inline">\(p(x_i | z_i; \theta) = p(x_i; \eta_{z_i})\)</span>, which we can generalise to <span class="math inline">\(p(x_i; f(\theta, z_i))\)</span> for some function <span class="math inline">\(f\)</span>. Using Beyes' theorem we can also write down <span class="math inline">\(p(z_i | x_i; \theta) = q(z_i; g(\theta, x_i))\)</span> for some function <span class="math inline">\(g\)</span>. So the problem becomes</p>
<p><span class="math display">\[\max_{\theta} \sum_i L(p(x_i; f(\theta, z_i)) p(z_i), q(z_i; g(\theta, x_i)))\]</span></p>
<p>In some cases <span class="math inline">\(g\)</span> can be hard to write down or compute. AEVB addresses this problem by replacing <span class="math inline">\(g(\theta, x_i)\)</span> with a neural network <span class="math inline">\(g_\phi(x_i)\)</span> with input <span class="math inline">\(x_i\)</span> and some separate parameters <span class="math inline">\(\phi\)</span>. It also replaces <span class="math inline">\(f(\theta, z_i)\)</span> with a neural network <span class="math inline">\(f_\theta(z_i)\)</span> with input <span class="math inline">\(z_i\)</span> and parameters <span class="math inline">\(\theta\)</span>. And now the problem becomes</p>
<p><span class="math display">\[\max_{\theta, \phi} \sum_i L(p(x_i; f_\theta(z_i)) p(z_i), q(z_i; g_\phi(x_i))).\]</span></p>
<p>The objective function can be written as</p>
<p><span class="math display">\[\sum_i \mathbb E_{q(z_i; g_\phi(x_i))} \log p(x_i; f_\theta(z_i)) - D(q(z_i; g_\phi(x_i)) || p(z_i)).\]</span></p>
<p>The first term is called the negative reconstruction error, like the <span class="math inline">\(- \|decoder(encoder(x)) - x\|\)</span> in autoencoders, which is where the "autoencoder" in the name comes from.</p>
<p>The second term is a regularisation term that penalises the posterior <span class="math inline">\(q(z_i)\)</span> that is very different from the prior <span class="math inline">\(p(z_i)\)</span>. We assume this term can be computed analytically.</p>
<p>So only the first term requires computing.</p>
<p>We can approximate the sum over <span class="math inline">\(i\)</span> in a similar fashion as SVI: pick <span class="math inline">\(j\)</span> uniformly randomly from <span class="math inline">\(\{1 ... m\}\)</span> and treat the whole dataset as <span class="math inline">\(m\)</span> replicates of <span class="math inline">\(x_j\)</span>, and approximate the expectation using Monte-Carlo:</p>
<p><span class="math display">\[U(x_i, \theta, \phi) := \sum_i \mathbb E_{q(z_i; g_\phi(x_i))} \log p(x_i; f_\theta(z_i)) \approx m \mathbb E_{q(z_j; g_\phi(x_j))} \log p(x_j; f_\theta(z_j)) \approx {m \over L} \sum_{\ell = 1}^L \log p(x_j; f_\theta(z_{j, \ell})),\]</span></p>
<p>where each <span class="math inline">\(z_{j, \ell}\)</span> is sampled from <span class="math inline">\(q(z_j; g_\phi(x_j))\)</span>.</p>
<p>But then it is not easy to approximate the gradient over <span class="math inline">\(\phi\)</span>. One can use the log trick as in policy gradients, but it has the problem of high variance. In policy gradients this is overcome by using baseline subtractions. In the AEVB paper it is tackled with the reparameterisation trick.</p>
<p>Assume there exists a transformation <span class="math inline">\(T_\phi\)</span> and a random variable <span class="math inline">\(\epsilon\)</span> with distribution independent of <span class="math inline">\(\phi\)</span> or <span class="math inline">\(\theta\)</span>, such that <span class="math inline">\(T_\phi(x_i, \epsilon)\)</span> has distribution <span class="math inline">\(q(z_i; g_\phi(x_i))\)</span>. In this case we can rewrite <span class="math inline">\(U(x, \phi, \theta)\)</span> as</p>
<p><span class="math display">\[\sum_i \mathbb E_{\epsilon \sim p(\epsilon)} \log p(x_i; f_\theta(T_\phi(x_i, \epsilon))),\]</span></p>
<p>This way one can use Monte-Carlo to approximate <span class="math inline">\(\nabla_\phi U(x, \phi, \theta)\)</span>:</p>
<p><span class="math display">\[\nabla_\phi U(x, \phi, \theta) \approx {m \over L} \sum_{\ell = 1 : L} \nabla_\phi \log p(x_j; f_\theta(T_\phi(x_j, \epsilon_\ell))),\]</span></p>
<p>where each <span class="math inline">\(\epsilon_{\ell}\)</span> is sampled from <span class="math inline">\(p(\epsilon)\)</span>. The approximation of <span class="math inline">\(U(x, \phi, \theta)\)</span> itself can be done similarly.</p>
<h3 id="vae">VAE</h3>
<p>As an example of AEVB, the paper introduces variational autoencoder (VAE), with the following instantiations:</p>
<ul>
<li>The prior <span class="math inline">\(p(z_i) = N(0, I)\)</span> is standard normal, thus independent of <span class="math inline">\(\theta\)</span>.</li>
<li>The distribution <span class="math inline">\(p(x_i; \eta)\)</span> is either Gaussian or categorical.</li>
<li>The distribution <span class="math inline">\(q(z_i; \mu, \Sigma)\)</span> is Gaussian with diagonal covariance matrix. So <span class="math inline">\(g_\phi(z_i) = (\mu_\phi(x_i), \text{diag}(\sigma^2_\phi(x_i)_{1 : d}))\)</span>. Thus in the reparameterisation trick <span class="math inline">\(\epsilon \sim N(0, I)\)</span> and <span class="math inline">\(T_\phi(x_i, \epsilon) = \epsilon \odot \sigma_\phi(x_i) + \mu_\phi(x_i)\)</span>, where <span class="math inline">\(\odot\)</span> is elementwise multiplication.</li>
<li>The KL divergence can be easily computed analytically as <span class="math inline">\(- D(q(z_i; g_\phi(x_i)) || p(z_i)) = {d \over 2} + \sum_{j = 1 : d} \log\sigma_\phi(x_i)_j - {1 \over 2} \sum_{j = 1 : d} (\mu_\phi(x_i)_j^2 + \sigma_\phi(x_i)_j^2)\)</span>.</li>
</ul>
<p>With this, one can use backprop to maximise the ELBO.</p>
<h3 id="fully-bayesian-aevb">Fully Bayesian AEVB</h3>
<p>Let us turn to fully Bayesian version of AEVB. Again, we first recall the ELBO of the fully Bayesian mixture models:</p>
<p><span class="math display">\[L(p(x, z, \pi, \eta; \alpha, \beta), q(z, \pi, \eta; r, \phi)) = L(p(x | z, \eta) p(z | \pi) p(\pi; \alpha) p(\eta; \beta), q(z; r) q(\eta; \phi^\eta) q(\pi; \phi^\pi)).\]</span></p>
<p>We write <span class="math inline">\(\theta = (\pi, \eta)\)</span>, rewrite <span class="math inline">\(\alpha := (\alpha, \beta)\)</span>, <span class="math inline">\(\phi := r\)</span>, and <span class="math inline">\(\gamma := (\phi^\eta, \phi^\pi)\)</span>. Furthermore, as in the half-Bayesian version we assume <span class="math inline">\(p(z | \theta) = p(z)\)</span>, i.e. <span class="math inline">\(z\)</span> does not depend on <span class="math inline">\(\theta\)</span>. Similarly we also assume <span class="math inline">\(p(\theta; \alpha) = p(\theta)\)</span>. Now we have</p>
<p><span class="math display">\[L(p(x, z, \theta; \alpha), q(z, \theta; \phi, \gamma)) = L(p(x | z, \theta) p(z) p(\theta), q(z; \phi) q(\theta; \gamma)).\]</span></p>
<p>And the objective is to maximise it over <span class="math inline">\(\phi\)</span> and <span class="math inline">\(\gamma\)</span>. We no longer maximise over <span class="math inline">\(\theta\)</span>, because it is now a random variable, like <span class="math inline">\(z\)</span>. Now let us transform it to a neural network model, as in the half-Bayesian case:</p>
<p><span class="math display">\[L\left(\left(\prod_{i = 1 : m} p(x_i; f_\theta(z_i))\right) \left(\prod_{i = 1 : m} p(z_i) \right) p(\theta), \left(\prod_{i = 1 : m} q(z_i; g_\phi(x_i))\right) q(\theta; h_\gamma(x))\right).\]</span></p>
<p>where <span class="math inline">\(f_\theta\)</span>, <span class="math inline">\(g_\phi\)</span> and <span class="math inline">\(h_\gamma\)</span> are neural networks. Again, by separating out KL-divergence terms, the above formula becomes</p>
<p><span class="math display">\[\sum_i \mathbb E_{q(\theta; h_\gamma(x))q(z_i; g_\phi(x_i))} \log p(x_i; f_\theta(z_i)) - \sum_i D(q(z_i; g_\phi(x_i)) || p(z_i)) - D(q(\theta; h_\gamma(x)) || p(\theta)).\]</span></p>
<p>Again, we assume the latter two terms can be computed analytically. Using reparameterisation trick, we write</p>
<p><span class="math display">\[\begin{aligned}
\theta &= R_\gamma(\zeta, x) \\
z_i &= T_\phi(\epsilon, x_i)
\end{aligned}\]</span></p>
<p>for some transformations <span class="math inline">\(R_\gamma\)</span> and <span class="math inline">\(T_\phi\)</span> and random variables <span class="math inline">\(\zeta\)</span> and <span class="math inline">\(\epsilon\)</span> so that the output has the desired distributions.</p>
<p>Then the first term can be written as</p>
<p><span class="math display">\[\mathbb E_{\zeta, \epsilon} \log p(x_i; f_{R_\gamma(\zeta, x)} (T_\phi(\epsilon, x_i))),\]</span></p>
<p>so that the gradients can be computed accordingly.</p>
<p>Again, one may use Monte-Carlo to approximate this expetation.</p>
<h2 id="references">References</h2>
<ul>
<li>Attias, Hagai. "A variational baysian framework for graphical models." In Advances in neural information processing systems, pp. 209-215. 2000.</li>
<li>Bishop, Christopher M. Neural networks for pattern recognition. Springer. 2006.</li>
<li>Blei, David M., and Michael I. Jordan. "Variational Inference for Dirichlet Process Mixtures." Bayesian Analysis 1, no. 1 (March 2006): 121--43. <a href="https://doi.org/10.1214/06-BA104" class="uri">https://doi.org/10.1214/06-BA104</a>.</li>
<li>Blei, David M., Andrew Y. Ng, and Michael I. Jordan. "Latent Dirichlet Allocation." Journal of Machine Learning Research 3, no. Jan (2003): 993--1022.</li>
<li>Hofmann, Thomas. "Latent Semantic Models for Collaborative Filtering." ACM Transactions on Information Systems 22, no. 1 (January 1, 2004): 89--115. <a href="https://doi.org/10.1145/963770.963774" class="uri">https://doi.org/10.1145/963770.963774</a>.</li>
<li>Hofmann, Thomas. "Learning the similarity of documents: An information-geometric approach to document retrieval and categorization." In Advances in neural information processing systems, pp. 914-920. 2000.</li>
<li>Hoffman, Matt, David M. Blei, Chong Wang, and John Paisley. "Stochastic Variational Inference." ArXiv:1206.7051 [Cs, Stat], June 29, 2012. <a href="http://arxiv.org/abs/1206.7051" class="uri">http://arxiv.org/abs/1206.7051</a>.</li>
<li>Kingma, Diederik P., and Max Welling. "Auto-Encoding Variational Bayes." ArXiv:1312.6114 [Cs, Stat], December 20, 2013. <a href="http://arxiv.org/abs/1312.6114" class="uri">http://arxiv.org/abs/1312.6114</a>.</li>
<li>Kurihara, Kenichi, Max Welling, and Nikos Vlassis. "Accelerated variational Dirichlet process mixtures." In Advances in neural information processing systems, pp. 761-768. 2007.</li>
<li>Sudderth, Erik Blaine. "Graphical models for visual object recognition and tracking." PhD diss., Massachusetts Institute of Technology, 2006.</li>
</ul>
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Discriminant analysisposts/2019-01-03-discriminant-analysis.html2019-01-03T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<ul>
<li><a href="#theory">Theory</a><ul>
<li><a href="#qda">QDA</a></li>
<li><a href="#vanilla-lda">Vanilla LDA</a></li>
<li><a href="#nearest-neighbour-classifier">Nearest neighbour classifier</a></li>
<li><a href="#dimensionality-reduction">Dimensionality reduction</a></li>
<li><a href="#fisher-discriminant-analysis">Fisher discriminant analysis</a></li>
<li><a href="#linear-model">Linear model</a></li>
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<li><a href="#implementation">Implementation</a><ul>
<li><a href="#fun-facts-about-lda">Fun facts about LDA</a></li>
</ul></li>
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<p>In this post I talk about the theory and implementation of linear and quadratic discriminant analysis, classical methods in statistical learning.</p>
<p><strong>Acknowledgement</strong>. Various sources were of great help to my understanding of the subject, including Chapter 4 of <a href="https://web.stanford.edu/~hastie/ElemStatLearn/">The Elements of Statistical Learning</a>, <a href="http://cs229.stanford.edu/notes/cs229-notes2.pdf">Stanford CS229 Lecture notes</a>, and <a href="https://github.com/scikit-learn/scikit-learn/blob/7389dba/sklearn/discriminant_analysis.py">the scikit-learn code</a>. Research was done while working at KTH mathematics department.</p>
<p><em>If you are reading on a mobile device, you may need to "request desktop site" for the equations to be properly displayed. This post is licensed under CC BY-SA and GNU FDL.</em></p>
<h2 id="theory">Theory</h2>
<p>Quadratic discriminant analysis (QDA) is a classical classification algorithm. It assumes that the data is generated by Gaussian distributions, where each class has its own mean and covariance.</p>
<p><span class="math display">\[(x | y = i) \sim N(\mu_i, \Sigma_i).\]</span></p>
<p>It also assumes a categorical class prior:</p>
<p><span class="math display">\[\mathbb P(y = i) = \pi_i\]</span></p>
<p>The log-likelihood is thus</p>
<p><span class="math display">\[\begin{aligned}
\log \mathbb P(y = i | x) &= \log \mathbb P(x | y = i) \log \mathbb P(y = i) + C\\
&= - {1 \over 2} \log \det \Sigma_i - {1 \over 2} (x - \mu_i)^T \Sigma_i^{-1} (x - \mu_i) + \log \pi_i + C', \qquad (0)
\end{aligned}\]</span></p>
<p>where <span class="math inline">\(C\)</span> and <span class="math inline">\(C'\)</span> are constants.</p>
<p>Thus the prediction is done by taking argmax of the above formula.</p>
<p>In training, let <span class="math inline">\(X\)</span>, <span class="math inline">\(y\)</span> be the input data, where <span class="math inline">\(X\)</span> is of shape <span class="math inline">\(m \times n\)</span>, and <span class="math inline">\(y\)</span> of shape <span class="math inline">\(m\)</span>. We adopt the convention that each row of <span class="math inline">\(X\)</span> is a sample <span class="math inline">\(x^{(i)T}\)</span>. So there are <span class="math inline">\(m\)</span> samples and <span class="math inline">\(n\)</span> features. Denote by <span class="math inline">\(m_i = \#\{j: y_j = i\}\)</span> be the number of samples in class <span class="math inline">\(i\)</span>. Let <span class="math inline">\(n_c\)</span> be the number of classes.</p>
<p>We estimate <span class="math inline">\(\mu_i\)</span> by the sample means, and <span class="math inline">\(\pi_i\)</span> by the frequencies:</p>
<p><span class="math display">\[\begin{aligned}
\mu_i &:= {1 \over m_i} \sum_{j: y_j = i} x^{(j)}, \\
\pi_i &:= \mathbb P(y = i) = {m_i \over m}.
\end{aligned}\]</span></p>
<p>Linear discriminant analysis (LDA) is a specialisation of QDA: it assumes all classes share the same covariance, i.e. <span class="math inline">\(\Sigma_i = \Sigma\)</span> for all <span class="math inline">\(i\)</span>.</p>
<p>Guassian Naive Bayes is a different specialisation of QDA: it assumes that all <span class="math inline">\(\Sigma_i\)</span> are diagonal, since all the features are assumed to be independent.</p>
<h3 id="qda">QDA</h3>
<p>We look at QDA.</p>
<p>We estimate <span class="math inline">\(\Sigma_i\)</span> by the variance mean:</p>
<p><span class="math display">\[\begin{aligned}
\Sigma_i &= {1 \over m_i - 1} \sum_{j: y_j = i} \hat x^{(j)} \hat x^{(j)T}.
\end{aligned}\]</span></p>
<p>where <span class="math inline">\(\hat x^{(j)} = x^{(j)} - \mu_{y_j}\)</span> are the centred <span class="math inline">\(x^{(j)}\)</span>. Plugging this into (0) we are done.</p>
<p>There are two problems that can break the algorithm. First, if one of the <span class="math inline">\(m_i\)</span> is <span class="math inline">\(1\)</span>, then <span class="math inline">\(\Sigma_i\)</span> is ill-defined. Second, one of <span class="math inline">\(\Sigma_i\)</span>'s might be singular.</p>
<p>In either case, there is no way around it, and the implementation should throw an exception.</p>
<p>This won't be a problem of the LDA, though, unless there is only one sample for each class.</p>
<h3 id="vanilla-lda">Vanilla LDA</h3>
<p>Now let us look at LDA.</p>
<p>Since all classes share the same covariance, we estimate <span class="math inline">\(\Sigma\)</span> using sample variance</p>
<p><span class="math display">\[\begin{aligned}
\Sigma &= {1 \over m - n_c} \sum_j \hat x^{(j)} \hat x^{(j)T},
\end{aligned}\]</span></p>
<p>where <span class="math inline">\(\hat x^{(j)} = x^{(j)} - \mu_{y_j}\)</span> and <span class="math inline">\({1 \over m - n_c}\)</span> comes from <a href="https://en.wikipedia.org/wiki/Bessel%27s_correction">Bessel's Correction</a>.</p>
<p>Let us write down the decision function (0). We can remove the first term on the right hand side, since all <span class="math inline">\(\Sigma_i\)</span> are the same, and we only care about argmax of that equation. Thus it becomes</p>
<p><span class="math display">\[- {1 \over 2} (x - \mu_i)^T \Sigma^{-1} (x - \mu_i) + \log\pi_i. \qquad (1)\]</span></p>
<p>Notice that we just walked around the problem of figuring out <span class="math inline">\(\log \det \Sigma\)</span> when <span class="math inline">\(\Sigma\)</span> is singular.</p>
<p>But how about <span class="math inline">\(\Sigma^{-1}\)</span>?</p>
<p>We sidestep this problem by using the pseudoinverse of <span class="math inline">\(\Sigma\)</span> instead. This can be seen as applying a linear transformation to <span class="math inline">\(X\)</span> to turn its covariance matrix to identity. And thus the model becomes a sort of a nearest neighbour classifier.</p>
<h3 id="nearest-neighbour-classifier">Nearest neighbour classifier</h3>
<p>More specifically, we want to transform the first term of (0) to a norm to get a classifier based on nearest neighbour modulo <span class="math inline">\(\log \pi_i\)</span>:</p>
<p><span class="math display">\[- {1 \over 2} \|A(x - \mu_i)\|^2 + \log\pi_i\]</span></p>
<p>To compute <span class="math inline">\(A\)</span>, we denote</p>
<p><span class="math display">\[X_c = X - M,\]</span></p>
<p>where the <span class="math inline">\(i\)</span>th row of <span class="math inline">\(M\)</span> is <span class="math inline">\(\mu_{y_i}^T\)</span>, the mean of the class <span class="math inline">\(x_i\)</span> belongs to, so that <span class="math inline">\(\Sigma = {1 \over m - n_c} X_c^T X_c\)</span>.</p>
<p>Let</p>
<p><span class="math display">\[{1 \over \sqrt{m - n_c}} X_c = U_x \Sigma_x V_x^T\]</span></p>
<p>be the SVD of <span class="math inline">\({1 \over \sqrt{m - n_c}}X_c\)</span>. Let <span class="math inline">\(D_x = \text{diag} (s_1, ..., s_r)\)</span> be the diagonal matrix with all the nonzero singular values, and rewrite <span class="math inline">\(V_x\)</span> as an <span class="math inline">\(n \times r\)</span> matrix consisting of the first <span class="math inline">\(r\)</span> columns of <span class="math inline">\(V_x\)</span>.</p>
<p>Then with an abuse of notation, the pseudoinverse of <span class="math inline">\(\Sigma\)</span> is</p>
<p><span class="math display">\[\Sigma^{-1} = V_x D_x^{-2} V_x^T.\]</span></p>
<p>So we just need to make <span class="math inline">\(A = D_x^{-1} V_x^T\)</span>. When it comes to prediction, just transform <span class="math inline">\(x\)</span> with <span class="math inline">\(A\)</span>, and find the nearest centroid <span class="math inline">\(A \mu_i\)</span> (again, modulo <span class="math inline">\(\log \pi_i\)</span>) and label the input with <span class="math inline">\(i\)</span>.</p>
<h3 id="dimensionality-reduction">Dimensionality reduction</h3>
<p>We can further simplify the prediction by dimensionality reduction. Assume <span class="math inline">\(n_c \le n\)</span>. Then the centroid spans an affine space of dimension <span class="math inline">\(p\)</span> which is at most <span class="math inline">\(n_c - 1\)</span>. So what we can do is to project both the transformed sample <span class="math inline">\(Ax\)</span> and centroids <span class="math inline">\(A\mu_i\)</span> to the linear subspace parallel to the affine space, and do the nearest neighbour classification there.</p>
<p>So we can perform SVD on the matrix <span class="math inline">\((M - \bar x) V_x D_x^{-1}\)</span> where <span class="math inline">\(\bar x\)</span>, a row vector, is the sample mean of all data i.e. average of rows of <span class="math inline">\(X\)</span>:</p>
<p><span class="math display">\[(M - \bar x) V_x D_x^{-1} = U_m \Sigma_m V_m^T.\]</span></p>
<p>Again, we let <span class="math inline">\(V_m\)</span> be the <span class="math inline">\(r \times p\)</span> matrix by keeping the first <span class="math inline">\(p\)</span> columns of <span class="math inline">\(V_m\)</span>.</p>
<p>The projection operator is thus <span class="math inline">\(V_m\)</span>. And so the final transformation is <span class="math inline">\(V_m^T D_x^{-1} V_x^T\)</span>.</p>
<p>There is no reason to stop here, and we can set <span class="math inline">\(p\)</span> even smaller, which will result in a lossy compression / regularisation equivalent to doing <a href="https://en.wikipedia.org/wiki/Principal_component_analysis">principle component analysis</a> on <span class="math inline">\((M - \bar x) V_x D_x^{-1}\)</span>.</p>
<p>Note that as of 2019-01-04, in the <a href="https://github.com/scikit-learn/scikit-learn/blob/master/sklearn/discriminant_analysis.py">scikit-learn implementation of LDA</a>, the prediction is done without any lossy compression, even if the parameter <code>n_components</code> is set to be smaller than dimension of the affine space spanned by the centroids. In other words, the prediction does not change regardless of <code>n_components</code>.</p>
<h3 id="fisher-discriminant-analysis">Fisher discriminant analysis</h3>
<p>The Fisher discriminant analysis involves finding an <span class="math inline">\(n\)</span>-dimensional vector <span class="math inline">\(a\)</span> that maximises between-class covariance with respect to within-class covariance:</p>
<p><span class="math display">\[{a^T M_c^T M_c a \over a^T X_c^T X_c a},\]</span></p>
<p>where <span class="math inline">\(M_c = M - \bar x\)</span> is the centred sample mean matrix.</p>
<p>As it turns out, this is (almost) equivalent to the derivation above, modulo a constant. In particular, <span class="math inline">\(a = c V_m^T D_x^{-1} V_x^T\)</span> where <span class="math inline">\(p = 1\)</span> for arbitrary constant <span class="math inline">\(c\)</span>.</p>
<p>To see this, we can first multiply the denominator with a constant <span class="math inline">\({1 \over m - n_c}\)</span> so that the matrix in the denominator becomes the covariance estimate <span class="math inline">\(\Sigma\)</span>.</p>
<p>We decompose <span class="math inline">\(a\)</span>: <span class="math inline">\(a = V_x D_x^{-1} b + \tilde V_x \tilde b\)</span>, where <span class="math inline">\(\tilde V_x\)</span> consists of column vectors orthogonal to the column space of <span class="math inline">\(V_x\)</span>.</p>
<p>We ignore the second term in the decomposition. In other words, we only consider <span class="math inline">\(a\)</span> in the column space of <span class="math inline">\(V_x\)</span>.</p>
<p>Then the problem is to find an <span class="math inline">\(r\)</span>-dimensional vector <span class="math inline">\(b\)</span> to maximise</p>
<p><span class="math display">\[{b^T (M_c V_x D_x^{-1})^T (M_c V_x D_x^{-1}) b \over b^T b}.\]</span></p>
<p>This is the problem of principle component analysis, and so <span class="math inline">\(b\)</span> is first column of <span class="math inline">\(V_m\)</span>.</p>
<p>Therefore, the solution to Fisher discriminant analysis is <span class="math inline">\(a = c V_x D_x^{-1} V_m\)</span> with <span class="math inline">\(p = 1\)</span>.</p>
<h3 id="linear-model">Linear model</h3>
<p>The model is called linear discriminant analysis because it is a linear model. To see this, let <span class="math inline">\(B = V_m^T D_x^{-1} V_x^T\)</span> be the matrix of transformation. Now we are comparing</p>
<p><span class="math display">\[- {1 \over 2} \| B x - B \mu_k\|^2 + \log \pi_k\]</span></p>
<p>across all <span class="math inline">\(k\)</span>s. Expanding the norm and removing the common term <span class="math inline">\(\|B x\|^2\)</span>, we see a linear form:</p>
<p><span class="math display">\[\mu_k^T B^T B x - {1 \over 2} \|B \mu_k\|^2 + \log\pi_k\]</span></p>
<p>So the prediction for <span class="math inline">\(X_{\text{new}}\)</span> is</p>
<p><span class="math display">\[\text{argmax}_{\text{axis}=0} \left(K B^T B X_{\text{new}}^T - {1 \over 2} \|K B^T\|_{\text{axis}=1}^2 + \log \pi\right)\]</span></p>
<p>thus the decision boundaries are linear.</p>
<p>This is how scikit-learn implements LDA, by inheriting from <code>LinearClassifierMixin</code> and redirecting the classification there.</p>
<h2 id="implementation">Implementation</h2>
<p>This is where things get interesting. How do I validate my understanding of the theory? By implementing and testing the algorithm.</p>
<p>I try to implement it as close as possible to the natural language / mathematical descriptions of the model, which means clarity over performance.</p>
<p>How about testing? Numerical experiments are harder to test than combinatorial / discrete algorithms in general because the output is less verifiable by hand. My shortcut solution to this problem is to test against output from the scikit-learn package.</p>
<p>It turned out to be harder than expected, as I had to dig into the code of scikit-learn when the outputs mismatch. Their code is quite well-written though.</p>
<p>The result is <a href="https://github.com/ycpei/machine-learning/tree/master/discriminant-analysis">here</a>.</p>
<h3 id="fun-facts-about-lda">Fun facts about LDA</h3>
<p>One property that can be used to test the LDA implementation is the fact that the scatter matrix <span class="math inline">\(B(X - \bar x)^T (X - \bar X) B^T\)</span> of the transformed centred sample is diagonal.</p>
<p>This can be derived by using another fun fact that the sum of the in-class scatter matrix and the between-class scatter matrix is the sample scatter matrix:</p>
<p><span class="math display">\[X_c^T X_c + M_c^T M_c = (X - \bar x)^T (X - \bar x) = (X_c + M_c)^T (X_c + M_c).\]</span></p>
<p>The verification is not very hard and left as an exercise.</p>
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Shapley, LIME and SHAPposts/2018-12-02-lime-shapley.html2018-12-02T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<ul>
<li><a href="#shapley-values">Shapley values</a></li>
<li><a href="#lime">LIME</a></li>
<li><a href="#shapley-values-and-lime">Shapley values and LIME</a></li>
<li><a href="#shap">SHAP</a></li>
<li><a href="#evaluating-shap">Evaluating SHAP</a></li>
<li><a href="#references">References</a></li>
</ul>
</div>
<p>In this post I explain LIME (Ribeiro et. al. 2016), the Shapley values (Shapley, 1953) and the SHAP values (Strumbelj-Kononenko, 2014; Lundberg-Lee, 2017).</p>
<p><strong>Acknowledgement</strong>. Thanks to Josef Lindman Hörnlund for bringing the LIME and SHAP papers to my attention. The research was done while working at KTH mathematics department.</p>
<p><em>If you are reading on a mobile device, you may need to "request desktop site" for the equations to be properly displayed. This post is licensed under CC BY-SA and GNU FDL.</em></p>
<h2 id="shapley-values">Shapley values</h2>
<p>A coalitional game <span class="math inline">\((v, N)\)</span> of <span class="math inline">\(n\)</span> players involves</p>
<ul>
<li>The set <span class="math inline">\(N = \{1, 2, ..., n\}\)</span> that represents the players.</li>
<li>A function <span class="math inline">\(v: 2^N \to \mathbb R\)</span>, where <span class="math inline">\(v(S)\)</span> is the worth of coalition <span class="math inline">\(S \subset N\)</span>.</li>
</ul>
<p>The Shapley values <span class="math inline">\(\phi_i(v)\)</span> of such a game specify a fair way to distribute the total worth <span class="math inline">\(v(N)\)</span> to the players. It is defined as (in the following, for a set <span class="math inline">\(S \subset N\)</span> we use the convention <span class="math inline">\(s = |S|\)</span> to be the number of elements of set <span class="math inline">\(S\)</span> and the shorthand <span class="math inline">\(S - i := S \setminus \{i\}\)</span> and <span class="math inline">\(S + i := S \cup \{i\}\)</span>)</p>
<p><span class="math display">\[\phi_i(v) = \sum_{S: i \in S} {(n - s)! (s - 1)! \over n!} (v(S) - v(S - i)).\]</span></p>
<p>It is not hard to see that <span class="math inline">\(\phi_i(v)\)</span> can be viewed as an expectation:</p>
<p><span class="math display">\[\phi_i(v) = \mathbb E_{S \sim \nu_i} (v(S) - v(S - i))\]</span></p>
<p>where <span class="math inline">\(\nu_i(S) = n^{-1} {n - 1 \choose s - 1}^{-1} 1_{i \in S}\)</span>, that is, first pick the size <span class="math inline">\(s\)</span> uniformly from <span class="math inline">\(\{1, 2, ..., n\}\)</span>, then pick <span class="math inline">\(S\)</span> uniformly from the subsets of <span class="math inline">\(N\)</span> that has size <span class="math inline">\(s\)</span> and contains <span class="math inline">\(i\)</span>.</p>
<p>The Shapley values satisfy some nice properties which are readily verified, including:</p>
<ul>
<li><strong>Efficiency</strong>. <span class="math inline">\(\sum_i \phi_i(v) = v(N) - v(\emptyset)\)</span>.</li>
<li><strong>Symmetry</strong>. If for some <span class="math inline">\(i, j \in N\)</span>, for all <span class="math inline">\(S \subset N\)</span>, we have <span class="math inline">\(v(S + i) = v(S + j)\)</span>, then <span class="math inline">\(\phi_i(v) = \phi_j(v)\)</span>.</li>
<li><strong>Null player</strong>. If for some <span class="math inline">\(i \in N\)</span>, for all <span class="math inline">\(S \subset N\)</span>, we have <span class="math inline">\(v(S + i) = v(S)\)</span>, then <span class="math inline">\(\phi_i(v) = 0\)</span>.</li>
<li><strong>Linearity</strong>. <span class="math inline">\(\phi_i\)</span> is linear in games. That is <span class="math inline">\(\phi_i(v) + \phi_i(w) = \phi_i(v + w)\)</span>, where <span class="math inline">\(v + w\)</span> is defined by <span class="math inline">\((v + w)(S) := v(S) + w(S)\)</span>.</li>
</ul>
<p>In the literature, an added assumption <span class="math inline">\(v(\emptyset) = 0\)</span> is often given, in which case the Efficiency property is defined as <span class="math inline">\(\sum_i \phi_i(v) = v(N)\)</span>. Here I discard this assumption to avoid minor inconsistencies across different sources. For example, in the LIME paper, the local model is defined without an intercept, even though the underlying <span class="math inline">\(v(\emptyset)\)</span> may not be <span class="math inline">\(0\)</span>. In the SHAP paper, an intercept <span class="math inline">\(\phi_0 = v(\emptyset)\)</span> is added which fixes this problem when making connections to the Shapley values.</p>
<p>Conversely, according to Strumbelj-Kononenko (2010), it was shown in Shapley's original paper (Shapley, 1953) that these four properties together with <span class="math inline">\(v(\emptyset) = 0\)</span> defines the Shapley values.</p>
<h2 id="lime">LIME</h2>
<p>LIME (Ribeiro et. al. 2016) is a model that offers a way to explain feature contributions of supervised learning models locally.</p>
<p>Let <span class="math inline">\(f: X_1 \times X_2 \times ... \times X_n \to \mathbb R\)</span> be a function. We can think of <span class="math inline">\(f\)</span> as a model, where <span class="math inline">\(X_j\)</span> is the space of <span class="math inline">\(j\)</span>th feature. For example, in a language model, <span class="math inline">\(X_j\)</span> may correspond to the count of the <span class="math inline">\(j\)</span>th word in the vocabulary, i.e. the bag-of-words model.</p>
<p>The output may be something like housing price, or log-probability of something.</p>
<p>LIME tries to assign a value to each feature <em>locally</em>. By locally, we mean that given a specific sample <span class="math inline">\(x \in X := \prod_{i = 1}^n X_i\)</span>, we want to fit a model around it.</p>
<p>More specifically, let <span class="math inline">\(h_x: 2^N \to X\)</span> be a function defined by</p>
<p><span class="math display">\[(h_x(S))_i =
\begin{cases}
x_i, & \text{if }i \in S; \\
0, & \text{otherwise.}
\end{cases}\]</span></p>
<p>That is, <span class="math inline">\(h_x(S)\)</span> masks the features that are not in <span class="math inline">\(S\)</span>, or in other words, we are perturbing the sample <span class="math inline">\(x\)</span>. Specifically, <span class="math inline">\(h_x(N) = x\)</span>. Alternatively, the <span class="math inline">\(0\)</span> in the "otherwise" case can be replaced by some kind of default value (see the section titled SHAP in this post).</p>
<p>For a set <span class="math inline">\(S \subset N\)</span>, let us denote <span class="math inline">\(1_S \in \{0, 1\}^n\)</span> to be an <span class="math inline">\(n\)</span>-bit where the <span class="math inline">\(k\)</span>th bit is <span class="math inline">\(1\)</span> if and only if <span class="math inline">\(k \in S\)</span>.</p>
<p>Basically, LIME samples <span class="math inline">\(S_1, S_2, ..., S_m \subset N\)</span> to obtain a set of perturbed samples <span class="math inline">\(x_i = h_x(S_i)\)</span> in the <span class="math inline">\(X\)</span> space, and then fits a linear model <span class="math inline">\(g\)</span> using <span class="math inline">\(1_{S_i}\)</span> as the input samples and <span class="math inline">\(f(h_x(S_i))\)</span> as the output samples:</p>
<p><strong>Problem</strong>(LIME). Find <span class="math inline">\(w = (w_1, w_2, ..., w_n)\)</span> that minimises</p>
<p><span class="math display">\[\sum_i (w \cdot 1_{S_i} - f(h_x(S_i)))^2 \pi_x(h_x(S_i))\]</span></p>
<p>where <span class="math inline">\(\pi_x(x')\)</span> is a function that penalises <span class="math inline">\(x'\)</span>s that are far away from <span class="math inline">\(x\)</span>. In the LIME paper the Gaussian kernel was used:</p>
<p><span class="math display">\[\pi_x(x') = \exp\left({- \|x - x'\|^2 \over \sigma^2}\right).\]</span></p>
<p>Then <span class="math inline">\(w_i\)</span> represents the importance of the <span class="math inline">\(i\)</span>th feature.</p>
<p>The LIME model has a more general framework, but the specific model considered in the paper is the one described above, with a Lasso for feature selection.</p>
<p><strong>Remark</strong>. One difference between our account here and the one in the LIME paper is: the dimension of the data space may differ from <span class="math inline">\(n\)</span> (see Section 3.1 of that paper). But in the case of text data, they do use bag-of-words (our <span class="math inline">\(X\)</span>) for an "intermediate" representation. So my understanding is, in their context, there is an "original" data space (let's call it <span class="math inline">\(X'\)</span>). And there is a one-one correspondence between <span class="math inline">\(X'\)</span> and <span class="math inline">\(X\)</span> (let's call it <span class="math inline">\(r: X' \to X\)</span>), so that given a sample <span class="math inline">\(x' \in X'\)</span>, we can compute the output of <span class="math inline">\(S\)</span> in the local model with <span class="math inline">\(f(r^{-1}(h_{r(x')}(S)))\)</span>. As an example, in the example of <span class="math inline">\(X\)</span> being the bag of words, <span class="math inline">\(X'\)</span> may be the embedding vector space, so that <span class="math inline">\(r(x') = A^{-1} x'\)</span>, where <span class="math inline">\(A\)</span> is the word embedding matrix. Therefore, without loss of generality, we assume the input space to be <span class="math inline">\(X\)</span> which is of dimension <span class="math inline">\(n\)</span>.</p>
<h2 id="shapley-values-and-lime">Shapley values and LIME</h2>
<p>The connection between the Shapley values and LIME is noted in Lundberg-Lee (2017), but the underlying connection goes back to 1988 (Charnes et. al.).</p>
<p>To see the connection, we need to modify LIME a bit.</p>
<p>First, we need to make LIME less efficient by considering <em>all</em> the <span class="math inline">\(2^n\)</span> subsets instead of the <span class="math inline">\(m\)</span> samples <span class="math inline">\(S_1, S_2, ..., S_{m}\)</span>.</p>
<p>Then we need to relax the definition of <span class="math inline">\(\pi_x\)</span>. It no longer needs to penalise samples that are far away from <span class="math inline">\(x\)</span>. In fact, we will see later than the choice of <span class="math inline">\(\pi_x(x')\)</span> that yields the Shapley values is high when <span class="math inline">\(x'\)</span> is very close or very far away from <span class="math inline">\(x\)</span>, and low otherwise. We further add the restriction that <span class="math inline">\(\pi_x(h_x(S))\)</span> only depends on the size of <span class="math inline">\(S\)</span>, thus we rewrite it as <span class="math inline">\(q(s)\)</span> instead.</p>
<p>We also denote <span class="math inline">\(v(S) := f(h_x(S))\)</span> and <span class="math inline">\(w(S) = \sum_{i \in S} w_i\)</span>.</p>
<p>Finally, we add the Efficiency property as a constraint: <span class="math inline">\(\sum_{i = 1}^n w_i = f(x) - f(h_x(\emptyset)) = v(N) - v(\emptyset)\)</span>.</p>
<p>Then the problem becomes a weighted linear regression:</p>
<p><strong>Problem</strong>. minimises <span class="math inline">\(\sum_{S \subset N} (w(S) - v(S))^2 q(s)\)</span> over <span class="math inline">\(w\)</span> subject to <span class="math inline">\(w(N) = v(N) - v(\emptyset)\)</span>.</p>
<p><strong>Claim</strong> (Charnes et. al. 1988). The solution to this problem is</p>
<p><span class="math display">\[w_i = {1 \over n} (v(N) - v(\emptyset)) + \left(\sum_{s = 1}^{n - 1} {n - 2 \choose s - 1} q(s)\right)^{-1} \sum_{S \subset N: i \in S} \left({n - s \over n} q(s) v(S) - {s - 1 \over n} q(s - 1) v(S - i)\right). \qquad (-1)\]</span></p>
<p>Specifically, if we choose</p>
<p><span class="math display">\[q(s) = c {n - 2 \choose s - 1}^{-1}\]</span></p>
<p>for any constant <span class="math inline">\(c\)</span>, then <span class="math inline">\(w_i = \phi_i(v)\)</span> are the Shapley values.</p>
<p><strong>Remark</strong>. Don't worry about this specific choice of <span class="math inline">\(q(s)\)</span> when <span class="math inline">\(s = 0\)</span> or <span class="math inline">\(n\)</span>, because <span class="math inline">\(q(0)\)</span> and <span class="math inline">\(q(n)\)</span> do not appear on the right hand side of (-1). Therefore they can be defined to be of any value. A common convention of the binomial coefficients is to set <span class="math inline">\({\ell \choose k} = 0\)</span> if <span class="math inline">\(k < 0\)</span> or <span class="math inline">\(k > \ell\)</span>, in which case <span class="math inline">\(q(0) = q(n) = \infty\)</span>.</p>
<p>In Lundberg-Lee (2017), <span class="math inline">\(c\)</span> is chosen to be <span class="math inline">\(1 / n\)</span>, see Theorem 2 there.</p>
<p>In Charnes et. al. 1988, the <span class="math inline">\(w_i\)</span>s defined in (-1) are called the generalised Shapley values.</p>
<p><strong>Proof</strong>. The Lagrangian is</p>
<p><span class="math display">\[L(w, \lambda) = \sum_{S \subset N} (v(S) - w(S))^2 q(s) - \lambda(w(N) - v(N) + v(\emptyset)).\]</span></p>
<p>and by making <span class="math inline">\(\partial_{w_i} L(w, \lambda) = 0\)</span> we have</p>
<p><span class="math display">\[{1 \over 2} \lambda = \sum_{S \subset N: i \in S} (w(S) - v(S)) q(s). \qquad (0)\]</span></p>
<p>Summing (0) over <span class="math inline">\(i\)</span> and divide by <span class="math inline">\(n\)</span>, we have</p>
<p><span class="math display">\[{1 \over 2} \lambda = {1 \over n} \sum_i \sum_{S: i \in S} (w(S) q(s) - v(S) q(s)). \qquad (1)\]</span></p>
<p>We examine each of the two terms on the right hand side.</p>
<p>Counting the terms involving <span class="math inline">\(w_i\)</span> and <span class="math inline">\(w_j\)</span> for <span class="math inline">\(j \neq i\)</span>, and using <span class="math inline">\(w(N) = v(N) - v(\emptyset)\)</span> we have</p>
<p><span class="math display">\[\begin{aligned}
&\sum_{S \subset N: i \in S} w(S) q(s) \\
&= \sum_{s = 1}^n {n - 1 \choose s - 1} q(s) w_i + \sum_{j \neq i}\sum_{s = 2}^n {n - 2 \choose s - 2} q(s) w_j \\
&= q(1) w_i + \sum_{s = 2}^n q(s) \left({n - 1 \choose s - 1} w_i + \sum_{j \neq i} {n - 2 \choose s - 2} w_j\right) \\
&= q(1) w_i + \sum_{s = 2}^n \left({n - 2 \choose s - 1} w_i + {n - 2 \choose s - 2} (v(N) - v(\emptyset))\right) q(s) \\
&= \sum_{s = 1}^{n - 1} {n - 2 \choose s - 1} q(s) w_i + \sum_{s = 2}^n {n - 2 \choose s - 2} q(s) (v(N) - v(\emptyset)). \qquad (2)
\end{aligned}\]</span></p>
<p>Summing (2) over <span class="math inline">\(i\)</span>, we obtain</p>
<p><span class="math display">\[\begin{aligned}
&\sum_i \sum_{S: i \in S} w(S) q(s)\\
&= \sum_{s = 1}^{n - 1} {n - 2 \choose s - 1} q(s) (v(N) - v(\emptyset)) + \sum_{s = 2}^n n {n - 2 \choose s - 2} q(s) (v(N) - v(\emptyset))\\
&= \sum_{s = 1}^n s{n - 1 \choose s - 1} q(s) (v(N) - v(\emptyset)). \qquad (3)
\end{aligned}\]</span></p>
<p>For the second term in (1), we have</p>
<p><span class="math display">\[\sum_i \sum_{S: i \in S} v(S) q(s) = \sum_{S \subset N} s v(S) q(s). \qquad (4)\]</span></p>
<p>Plugging (3)(4) in (1), we have</p>
<p><span class="math display">\[{1 \over 2} \lambda = {1 \over n} \left(\sum_{S \subset N} s q(s) v(S) - \sum_{s = 1}^n s {n - 1 \choose s - 1} q(s) (v(N) - v(\emptyset))\right). \qquad (5)\]</span></p>
<p>Plugging (5)(2) in (0) and solve for <span class="math inline">\(w_i\)</span>, we have</p>
<p><span class="math display">\[w_i = {1 \over n} (v(N) - v(\emptyset)) + \left(\sum_{s = 1}^{n - 1} {n - 2 \choose s - 1} q(s) \right)^{-1} \left( \sum_{S: i \in S} q(s) v(S) - {1 \over n} \sum_{S \subset N} s q(s) v(S) \right). \qquad (6)\]</span></p>
<p>By splitting all subsets of <span class="math inline">\(N\)</span> into ones that contain <span class="math inline">\(i\)</span> and ones that do not and pair them up, we have</p>
<p><span class="math display">\[\sum_{S \subset N} s q(s) v(S) = \sum_{S: i \in S} (s q(s) v(S) + (s - 1) q(s - 1) v(S - i)).\]</span></p>
<p>Plugging this back into (6) we get the desired result. <span class="math inline">\(\square\)</span></p>
<h2 id="shap">SHAP</h2>
<p>The paper that coined the term "SHAP values" (Lundberg-Lee 2017) is not clear in its definition of the "SHAP values" and its relation to LIME, so the following is my interpretation of their interpretation model, which coincide with a model studied in Strumbelj-Kononenko 2014.</p>
<p>Recall that we want to calculate feature contributions to a model <span class="math inline">\(f\)</span> at a sample <span class="math inline">\(x\)</span>.</p>
<p>Let <span class="math inline">\(\mu\)</span> be a probability density function over the input space <span class="math inline">\(X = X_1 \times ... \times X_n\)</span>. A natural choice would be the density that generates the data, or one that approximates such density (e.g. empirical distribution).</p>
<p>The feature contribution (SHAP value) is thus defined as the Shapley value <span class="math inline">\(\phi_i(v)\)</span>, where</p>
<p><span class="math display">\[v(S) = \mathbb E_{z \sim \mu} (f(z) | z_S = x_S). \qquad (7)\]</span></p>
<p>So it is a conditional expectation where <span class="math inline">\(z_i\)</span> is clamped for <span class="math inline">\(i \in S\)</span>. In fact, the definition of feature contributions in this form predates Lundberg-Lee 2017. For example, it can be found in Strumbelj-Kononenko 2014.</p>
<p>One simplification is to assume the <span class="math inline">\(n\)</span> features are independent, thus <span class="math inline">\(\mu = \mu_1 \times \mu_2 \times ... \times \mu_n\)</span>. In this case, (7) becomes</p>
<p><span class="math display">\[v(S) = \mathbb E_{z_{N \setminus S} \sim \mu_{N \setminus S}} f(x_S, z_{N \setminus S}) \qquad (8)\]</span></p>
<p>For example, Strumbelj-Kononenko (2010) considers this scenario where <span class="math inline">\(\mu\)</span> is the uniform distribution over <span class="math inline">\(X\)</span>, see Definition 4 there.</p>
<p>A further simplification is model linearity, which means <span class="math inline">\(f\)</span> is linear. In this case, (8) becomes</p>
<p><span class="math display">\[v(S) = f(x_S, \mathbb E_{\mu_{N \setminus S}} z_{N \setminus S}). \qquad (9)\]</span></p>
<p>It is worth noting that to make the modified LIME model considered in the previous section fall under the linear SHAP framework (9), we need to make two further specialisations, the first is rather cosmetic: we need to change the definition of <span class="math inline">\(h_x(S)\)</span> to</p>
<p><span class="math display">\[(h_x(S))_i =
\begin{cases}
x_i, & \text{if }i \in S; \\
\mathbb E_{\mu_i} z_i, & \text{otherwise.}
\end{cases}\]</span></p>
<p>But we also need to boldly assume the original <span class="math inline">\(f\)</span> to be linear, which in my view, defeats the purpose of interpretability, because linear models are interpretable by themselves.</p>
<p>One may argue that perhaps we do not need linearity to define <span class="math inline">\(v(S)\)</span> as in (9). If we do so, however, then (9) loses mathematical meaning. A bigger question is: how effective is SHAP? An even bigger question: in general, how do we evaluate models of interpretation?</p>
<h2 id="evaluating-shap">Evaluating SHAP</h2>
<p>The quest of the SHAP paper can be decoupled into two independent components: showing the niceties of Shapley values and choosing the coalitional game <span class="math inline">\(v\)</span>.</p>
<p>The SHAP paper argues that Shapley values <span class="math inline">\(\phi_i(v)\)</span> are a good measurement because they are the only values satisfying the some nice properties including the Efficiency property mentioned at the beginning of the post, invariance under permutation and monotonicity, see the paragraph below Theorem 1 there, which refers to Theorem 2 of Young (1985).</p>
<p>Indeed, both efficiency (the "additive feature attribution methods" in the paper) and monotonicity are meaningful when considering <span class="math inline">\(\phi_i(v)\)</span> as the feature contribution of the <span class="math inline">\(i\)</span>th feature.</p>
<p>The question is thus reduced to the second component: what constitutes a nice choice of <span class="math inline">\(v\)</span>?</p>
<p>The SHAP paper answers this question with 3 options with increasing simplification: (7)(8)(9) in the previous section of this post (corresponding to (9)(11)(12) in the paper). They are intuitive, but it will be interesting to see more concrete (or even mathematical) justifications of such choices.</p>
<h2 id="references">References</h2>
<ul>
<li>Charnes, A., B. Golany, M. Keane, and J. Rousseau. "Extremal Principle Solutions of Games in Characteristic Function Form: Core, Chebychev and Shapley Value Generalizations." In Econometrics of Planning and Efficiency, edited by Jati K. Sengupta and Gopal K. Kadekodi, 123--33. Dordrecht: Springer Netherlands, 1988. <a href="https://doi.org/10.1007/978-94-009-3677-5_7" class="uri">https://doi.org/10.1007/978-94-009-3677-5_7</a>.</li>
<li>Lundberg, Scott, and Su-In Lee. "A Unified Approach to Interpreting Model Predictions." ArXiv:1705.07874 [Cs, Stat], May 22, 2017. <a href="http://arxiv.org/abs/1705.07874" class="uri">http://arxiv.org/abs/1705.07874</a>.</li>
<li>Ribeiro, Marco Tulio, Sameer Singh, and Carlos Guestrin. "'Why Should I Trust You?': Explaining the Predictions of Any Classifier." ArXiv:1602.04938 [Cs, Stat], February 16, 2016. <a href="http://arxiv.org/abs/1602.04938" class="uri">http://arxiv.org/abs/1602.04938</a>.</li>
<li>Shapley, L. S. "17. A Value for n-Person Games." In Contributions to the Theory of Games (AM-28), Volume II, Vol. 2. Princeton: Princeton University Press, 1953. <a href="https://doi.org/10.1515/9781400881970-018" class="uri">https://doi.org/10.1515/9781400881970-018</a>.</li>
<li>Strumbelj, Erik, and Igor Kononenko. "An Efficient Explanation of Individual Classifications Using Game Theory." J. Mach. Learn. Res. 11 (March 2010): 1--18.</li>
<li>Strumbelj, Erik, and Igor Kononenko. “Explaining Prediction Models and Individual Predictions with Feature Contributions.” Knowledge and Information Systems 41, no. 3 (December 2014): 647–65. <a href="https://doi.org/10.1007/s10115-013-0679-x" class="uri">https://doi.org/10.1007/s10115-013-0679-x</a>.</li>
<li>Young, H. P. “Monotonic Solutions of Cooperative Games.” International Journal of Game Theory 14, no. 2 (June 1, 1985): 65–72. <a href="https://doi.org/10.1007/BF01769885" class="uri">https://doi.org/10.1007/BF01769885</a>.</li>
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Automatic differentiationposts/2018-06-03-automatic_differentiation.html2018-06-03T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>This post serves as a note and explainer of autodiff. It is licensed under <a href="https://www.gnu.org/licenses/fdl.html">GNU FDL</a>.</p>
<p>For my learning I benefited a lot from <a href="http://www.cs.toronto.edu/%7Ergrosse/courses/csc321_2018/slides/lec10.pdf">Toronto CSC321 slides</a> and the <a href="https://github.com/mattjj/autodidact/">autodidact</a> project which is a pedagogical implementation of <a href="https://github.com/hips/autograd">Autograd</a>. That said, any mistakes in this note are mine (especially since some of the knowledge is obtained from interpreting slides!), and if you do spot any I would be grateful if you can let me know.</p>
<p>Automatic differentiation (AD) is a way to compute derivatives. It does so by traversing through a computational graph using the chain rule.</p>
<p>There are the forward mode AD and reverse mode AD, which are kind of symmetric to each other and understanding one of them results in little to no difficulty in understanding the other.</p>
<p>In the language of neural networks, one can say that the forward mode AD is used when one wants to compute the derivatives of functions at all layers with respect to input layer weights, whereas the reverse mode AD is used to compute the derivatives of output functions with respect to weights at all layers. Therefore reverse mode AD (rmAD) is the one to use for gradient descent, which is the one we focus in this post.</p>
<p>Basically rmAD requires the computation to be sufficiently decomposed, so that in the computational graph, each node as a function of its parent nodes is an elementary function that the AD engine has knowledge about.</p>
<p>For example, the Sigmoid activation <span class="math inline">\(a' = \sigma(w a + b)\)</span> is quite simple, but it should be decomposed to simpler computations:</p>
<ul>
<li><span class="math inline">\(a' = 1 / t_1\)</span></li>
<li><span class="math inline">\(t_1 = 1 + t_2\)</span></li>
<li><span class="math inline">\(t_2 = \exp(t_3)\)</span></li>
<li><span class="math inline">\(t_3 = - t_4\)</span></li>
<li><span class="math inline">\(t_4 = t_5 + b\)</span></li>
<li><span class="math inline">\(t_5 = w a\)</span></li>
</ul>
<p>Thus the function <span class="math inline">\(a'(a)\)</span> is decomposed to elementary operations like addition, subtraction, multiplication, reciprocitation, exponentiation, logarithm etc. And the rmAD engine stores the Jacobian of these elementary operations.</p>
<p>Since in neural networks we want to find derivatives of a single loss function <span class="math inline">\(L(x; \theta)\)</span>, we can omit <span class="math inline">\(L\)</span> when writing derivatives and denote, say <span class="math inline">\(\bar \theta_k := \partial_{\theta_k} L\)</span>.</p>
<p>In implementations of rmAD, one can represent the Jacobian as a transformation <span class="math inline">\(j: (x \to y) \to (y, \bar y, x) \to \bar x\)</span>. <span class="math inline">\(j\)</span> is called the <em>Vector Jacobian Product</em> (VJP). For example, <span class="math inline">\(j(\exp)(y, \bar y, x) = y \bar y\)</span> since given <span class="math inline">\(y = \exp(x)\)</span>,</p>
<p><span class="math inline">\(\partial_x L = \partial_x y \cdot \partial_y L = \partial_x \exp(x) \cdot \partial_y L = y \bar y\)</span></p>
<p>as another example, <span class="math inline">\(j(+)(y, \bar y, x_1, x_2) = (\bar y, \bar y)\)</span> since given <span class="math inline">\(y = x_1 + x_2\)</span>, <span class="math inline">\(\bar{x_1} = \bar{x_2} = \bar y\)</span>.</p>
<p>Similarly,</p>
<ol style="list-style-type: decimal">
<li><span class="math inline">\(j(/)(y, \bar y, x_1, x_2) = (\bar y / x_2, - \bar y x_1 / x_2^2)\)</span></li>
<li><span class="math inline">\(j(\log)(y, \bar y, x) = \bar y / x\)</span></li>
<li><span class="math inline">\(j((A, \beta) \mapsto A \beta)(y, \bar y, A, \beta) = (\bar y \otimes \beta, A^T \bar y)\)</span>.</li>
<li>etc...</li>
</ol>
<p>In the third one, the function is a matrix <span class="math inline">\(A\)</span> multiplied on the right by a column vector <span class="math inline">\(\beta\)</span>, and <span class="math inline">\(\bar y \otimes \beta\)</span> is the tensor product which is a fancy way of writing <span class="math inline">\(\bar y \beta^T\)</span>. See <a href="https://github.com/mattjj/autodidact/blob/master/autograd/numpy/numpy_vjps.py">numpy_vjps.py</a> for the implementation in autodidact.</p>
<p>So, given a node say <span class="math inline">\(y = y(x_1, x_2, ..., x_n)\)</span>, and given the value of <span class="math inline">\(y\)</span>, <span class="math inline">\(x_{1 : n}\)</span> and <span class="math inline">\(\bar y\)</span>, rmAD computes the values of <span class="math inline">\(\bar x_{1 : n}\)</span> by using the Jacobians.</p>
<p>This is the gist of rmAD. It stores the values of each node in a forward pass, and computes the derivatives of each node exactly once in a backward pass.</p>
<p>It is a nice exercise to derive the backpropagation in the fully connected feedforward neural networks (e.g. <a href="http://neuralnetworksanddeeplearning.com/chap2.html#the_four_fundamental_equations_behind_backpropagation">the one for MNIST in Neural Networks and Deep Learning</a>) using rmAD.</p>
<p>AD is an approach lying between the extremes of numerical approximation (e.g. finite difference) and symbolic evaluation. It uses exact formulas (VJP) at each elementary operation like symbolic evaluation, while evaluates each VJP numerically rather than lumping all the VJPs into an unwieldy symbolic formula.</p>
<p>Things to look further into: the higher-order functional currying form <span class="math inline">\(j: (x \to y) \to (y, \bar y, x) \to \bar x\)</span> begs for a functional programming implementation.</p>
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Updates on open researchposts/2018-04-10-update-open-research.html2018-04-29T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<li><a href="#freedom-and-community">Freedom and community</a></li>
<li><a href="#tools-for-open-research">Tools for open research</a></li>
<li><a href="#an-anecdote-from-the-workshop">An anecdote from the workshop</a></li>
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<p>It has been 9 months since I last wrote about open (maths) research. Since then two things happened which prompted me to write an update.</p>
<p>As always I discuss open research only in mathematics, not because I think it should not be applied to other disciplines, but simply because I do not have experience nor sufficient interests in non-mathematical subjects.</p>
<p>First, I read about Richard Stallman the founder of the free software movement, in <a href="http://shop.oreilly.com/product/9780596002879.do">his biography by Sam Williams</a> and his own collection of essays <a href="https://shop.fsf.org/books-docs/free-software-free-society-selected-essays-richard-m-stallman-3rd-edition"><em>Free software, free society</em></a>, from which I learned a bit more about the context and philosophy of free software and its relation to that of open source software. For anyone interested in open research, I highly recommend having a look at these two books. I am also reading Levy's <a href="http://www.stevenlevy.com/index.php/books/hackers">Hackers</a>, which documented the development of the hacker culture predating Stallman. I can see the connection of ideas from the hacker ethic to the free software philosophy and to the open source philosophy. My guess is that the software world is fortunate to have pioneers who advocated for various kinds of freedom and openness from the beginning, whereas for academia which has a much longer history, credit protection has always been a bigger concern.</p>
<p>Also a month ago I attended a workshop called <a href="https://www.perimeterinstitute.ca/conferences/open-research-rethinking-scientific-collaboration">Open research: rethinking scientific collaboration</a>. That was the first time I met a group of people (mostly physicists) who also want open research to happen, and we had some stimulating discussions. Many thanks to the organisers at Perimeter Institute for organising the event, and special thanks to <a href="https://www.perimeterinstitute.ca/people/matteo-smerlak">Matteo Smerlak</a> and <a href="https://www.perimeterinstitute.ca/people/ashley-milsted">Ashley Milsted</a> for invitation and hosting.</p>
<p>From both of these I feel like I should write an updated post on open research.</p>
<h3 id="freedom-and-community">Freedom and community</h3>
<p>Ideals matter. Stallman's struggles stemmed from the frustration of denied request of source code (a frustration I shared in academia except source code is replaced by maths knowledge), and revolved around two things that underlie the free software movement: freedom and community. That is, the freedom to use, modify and share a work, and by sharing, to help the community.</p>
<p>Likewise, as for open research, apart from the utilitarian view that open research is more efficient / harder for credit theft, we should not ignore the ethical aspect that open research is right and fair. In particular, I think freedom and community can also serve as principles in open research. One way to make this argument more concrete is to describe what I feel are the central problems: NDAs (non-disclosure agreements) and reproducibility.</p>
<p><strong>NDAs</strong>. It is assumed that when establishing a research collaboration, or just having a discussion, all those involved own the joint work in progress, and no one has the freedom to disclose any information e.g. intermediate results without getting permission from all collaborators. In effect this amounts to signing an NDA. NDAs are harmful because they restrict people's freedom from sharing information that can benefit their own or others' research. Considering that in contrast to the private sector, the primary goal of academia is knowledge but not profit, NDAs in research are unacceptable.</p>
<p><strong>Reproducibility</strong>. Research papers written down are not necessarily reproducible, even though they appear on peer-reviewed journals. This is because the peer-review process is opaque and the proofs in the papers may not be clear to everyone. To make things worse, there are no open channels to discuss results in these papers and one may have to rely on interacting with the small circle of the informed. One example is folk theorems. Another is trade secrets required to decipher published works.</p>
<p>I should clarify that freedom works both ways. One should have the freedom to disclose maths knowledge, but they should also be free to withhold any information that does not hamper the reproducibility of published works (e.g. results in ongoing research yet to be published), even though it may not be nice to do so when such information can help others with their research.</p>
<p>Similar to the solution offered by the free software movement, we need a community that promotes and respects free flow of maths knowledge, in the spirit of the <a href="https://www.gnu.org/philosophy/">four essential freedoms</a>, a community that rejects NDAs and upholds reproducibility.</p>
<p>Here are some ideas on how to tackle these two problems and build the community:</p>
<ol style="list-style-type: decimal">
<li>Free licensing. It solves NDA problem - free licenses permit redistribution and modification of works, so if you adopt them in your joint work, then you have the freedom to modify and distribute the work; it also helps with reproducibility - if a paper is not clear, anyone can write their own version and publish it. Bonus points with the use of copyleft licenses like <a href="https://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Share-Alike</a> or the <a href="https://www.gnu.org/licenses/fdl.html">GNU Free Documentation License</a>.</li>
<li>A forum for discussions of mathematics. It helps solve the reproducibility problem - public interaction may help quickly clarify problems. By the way, Math Overflow is not a forum.</li>
<li>An infrastructure of mathematical knowledge. Like the GNU system, a mathematics encyclopedia under a copyleft license maintained in the Github-style rather than Wikipedia-style by a "Free Mathematics Foundation", and drawing contributions from the public (inside or outside of the academia). To begin with, crowd-source (again, Github-style) the proofs of say 1000 foundational theorems covered in the curriculum of a bachelor's degree. Perhaps start with taking contributions from people with some credentials (e.g. having a bachelor degree in maths) and then expand the contribution permission to the public, or taking advantage of existing corpus under free license like Wikipedia.</li>
<li>Citing with care: if a work is considered authorative but you couldn't reproduce the results, whereas another paper which tries to explain or discuss similar results makes the first paper understandable to you, give both papers due attribution (something like: see [1], but I couldn't reproduce the proof in [1], and the proofs in [2] helped clarify it). No one should be offended if you say you can not reproduce something - there may be causes on both sides, whereas citing [2] is fairer and helps readers with a similar background.</li>
</ol>
<h3 id="tools-for-open-research">Tools for open research</h3>
<p>The open research workshop revolved around how to lead academia towards a more open culture. There were discussions on open research tools, improving credit attributions, the peer-review process and the path to adoption.</p>
<p>During the workshop many efforts for open research were mentioned, and afterwards I was also made aware by more of them, like the following:</p>
<ul>
<li><a href="https://osf.io">OSF</a>, an online research platform. It has a clean and simple interface with commenting, wiki, citation generation, DOI generation, tags, license generation etc. Like Github it supports private and public repositories (but defaults to private), version control, with the ability to fork or bookmark a project.</li>
<li><a href="https://scipost.org/">SciPost</a>, physics journals whose peer review reports and responses are public (peer-witnessed refereeing), and allows comments (post-publication evaluation). Like arXiv, it requires some academic credential (PhD or above) to register.</li>
<li><a href="https://knowen.org/">Knowen</a>, a platform to organise knowledge in directed acyclic graphs. Could be useful for building the infrastructure of mathematical knowledge.</li>
<li><a href="https://fermatslibrary.com/">Fermat's Library</a>, the journal club website that crowd-annotates one notable paper per week released a Chrome extension <a href="https://fermatslibrary.com/librarian">Librarian</a> that overlays a commenting interface on arXiv. As an example Ian Goodfellow did an <a href="https://fermatslibrary.com/arxiv_comments?url=https://arxiv.org/pdf/1406.2661.pdf">AMA (ask me anything) on his GAN paper</a>.</li>
<li><a href="https://polymathprojects.org/">The Polymath project</a>, the famous massive collaborative mathematical project. Not exactly new, the Polymath project is the only open maths research project that has gained some traction and recognition. However, it does not have many active projects (<a href="http://michaelnielsen.org/polymath1/index.php?title=Main_Page">currently only one active project</a>).</li>
<li><a href="https://stacks.math.columbia.edu/">The Stacks Project</a>. I was made aware of this project by <a href="https://people.kth.se/~yitingl/">Yiting</a>. Its data is hosted on github and accepts contributions via pull requests and is licensed under the GNU Free Documentation License, ticking many boxes of the free and open source model.</li>
</ul>
<h3 id="an-anecdote-from-the-workshop">An anecdote from the workshop</h3>
<p>In a conversation during the workshop, one of the participants called open science "normal science", because reproducibility, open access, collaborations, and fair attributions are all what science is supposed to be, and practices like treating the readers as buyers rather than users should be called "bad science", rather than "closed science".</p>
<p>To which an organiser replied: maybe we should rename the workshop "Not-bad science".</p>
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The Mathematical Bazaarposts/2017-08-07-mathematical_bazaar.html2017-08-07T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<li><a href="#problems-of-academia">problems of academia</a></li>
<li><a href="#open-source-collaborations-on-github">open source collaborations on Github</a></li>
<li><a href="#open-research-in-mathematics">open research in mathematics</a></li>
<li><a href="#related-readings">related readings</a></li>
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<p>In this essay I describe some problems in academia of mathematics and propose an open source model, which I call open research in mathematics.</p>
<p>This essay is a work in progress - comments and criticisms are welcome! <a href="#fn1" class="footnoteRef" id="fnref1"><sup>1</sup></a></p>
<p>Before I start I should point out that</p>
<ol style="list-style-type: decimal">
<li>Open research is <em>not</em> open access. In fact the latter is a prerequisite to the former.</li>
<li>I am not proposing to replace the current academic model with the open model - I know academia works well for many people and I am happy for them, but I think an open research community is long overdue since the wide adoption of the World Wide Web more than two decades ago. In fact, I fail to see why an open model can not run in tandem with the academia, just like open source and closed source software development coexist today.</li>
</ol>
<h2 id="problems-of-academia">problems of academia</h2>
<p>Open source projects are characterised by publicly available source codes as well as open invitations for public collaborations, whereas closed source projects do not make source codes accessible to the public. How about mathematical academia then, is it open source or closed source? The answer is neither.</p>
<p>Compared to some other scientific disciplines, mathematics does not require expensive equipments or resources to replicate results; compared to programming in conventional software industry, mathematical findings are not meant to be commercial, as credits and reputation rather than money are the direct incentives (even though the former are commonly used to trade for the latter). It is also a custom and common belief that mathematical derivations and theorems shouldn't be patented. Because of this, mathematical research is an open source activity in the sense that proofs to new results are all available in papers, and thanks to open access e.g. the arXiv preprint repository most of the new mathematical knowledge is accessible for free.</p>
<p>Then why, you may ask, do I claim that maths research is not open sourced? Well, this is because 1. mathematical arguments are not easily replicable and 2. mathematical research projects are mostly not open for public participation.</p>
<p>Compared to computer programs, mathematical arguments are not written in an unambiguous language, and they are terse and not written in maximum verbosity (this is especially true in research papers as journals encourage limiting the length of submissions), so the understanding of a proof depends on whether the reader is equipped with the right background knowledge, and the completeness of a proof is highly subjective. More generally speaking, computer programs are mostly portable because all machines with the correct configurations can understand and execute a piece of program, whereas humans are subject to their environment, upbringings, resources etc. to have a brain ready to comprehend a proof that interests them. (these barriers are softer than the expensive equipments and resources in other scientific fields mentioned before because it is all about having access to the right information)</p>
<p>On the other hand, as far as the pursuit of reputation and prestige (which can be used to trade for the scarce resource of research positions and grant money) goes, there is often little practical motivation for career mathematicians to explain their results to the public carefully. And so the weird reality of the mathematical academia is that it is not an uncommon practice to keep trade secrets in order to protect one's territory and maintain a monopoly. This is doable because as long as a paper passes the opaque and sometimes political peer review process and is accepted by a journal, it is considered work done, accepted by the whole academic community and adds to the reputation of the author(s). Just like in the software industry, trade secrets and monopoly hinder the development of research as a whole, as well as demoralise outsiders who are interested in participating in related research.</p>
<p>Apart from trade secrets and territoriality, another reason to the nonexistence of open research community is an elitist tradition in the mathematical academia, which goes as follows:</p>
<ul>
<li>Whoever is not good at mathematics or does not possess a degree in maths is not eligible to do research, or else they run high risks of being labelled a crackpot.</li>
<li>Mistakes made by established mathematicians are more tolerable than those less established.</li>
<li>Good mathematical writings should be deep, and expositions of non-original results are viewed as inferior work and do not add to (and in some cases may even damage) one's reputation.</li>
</ul>
<p>All these customs potentially discourage public participations in mathematical research, and I do not see them easily go away unless an open source community gains momentum.</p>
<p>To solve the above problems, I propose a open source model of mathematical research, which has high levels of openness and transparency and also has some added benefits listed in the last section of this essay. This model tries to achieve two major goals:</p>
<ul>
<li>Open and public discussions and collaborations of mathematical research projects online</li>
<li>Open review to validate results, where author name, reviewer name, comments and responses are all publicly available online.</li>
</ul>
<p>To this end, a Github model is fitting. Let me first describe how open source collaboration works on Github.</p>
<h2 id="open-source-collaborations-on-github">open source collaborations on Github</h2>
<p>On <a href="https://github.com">Github</a>, every project is publicly available in a repository (we do not consider private repos). The owner can update the project by "committing" changes, which include a message of what has been changed, the author of the changes and a timestamp. Each project has an issue tracker, which is basically a discussion forum about the project, where anyone can open an issue (start a discussion), and the owner of the project as well as the original poster of the issue can close it if it is resolved, e.g. bug fixed, feature added, or out of the scope of the project. Closing the issue is like ending the discussion, except that the thread is still open to more posts for anyone interested. People can react to each issue post, e.g. upvote, downvote, celebration, and importantly, all the reactions are public too, so you can find out who upvoted or downvoted your post.</p>
<p>When one is interested in contributing code to a project, they fork it, i.e. make a copy of the project, and make the changes they like in the fork. Once they are happy with the changes, they submit a pull request to the original project. The owner of the original project may accept or reject the request, and they can comment on the code in the pull request, asking for clarification, pointing out problematic part of the code etc and the author of the pull request can respond to the comments. Anyone, not just the owner can participate in this review process, turning it into a public discussion. In fact, a pull request is a special issue thread. Once the owner is happy with the pull request, they accept it and the changes are merged into the original project. The author of the changes will show up in the commit history of the original project, so they get the credits.</p>
<p>As an alternative to forking, if one is interested in a project but has a different vision, or that the maintainer has stopped working on it, they can clone it and make their own version. This is a more independent kind of fork because there is no longer intention to contribute back to the original project.</p>
<p>Moreover, on Github there is no way to send private messages, which forces people to interact publicly. If say you want someone to see and reply to your comment in an issue post or pull request, you simply mention them by <code>@someone</code>.</p>
<h2 id="open-research-in-mathematics">open research in mathematics</h2>
<p>All this points to a promising direction of open research. A maths project may have a wiki / collection of notes, the paper being written, computer programs implementing the results etc. The issue tracker can serve as a discussion forum about the project as well as a platform for open review (bugs are analogous to mistakes, enhancements are possible ways of improving the main results etc.), and anyone can make their own version of the project, and (optionally) contribute back by making pull requests, which will also be openly reviewed. One may want to add an extra "review this project" functionality, so that people can comment on the original project like they do in a pull request. This may or may not be necessary, as anyone can make comments or point out mistakes in the issue tracker.</p>
<p>One may doubt this model due to concerns of credits because work in progress is available to anyone. Well, since all the contributions are trackable in project commit history and public discussions in issues and pull request reviews, there is in fact <em>less</em> room for cheating than the current model in academia, where scooping can happen without any witnesses. What we need is a platform with a good amount of trust like arXiv, so that the open research community honours (and can not ignore) the commit history, and the chance of mis-attribution can be reduced to minimum.</p>
<p>Compared to the academic model, open research also has the following advantages:</p>
<ul>
<li>Anyone in the world with Internet access will have a chance to participate in research, whether they are affiliated to a university, have the financial means to attend conferences, or are colleagues of one of the handful experts in a specific field.</li>
<li>The problem of replicating / understanding maths results will be solved, as people help each other out. This will also remove the burden of answering queries about one's research. For example, say one has a project "Understanding the fancy results in [paper name]", they write up some initial notes but get stuck understanding certain arguments. In this case they can simply post the questions on the issue tracker, and anyone who knows the answer, or just has a speculation can participate in the discussion. In the end the problem may be resolved without the authors of the paper being bothered, who may be too busy to answer.</li>
<li>Similarly, the burden of peer review can also be shifted from a few appointed reviewers to the crowd.</li>
</ul>
<h2 id="related-readings">related readings</h2>
<ul>
<li><a href="http://www.catb.org/esr/writings/cathedral-bazaar/">The Cathedral and the Bazaar by Eric Raymond</a></li>
<li><a href="http://michaelnielsen.org/blog/doing-science-online/">Doing sience online by Michael Nielson</a></li>
<li><a href="https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/">Is massively collaborative mathematics possible? by Timothy Gowers</a></li>
</ul>
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<hr />
<ol>
<li id="fn1"><p>Please send your comments to my email address - I am still looking for ways to add a comment functionality to this website.<a href="#fnref1">↩</a></p></li>
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Open mathematical research and launching toywikiposts/2017-04-25-open_research_toywiki.html2017-04-25T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>As an experimental project, I am launching toywiki.</p>
<p>It hosts a collection of my research notes.</p>
<p>It takes some ideas from the open source culture and apply them to mathematical research: 1. It uses a very permissive license (CC-BY-SA). For example anyone can fork the project and make their own version if they have a different vision and want to build upon the project. 2. All edits will done with maximum transparency, and discussions of any of notes should also be as public as possible (e.g. Github issues) 3. Anyone can suggest changes by opening issues and submitting pull requests</p>
<p>Here are the links: <a href="http://toywiki.xyz">toywiki</a> and <a href="https://github.com/ycpei/toywiki">github repo</a>.</p>
<p>Feedbacks are welcome by email.</p>
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A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymerposts/2016-10-13-q-robinson-schensted-knuth-polymer.html2016-10-13T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>(Latest update: 2017-01-12) In <a href="http://arxiv.org/abs/1504.00666">Matveev-Petrov 2016</a> a \(q\)-deformed Robinson-Schensted-Knuth algorithm (\(q\)RSK) was introduced. In this article we give reformulations of this algorithm in terms of Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pair are swapped in a sense of distribution when the input matrix is transposed. We also formulate a \(q\)-polymer model based on the \(q\)RSK and prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary boundary conditions and \(q\)-geometric weights. We use the \(q\)-local moves to define a generalisation of the \(q\)RSK taking a Young diagram-shape of array as the input. We write down the joint distribution of partition functions in the space-like direction of the \(q\)-polymer in \(q\)-geometric environment, formulate a \(q\)-version of the multilayer polynuclear growth model (\(q\)PNG) and write down the joint distribution of the \(q\)-polymer partition functions at a fixed time.</p>
<p>This article is available at <a href="https://arxiv.org/abs/1610.03692">arXiv</a>. It seems to me that one difference between arXiv and Github is that on arXiv each preprint has a few versions only. In Github many projects have a "dev" branch hosting continuous updates, whereas the master branch is where the stable releases live.</p>
<p><a href="%7B%7B%20site.url%20%7D%7D/assets/resources/qrsklatest.pdf">Here</a> is a "dev" version of the article, which I shall push to arXiv when it stablises. Below is the changelog.</p>
<ul>
<li>2017-01-12: Typos and grammar, arXiv v2.</li>
<li>2016-12-20: Added remarks on the geometric \(q\)-pushTASEP. Added remarks on the converse of the Burke property. Added natural language description of the \(q\)RSK. Fixed typos.</li>
<li>2016-11-13: Fixed some typos in the proof of Theorem 3.</li>
<li>2016-11-07: Fixed some typos. The \(q\)-Burke property is now stated in a more symmetric way, so is the law of large numbers Theorem 2.</li>
<li>2016-10-20: Fixed a few typos. Updated some references. Added a reference: <a href="http://web.mit.edu/~shopkins/docs/rsk.pdf">a set of notes titled "RSK via local transformations"</a>. It is written by <a href="http://web.mit.edu/~shopkins/">Sam Hopkins</a> in 2014 as an expository article based on MIT combinatorics preseminar presentations of Alex Postnikov. It contains some idea (applying local moves to a general Young-diagram shaped array in the order that matches any growth sequence of the underlying Young diagram) which I thought I was the first one to write down.</li>
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AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieuposts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html2015-07-15T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>A Macdonald superpolynomial (introduced in [O. Blondeau-Fournier et al., Lett. Math. Phys. <span class="bf">101</span> (2012), no. 1, 27–47; <a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=2935476&loc=fromrevtext">MR2935476</a>; J. Comb. <span class="bf">3</span> (2012), no. 3, 495–561; <a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=3029444&loc=fromrevtext">MR3029444</a>]) in \(N\) Grassmannian variables indexed by a superpartition \(\Lambda\) is said to be stable if \({m (m + 1) \over 2} \ge |\Lambda|\) and \(N \ge |\Lambda| - {m (m - 3) \over 2}\) , where \(m\) is the fermionic degree. A stable Macdonald superpolynomial (corresponding to a bisymmetric polynomial) is also called a double Macdonald polynomial (dMp). The main result of this paper is the factorisation of a dMp into plethysms of two classical Macdonald polynomials (Theorem 5). Based on this result, this paper</p>
<ol style="list-style-type: decimal">
<li><p>shows that the dMp has a unique decomposition into bisymmetric monomials;</p></li>
<li><p>calculates the norm of the dMp;</p></li>
<li><p>calculates the kernel of the Cauchy-Littlewood-type identity of the dMp;</p></li>
<li><p>shows the specialisation of the aforementioned factorisation to the Jack, Hall-Littlewood and Schur cases. One of the three Schur specialisations, denoted as \(s_{\lambda, \mu}\), also appears in (7) and (9) below;</p></li>
<li><p>defines the \(\omega\) -automorphism in this setting, which was used to prove an identity involving products of four Littlewood-Richardson coefficients;</p></li>
<li><p>shows an explicit evaluation of the dMp motivated by the most general evaluation of the usual Macdonald polynomials;</p></li>
<li><p>relates dMps with the representation theory of the hyperoctahedral group \(B_n\) via the double Kostka coefficients (which are defined as the entries of the transition matrix from the bisymmetric Schur functions \(s_{\lambda, \mu}\) to the modified dMps);</p></li>
<li><p>shows that the double Kostka coefficients have the positivity and the symmetry property, and can be written as sums of products of the usual Kostka coefficients;</p></li>
<li><p>defines an operator \(\nabla^B\) as an analogue of the nabla operator \(\nabla\) introduced in [F. Bergeron and A. M. Garsia, in <em>Algebraic methods and \(q\)-special functions</em> (Montréal, QC, 1996), 1–52, CRM Proc. Lecture Notes, 22, Amer. Math. Soc., Providence, RI, 1999; <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1726826&loc=fromrevtext">MR1726826</a>]. The action of \(\nabla^B\) on the bisymmetric Schur function \(s_{\lambda, \mu}\) yields the dimension formula \((h + 1)^r\) for the corresponding representation of \(B_n\) , where \(h\) and \(r\) are the Coxeter number and the rank of \(B_n\) , in the same way that the action of \(\nabla\) on the \(n\) th elementary symmetric function leads to the same formula for the group of type \(A_n\) .</p></li>
</ol>
<p>Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3306078, its copyright owned by the AMS.</p>
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On a causal quantum double product integral related to Lévy stochastic area.posts/2015-07-01-causal-quantum-product-levy-area.html2015-07-01T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>In <a href="https://arxiv.org/abs/1506.04294">this paper</a> with <a href="http://homepages.lboro.ac.uk/~marh3/">Robin</a> we study the family of causal double product integrals \[ \prod_{a < x < y < b}\left(1 + i{\lambda \over 2}(dP_x dQ_y - dQ_x dP_y) + i {\mu \over 2}(dP_x dP_y + dQ_x dQ_y)\right) \]</p>
<p>where <span class="math inline">\(P\)</span> and <span class="math inline">\(Q\)</span> are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in <a href="http://www.actaphys.uj.edu.pl/findarticle?series=Reg&vol=46&page=1851">(Hudson-Pei2015)</a>. The main problem solved in this paper is the explicit evaluation of the continuum limit <span class="math inline">\(W\)</span> of the latter, and showing that <span class="math inline">\(W\)</span> is a unitary operator. The kernel of <span class="math inline">\(W\)</span> is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.</p>
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AMS review of 'Infinite binary words containing repetitions of odd period' by Badkobeh and Crochemoreposts/2015-05-30-infinite-binary-words-containing-repetitions-odd-periods.html2015-05-30T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>This paper is about the existence of pattern-avoiding infinite binary words, where the patterns are squares, cubes and \(3^+\)-powers. There are mainly two kinds of results, positive (existence of an infinite binary word avoiding a certain pattern) and negative (non-existence of such a word). Each positive result is proved by the construction of a word with finitely many squares and cubes which are listed explicitly. First a synchronising (also known as comma-free) uniform morphism \(g\: \Sigma_3^* \to \Sigma_2^*\)</p>
<p>is constructed. Then an argument is given to show that the length of squares in the code \(g(w)\) for a squarefree \(w\) is bounded, hence all the squares can be obtained by examining all \(g(s)\) for \(s\) of bounded lengths. The argument resembles that of the proof of, e.g., Theorem 1, Lemma 2, Theorem 3 and Lemma 4 in [N. Rampersad, J. O. Shallit and M. Wang, Theoret. Comput. Sci. <strong>339</strong> (2005), no. 1, 19–34; <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=2142071&loc=fromrevtext">MR2142071</a>]. The negative results are proved by traversing all possible finite words satisfying the conditions.</p>
<p> Let \(L(n_2, n_3, S)\) be the maximum length of a word with \(n_2\) distinct squares, \(n_3\) distinct cubes and that the periods of the squares can take values only in \(S\) , where \(n_2, n_3 \in \Bbb N \cup \{\infty, \omega\}\) and \(S \subset \Bbb N_+\) . \(n_k = 0\) corresponds to \(k\)-free, \(n_k = \infty\) means no restriction on the number of distinct \(k\)-powers, and \(n_k = \omega\) means \(k^+\)-free.</p>
<p> Below is the summary of the positive and negative results:</p>
<ol style="list-style-type: decimal">
<li><p>(Negative) \(L(\infty, \omega, 2 \Bbb N) < \infty\) : \(\nexists\) an infinite \(3^+\) -free binary word avoiding all squares of odd periods. (Proposition 1)</p></li>
<li><p>(Negative) \(L(\infty, 0, 2 \Bbb N + 1) \le 23\) : \(\nexists\) an infinite 3-free binary word, avoiding squares of even periods. The longest one has length \(\le 23\) (Proposition 2).</p></li>
<li>(Positive) \(L(\infty, \omega, 2 \Bbb N +
<ol style="list-style-type: decimal">
<li><dl>
<dt>= \infty\)</dt>
<dd>\(\exists\) an infinite \(3^+\) -free binary word avoiding squares of even periods (Theorem 1).
</dd>
</dl></li>
</ol></li>
<li><p>(Positive) \(L(\infty, \omega, \{1, 3\}) = \infty\) : \(\exists\) an infinite \(3^+\) -free binary word containing only squares of period 1 or 3 (Theorem 2).</p></li>
<li><p>(Negative) \(L(6, 1, 2 \Bbb N + 1) = 57\) : \(\nexists\) an infinite binary word avoiding squares of even period containing \(< 7\) squares and \(< 2\) cubes. The longest one containing 6 squares and 1 cube has length 57 (Proposition 6).</p></li>
<li><p>(Positive) \(L(7, 1, 2 \Bbb N + 1) = \infty\) : \(\exists\) an infinite \(3^+\) -free binary word avoiding squares of even period with 1 cube and 7 squares (Theorem 3).</p></li>
<li><p>(Positive) \(L(4, 2, 2 \Bbb N + 1) = \infty\) : \(\exists\) an infinite \(3^+\) -free binary words avoiding squares of even period and containing 2 cubes and 4 squares (Theorem 4).</p></li>
</ol>
<p>Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3313467, its copyright owned by the AMS.</p>
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jstposts/2015-04-02-juggling-skill-tree.html2015-04-02T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>jst = juggling skill tree</p>
<p>If you have ever played a computer role playing game, you may have noticed the protagonist sometimes has a skill "tree" (most of the time it is actually a directed acyclic graph), where certain skills leads to others. For example, <a href="http://hydra-media.cursecdn.com/diablo.gamepedia.com/3/37/Sorceress_Skill_Trees_%28Diablo_II%29.png?version=b74b3d4097ef7ad4e26ebee0dcf33d01">here</a> is the skill tree of sorceress in <a href="https://en.wikipedia.org/wiki/Diablo_II">Diablo II</a>.</p>
<p>Now suppose our hero embarks on a quest for learning all the possible juggling patterns. Everyone would agree she should start with cascade, the simplest nontrivial 3-ball pattern, but what afterwards? A few other accessible patterns for beginners are juggler's tennis, two in one and even reverse cascade, but what to learn after that? The encyclopeadic <a href="http://libraryofjuggling.com/">Library of Juggling</a> serves as a good guide, as it records more than 160 patterns, some of which very aesthetically appealing. On this website almost all the patterns have a "prerequisite" section, indicating what one should learn beforehand. I have therefore written a script using <a href="http://python.org">Python</a>, <a href="http://www.crummy.com/software/BeautifulSoup/">BeautifulSoup</a> and <a href="http://pygraphviz.github.io/">pygraphviz</a> to generate a jst (graded by difficulties, which is the leftmost column) from the Library of Juggling (click the image for the full size):</p>
<p><a href="../assets/resources/juggling.png"><img src="../assets/resources/juggling.png" alt="The juggling skill tree" style="width:38em" /></a></p>
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Unitary causal quantum stochastic double products as universal interactions Iposts/2015-04-01-unitary-double-products.html2015-04-01T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>In <a href="http://www.actaphys.uj.edu.pl/findarticle?series=Reg&vol=46&page=1851">this paper</a> with <a href="http://homepages.lboro.ac.uk/~marh3/">Robin</a> we show the explicit formulae for a family of unitary triangular and rectangular double product integrals which can be described as second quantisations.</p>
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AMS review of 'A weighted interpretation for the super Catalan numbers' by Allen and Gheorghiciucposts/2015-01-20-weighted-interpretation-super-catalan-numbers.html2015-01-20T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>The super Catalan numbers are defined as $$ T(m,n) = {(2 m)! (2 n)! 2 m! n! (m + n)!}. $$</p>
<p> This paper has two main results. First a combinatorial interpretation of the super Catalan numbers is given: $$ T(m,n) = P(m,n) - N(m,n) $$ where \(P(m,n)\) enumerates the number of 2-Motzkin paths whose \(m\) -th step begins at an even level (called \(m\)-positive paths) and \(N(m,n)\) those with \(m\)-th step beginning at an odd level (\(m\)-negative paths). The proof uses a recursive argument on the number of \(m\)-positive and -negative paths, based on a recursion of the super Catalan numbers appearing in [I. M. Gessel, J. Symbolic Comput. <strong>14</strong> (1992), no. 2-3, 179–194; <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1187230&loc=fromrevtext">MR1187230</a>]: $$ 4T(m,n) = T(m+1, n) + T(m, n+1). $$ This result gives an expression for the super Catalan numbers in terms of numbers counting the so-called ballot paths. The latter sometimes are also referred to as the generalised Catalan numbers forming the entries of the Catalan triangle.</p>
<p> Based on the first result, the second result is a combinatorial interpretation of the super Catalan numbers \(T(2,n)\) in terms of counting certain Dyck paths. This is equivalent to a theorem, which represents \(T(2,n)\) as counting of certain pairs of Dyck paths, in [I. M. Gessel and G. Xin, J. Integer Seq. <strong>8</strong> (2005), no. 2, Article 05.2.3, 13 pp.; <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=2134162&loc=fromrevtext">MR2134162</a>], and the equivalence is explained at the end of the paper by a bijection between the Dyck paths and the pairs of Dyck paths. The proof of the theorem itself is also done by constructing two bijections between Dyck paths satisfying certain conditions. All the three bijections are formulated by locating, removing and adding steps.</p>
<p>Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3275875, its copyright owned by the AMS.</p>
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Symmetry property of \(q\)-weighted Robinson-Schensted algorithms and branching algorithmsposts/2014-04-01-q-robinson-schensted-symmetry-paper.html2014-04-01T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>In <a href="http://link.springer.com/article/10.1007/s10801-014-0505-x">this paper</a> a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm has been shown for the \(q\)-weighted Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin. This approach, which uses "growth graphs", can also be applied to a wider class of insertion algorithms which have a branching structure.</p>
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<img src="../assets/resources/1423graph.jpg" alt="Growth graph of q-RS for 1423" />
<p class="caption">Growth graph of q-RS for 1423</p>
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<p>Above is the growth graph of the \(q\)-weighted Robinson-Schensted algorithm for the permutation \({1 2 3 4\choose1 4 2 3}\).</p>
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A \(q\)-weighted Robinson-Schensted algorithmposts/2013-06-01-q-robinson-schensted-paper.html2013-06-01T00:00:00ZYuchen Pei<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<p>In <a href="https://projecteuclid.org/euclid.ejp/1465064320">this paper</a> with <a href="http://www.bristol.ac.uk/maths/people/neil-m-oconnell/">Neil</a> we construct a \(q\)-version of the Robinson-Schensted algorithm with column insertion. Like the <a href="http://en.wikipedia.org/wiki/Robinson–Schensted_correspondence">usual RS correspondence</a> with column insertion, this algorithm could take words as input. Unlike the usual RS algorithm, the output is a set of weighted pairs of semistandard and standard Young tableaux \((P,Q)\) with the same shape. The weights are rational functions of indeterminant \(q\).</p>
<p>If \(q\in[0,1]\), the algorithm can be considered as a randomised RS algorithm, with 0 and 1 being two interesting cases. When \(q\to0\), it is reduced to the latter usual RS algorithm; while when \(q\to1\) with proper scaling it should scale to directed random polymer model in <a href="http://arxiv.org/abs/0910.0069">(O'Connell 2012)</a>. When the input word \(w\) is a random walk:</p>
<p>\begin{align*}\mathbb P(w=v)=\prod_{i=1}^na_{v_i},\qquad\sum_ja_j=1\end{align*}</p>
<p>the shape of output evolves as a Markov chain with kernel related to \(q\)-Whittaker functions, which are Macdonald functions when \(t=0\) with a factor.</p>
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