Yuchen Pei's Blog

Great but Manageable Expectations

2019-03-14T00:00:00Z

This is Part 2 of a two-part blog post on differential privacy. Continuing from Part 1, I discuss the Rényi differential privacy, corresponding to the Rényi divergence, a study of the moment generating functions the divergence between probability measures to derive the tail bounds.

Like in Part 1, I prove a composition theorem and a subsampling theorem.

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.

After that I explain the Tensorflow implementation of differential privacy in its Privacy module, which focuses on the differentially private stochastic gradient descent algorithm (DP-SGD).

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.

If you are confused by any notations, ask me or try this.

Rényi divergence and differential privacy

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.

So far we have seen several notions of divergence used in differential privacy: the max divergence which is $\epsilon$-ind in disguise:

\[D_\infty(p || q) := \max_y \log {p(y) \over q(y)},\]

the $\delta$-approximate max divergence that defines the $(\epsilon, \delta)$-ind:

\[D_\infty^\delta(p || q) := \max_y \log{p(y) - \delta \over q(y)},\]

and the KL-divergence which is the expectation of the divergence variable:

\[D(p || q) = \mathbb E L(p || q) = \int \log {p(y) \over q(y)} p(y) dy.\]

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:

\[D_\lambda(p || q) = (\lambda - 1)^{-1} \log \mathbb E \exp((\lambda - 1) L(p || q)) = (\lambda - 1)^{-1} \log \int {p(y)^\lambda \over q(y)^{\lambda - 1}} dx.\]

Indeed, when $\lambda \to \infty$ we recover the max divergence, and when $\lambda \to 1$, by recognising $D_\lambda$ as a derivative in $\lambda$ at $\lambda = 1$, we recover the KL divergence. In this post we only consider $\lambda > 1$.

Using the Rényi divergence we may define:

Definition (Rényi differential privacy) (Mironov 2017). An mechanism $M$ is $(\lambda, \rho)$-Rényi differentially private ($(\lambda, \rho)$-rdp) if for all $x$ and $x'$ with distance $1$,

\[D_\lambda(M(x) || M(x')) \le \rho.\]

For convenience we also define two related notions, $G_\lambda (f || g)$ and $\kappa_{f, g} (t)$ for $\lambda > 1$, $t > 0$ and positive functions $f$ and $g$:

\[G_\lambda(f || g) = \int f(y)^{\lambda} g(y)^{1 - \lambda} dy; \qquad \kappa_{f, g} (t) = \log G_{t + 1}(f || g).\]

For probability densities $p$ and $q$, $G_{t + 1}(p || q)$ and $\kappa_{p, q}(t)$ are the $t$th moment generating function and cumulant of the divergence variable $L(p || q)$, and

\[D_\lambda(p || q) = (\lambda - 1)^{-1} \kappa_{p, q}(\lambda - 1).\]

In the following, whenever you see $t$, think of it as $\lambda - 1$.

Example 1 (RDP for the Gaussian mechanism). Using the scaling and translation invariance of $L$ (6.1), we have that the divergence variable for two Gaussians with the same variance is

\[L(N(\mu_1, \sigma^2 I) || N(\mu_2, \sigma^2 I)) \overset{d}{=} L(N(0, I) || N((\mu_2 - \mu_1) / \sigma, I)).\]

With this we get

\[D_\lambda(N(\mu_1, \sigma^2 I) || N(\mu_2, \sigma^2 I)) = {\lambda \|\mu_2 - \mu_1\|_2^2 \over 2 \sigma^2} = D_\lambda(N(\mu_2, \sigma^2 I) || N(\mu_1, \sigma^2 I)).\]

Again due to the scaling invariance of $L$, we only need to consider $f$ with sensitivity $1$, see the discussion under (6.1). The Gaussian mechanism on query $f$ is thus $(\lambda, \lambda / 2 \sigma^2)$-rdp for any $\lambda > 1$.

From the example of Gaussian mechanism, we see that the relation between $\lambda$ and $\rho$ is like that between $\epsilon$ and $\delta$. Given $\lambda$ (resp. $\rho$) and parameters like variance of the noise and the sensitivity of the query, we can write $\rho = \rho(\lambda)$ (resp. $\lambda = \lambda(\rho)$).

Using the Chernoff bound (6.7), we can bound the divergence variable:

\[\mathbb P(L(p || q) \ge \epsilon) \le {\mathbb E \exp(t L(p || q)) \over \exp(t \epsilon))} = \exp (\kappa_{p, q}(t) - \epsilon t). \qquad (7.7)\]

For a function $f: I \to \mathbb R$, denote its Legendre transform by

\[f^*(\epsilon) := \sup_{t \in I} (\epsilon t - f(t)).\]

By taking infimum on the RHS of (7.7), we obtain

Claim 20. Two probability densities $p$ and $q$ are $(\epsilon, \exp(-\kappa_{p, q}^*(\epsilon)))$-ind.

Given a mechanism $M$, let $\kappa_M(t)$ denote an upper bound for the cumulant of its privacy loss:

\[\log \mathbb E \exp(t L(M(x) || M(x'))) \le \kappa_M(t), \qquad \forall x, x'\text{ with } d(x, x') = 1.\]

For example, we can set $\kappa_M(t) = t \rho(t + 1)$. Using the same argument we have the following:

Claim 21. If $M$ is $(\lambda, \rho)$-rdp, then

it is also $(\epsilon, \exp((\lambda - 1) (\rho - \epsilon)))$-dp for any $\epsilon \ge \rho$.
Alternatively, $M$ is $(\epsilon, - \exp(\kappa_M^*(\epsilon)))$-dp for any $\epsilon > 0$.
Alternatively, for any $0 < \delta \le 1$, $M$ is $(\rho + (\lambda - 1)^{-1} \log \delta^{-1}, \delta)$-dp.

Example 2 (Gaussian mechanism). We can apply the above argument to the Gaussian mechanism on query $f$ and get:

\[\delta \le \inf_{\lambda > 1} \exp((\lambda - 1) ({\lambda \over 2 \sigma^2} - \epsilon))\]

By assuming $\sigma^2 > (2 \epsilon)^{-1}$ we have that the infimum is achieved when $\lambda = (1 + 2 \epsilon / \sigma^2) / 2$ and

\[\delta \le \exp(- ((2 \sigma)^{-1} - \epsilon \sigma)^2 / 2)\]

which is the same result as (6.8), obtained using the Chernoff bound of the noise.

However, as we will see later, compositions will yield different results from those obtained from methods in Part 1 when considering Rényi dp.

Claim 22 (Moment Composition Theorem). Let $M$ be the adaptive composition of $M_{1 : k}$. Suppose for any $y_{< i}$, $M_i(y_{< i})$ is $(\lambda, \rho)$-rdp. Then $M$ is $(\lambda, k\rho)$-rdp.

Proof. Rather straightforward. As before let $p_i$ and $q_i$ be the conditional laws of adpative composition of $M_{1 : i}$ at $x$ and $x'$ respectively, and $p^i$ and $q^i$ be the joint laws of $M_{1 : i}$ at $x$ and $x'$ respectively. Denote

\[D_i = \mathbb E \exp((\lambda - 1)\log {p^i(\xi_{1 : i}) \over q^i(\xi_{1 : i})})\]

Then

\[\begin{aligned} D_i &= \mathbb E\mathbb E \left(\exp((\lambda - 1)\log {p_i(\xi_i | \xi_{< i}) \over q_i(\xi_i | \xi_{< i})}) \exp((\lambda - 1)\log {p^{i - 1}(\xi_{< i}) \over q^{i - 1}(\xi_{< i})}) \big| \xi_{< i}\right) \\ &= \mathbb E \mathbb E \left(\exp((\lambda - 1)\log {p_i(\xi_i | \xi_{< i}) \over q_i(\xi_i | \xi_{< i})}) | \xi_{< i}\right) \exp\left((\lambda - 1)\log {p^{i - 1}(\xi_{< i}) \over q^{i - 1}(\xi_{< i})}\right)\\ &\le \mathbb E \exp((\lambda - 1) \rho) \exp\left((\lambda - 1)\log {p^{i - 1}(\xi_{< i}) \over q^{i - 1}(\xi_{< i})}\right)\\ &= \exp((\lambda - 1) \rho) D_{i - 1}. \end{aligned}\]

Applying this recursively we have

\[D_k \le \exp(k(\lambda - 1) \rho),\]

and so

\[(\lambda - 1)^{-1} \log \mathbb E \exp((\lambda - 1)\log {p^k(\xi_{1 : i}) \over q^k(\xi_{1 : i})}) = (\lambda - 1)^{-1} \log D_k \le k \rho.\]

Since this holds for all $x$ and $x'$, we are done. $\square$

This, together with the scaling property of the legendre transformation:

\[(k f)^*(x) = k f^*(x / k)\]

yields

Claim 23. The $k$-fold adaptive composition of $(\lambda, \rho(\lambda))$-rdp mechanisms is $(\epsilon, \exp(- k \kappa^*(\epsilon / k)))$-dp, where $\kappa(t) := t \rho(t + 1)$.

Example 3 (Gaussian mechanism). We can apply the above claim to Gaussian mechanism. Again, without loss of generality we assume $S_f = 1$. 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 $M_i$ is $(\lambda, (2 \sigma^2)^{-1} \lambda)$-rdp, the composition $M$ is $(\lambda, (2 \sigma^2)^{-1} k \lambda)$-rdp. Processing this using the Chernoff bound as in the previous example, we have

\[\delta = \exp(- ((2 \sigma / \sqrt k)^{-1} - \epsilon \sigma / \sqrt k)^2 / 2),\]

Substituting $\sigma$ with $\sigma / \sqrt k$ in (6.81), we conclude that if

\[\sigma > \sqrt k \left(\epsilon^{-1} \sqrt{2 \log \delta^{-1}} + (2 \epsilon)^{- {1 \over 2}}\right)\]

then the composition $M$ is $(\epsilon, \delta)$-dp.

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).

We also have a subsampling theorem for the Rényi dp.

Claim 24. Fix $r \in [0, 1]$. Let $m \le n$ be two nonnegative integers with $m = r n$. Let $N$ be a $(\lambda, \rho)$-rdp machanism on $X^m$. Let $\mathcal I := \{J \subset [n]: |J| = m\}$ be the set of subsets of $[n]$ of size $m$. Define mechanism $M$ on $X^n$ by

\[M(x) = N(x_\gamma)\]

where $\gamma$ is sampled uniformly from $\mathcal I$. Then $M$ is $(\lambda, {1 \over \lambda - 1} \log (1 + r(e^{(\lambda - 1) \rho} - 1)))$-rdp.

To prove Claim 24, we need a useful lemma:

Claim 25. Let $p_{1 : n}$ and $q_{1 : n}$ be nonnegative integers, and $\lambda > 1$. Then

\[{(\sum p_i)^\lambda \over (\sum q_i)^{\lambda - 1}} \le \sum_i {p_i^\lambda \over q_i^{\lambda - 1}}. \qquad (8)\]

Proof. Let

\[r(i) := p_i / P, \qquad u(i) := q_i / Q\]

where

\[P := \sum p_i, \qquad Q := \sum q_i\]

then $r$ and $u$ are probability mass functions. Plugging in $p_i = r(i) P$ and $q_i = u(i) Q$ into the objective (8), it suffices to show

\[1 \le \sum_i {r(i)^\lambda \over u(i)^{\lambda - 1}} = \mathbb E_{\xi \sim u} \left({r(\xi) \over u(\xi)}\right)^\lambda\]

This is true due to Jensen's Inequality:

\[\mathbb E_{\xi \sim u} \left({r(\xi) \over u(\xi)}\right)^\lambda \ge \left(\mathbb E_{\xi \sim u} {r(\xi) \over u(\xi)} \right)^\lambda = 1.\]

$\square$

Proof of Claim 24. Define $\mathcal I$ as before.

Let $p$ and $q$ be the laws of $M(x)$ and $M(x')$ respectively. For any $I \in \mathcal I$, let $p_I$ and $q_I$ be the laws of $N(x_I)$ and $N(x_I')$ respectively. Then we have

\[\begin{aligned} p(y) &= n^{-1} \sum_{I \in \mathcal I} p_I(y) \\ q(y) &= n^{-1} \sum_{I \in \mathcal I} q_I(y), \end{aligned}\]

where $n = |\mathcal I|$.

The MGF of $L(p || q)$ is thus

\[\mathbb E((\lambda - 1) L(p || q)) = n^{-1} \int {(\sum_I p_I(y))^\lambda \over (\sum_I q_I(y))^{\lambda - 1}} dy \le n^{-1} \sum_I \int {p_I(y)^\lambda \over q_I(y)^{\lambda - 1}} dy \qquad (9)\]

where in the last step we used Claim 25. As in the proof of Claim 19, we divide $\mathcal I$ into disjoint sets $\mathcal I_\in$ and $\mathcal I_\notin$. Furthermore we denote by $n_\in$ and $n_\notin$ their cardinalities. Then the right hand side of (9) becomes

\[n^{-1} \sum_{I \in \mathcal I_\in} \int {p_I(y)^\lambda \over q_I(y)^{\lambda - 1}} dy + n^{-1} \sum_{I \in \mathcal I_\notin} \int {p_I(y)^\lambda \over q_I(y)^{\lambda - 1}} dy\]

The summands in the first are the MGF of $L(p_I || q_I)$, and the summands in the second term are $1$, so

\[\begin{aligned} \mathbb E((\lambda - 1) L(p || q)) &\le n^{-1} \sum_{I \in \mathcal I_\in} \mathbb E \exp((\lambda - 1) L(p_I || q_I)) + (1 - r) \\ &\le n^{-1} \sum_{I \in \mathcal I_\in} \exp((\lambda - 1) D_\lambda(p_I || q_I)) + (1 - r) \\ &\le r \exp((\lambda - 1) \rho) + (1 - r). \end{aligned}\]

Taking log and dividing by $(\lambda - 1)$ on both sides we have

\[D_\lambda(p || q) \le (\lambda - 1)^{-1} \log (1 + r(\exp((\lambda - 1) \rho) - 1)).\]

$\square$

As before, we can rewrite the conclusion of Lemma 6 using $1 + z \le e^z$ and obtain $(\lambda, (\lambda - 1)^{-1} r (e^{(\lambda - 1) \rho} - 1))$-rdp, which further gives $(\lambda, \alpha^{-1} (e^\alpha - 1) r \rho)$-rdp (or $(\lambda, O(r \rho))$-rdp) if $(\lambda - 1) \rho < \alpha$ for some $\alpha$.

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.

Example 4 (Gaussian mechanism). Applying the moment subsampling theorem to the Gaussian mechanism, we obtain $(\lambda, O(r \lambda / \sigma^2))$-rdp for a subsampled Gaussian mechanism with rate $r$. Abadi-Chu-Goodfellow-McMahan-Mironov-Talwar-Zhang 2016 (ACGMMTZ16 in the following), however, gains an extra $r$ in the bound given certain assumptions.

ACGMMTZ16

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.

Claim 26. Assuming Conjecture 1 (see below) is true. For a subsampled Gaussian mechanism with ratio $r$, if $r = O(\sigma^{-1})$ and $\lambda = O(\sigma^2)$, then we have $(\lambda, O(r^2 \lambda / \sigma^2))$-rdp.

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:

Conjecture 1. 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.

This conjecture is equivalent to its special case when $n = 2$ by an induction argument (replacing one pair of densities at a time).

Recall the definition of $G_\lambda$ under the definition of Rényi differential privacy. The following Claim will be useful.

Claim 27. Let $\lambda$ be a positive integer, then

\[G_\lambda(r p + (1 - r) q || q) = \sum_{k = 1 : \lambda} {\lambda \choose k} r^k (1 - r)^{\lambda - k} G_k(p || q).\]

Proof. Quite straightforward, by expanding the numerator $(r p + (1 - r) q)^\lambda$ using binomial expansion. $\square$

Proof of Claim 26. Let $M$ be the Gaussian mechanism with subsampling rate $r$, and $p$ and $q$ be the laws of $M(x)$ and $M(x')$ respectively, where $d(x, x') = 1$. I will break the proof into two parts:

The MGF of the privacy loss $L(p || q)$ is bounded by that of $L(r \mu_1 + (1 - r) \mu_0 || \mu_0)$ where $\mu_i = N(i, \sigma^2)$.
If $r \le c_1 \sigma^{-1}$ and $\lambda \le c_2 \sigma^2$, then there exists $C = C(c_1, c_2)$ such that $G_\lambda (r \mu_1 + (1 - r) \mu_0 || \mu_0) \le C$ (since $O(r^2 \lambda^2 / \sigma^2) = O(1)$).

Remark in the proof. Note that the choice of $c_1$, $c_2$ and the function $C(c_1, c_2)$ are important to the practicality and usefulness of Conjecture 0.

Part 1 can be derived using Conjecture 1. 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.

Now let us verify Part 2.

Using Claim 27 and Example 1, we have

\[\begin{aligned} G_\lambda(r \mu_1 + (1 - r) \mu_0 || \mu_0)) &= \sum_{j = 0 : \lambda} {\lambda \choose j} r^j (1 - r)^{\lambda - j} G_j(\mu_1 || \mu_0)\\ &=\sum_{j = 0 : \lambda} {\lambda \choose j} r^j (1 - r)^{\lambda - j} \exp(j (j - 1) / 2 \sigma^2). \qquad (9.5) \end{aligned}\]

Denote by $n = \lceil \sigma^2 \rceil$. It suffices to show

\[\sum_{j = 0 : n} {n \choose j} (c_1 n^{- 1 / 2})^j (1 - c_1 n^{- 1 / 2})^{n - j} \exp(c_2 j (j - 1) / 2 n) \le C\]

Note that we can discard the linear term $- c_2 j / \sigma^2$ in the exponential term since we want to bound the sum from above.

We examine the asymptotics of this sum when $n$ is large, and treat the sum as an approximation to an integration of a function $\phi: [0, 1] \to \mathbb R$. For $j = x n$, where $x \in (0, 1)$, $\phi$ is thus defined as (note we multiply the summand with $n$ to compensate the uniform measure on $1, ..., n$:

\[\begin{aligned} \phi_n(x) &:= n {n \choose j} (c_1 n^{- 1 / 2})^j (1 - c_1 n^{- 1 / 2})^{n - j} \exp(c_2 j^2 / 2 n) \\ &= n {n \choose x n} (c_1 n^{- 1 / 2})^{x n} (1 - c_1 n^{- 1 / 2})^{(1 - x) n} \exp(c_2 x^2 n / 2) \end{aligned}\]

Using Stirling's approximation

\[n! \approx \sqrt{2 \pi n} n^n e^{- n},\]

we can approach the binomial coefficient:

\[{n \choose x n} \approx (\sqrt{2 \pi x (1 - x)} x^{x n} (1 - x)^{(1 - x) n})^{-1}.\]

We also approximate

\[(1 - c_1 n^{- 1 / 2})^{(1 - x) n} \approx \exp(- c_1 \sqrt{n} (1 - x)).\]

With these we have

\[\phi_n(x) \approx {1 \over \sqrt{2 \pi x (1 - x)}} \exp\left(- {1 \over 2} x n \log n + (x \log c_1 - x \log x - (1 - x) \log (1 - x) + {1 \over 2} c_2 x^2) n + {1 \over 2} \log n\right).\]

This vanishes as $n \to \infty$, and since $\phi_n(x)$ is bounded above by the integrable function ${1 \over \sqrt{2 \pi x (1 - x)}}$ (c.f. the arcsine law), and below by $0$, we may invoke the dominant convergence theorem and exchange the integral with the limit and get

\[\begin{aligned} \lim_{n \to \infty} &G_n (r \mu_1 + (1 - r) \mu_0 || \mu_0)) \\ &\le \lim_{n \to \infty} \int \phi_n(x) dx = \int \lim_{n \to \infty} \phi_n(x) dx = 0. \end{aligned}\]

Thus we have that the generating function of the divergence variable $L(r \mu_1 + (1 - r) \mu_0 || \mu_0)$ is bounded.

Can this be true for better orders

\[r \le c_1 \sigma^{- d_r},\qquad \lambda \le c_2 \sigma^{d_\lambda}\]

for some $d_r \in (0, 1]$ and $d_\lambda \in [2, \infty)$? If we follow the same approximation using these exponents, then letting $n = c_2 \sigma^{d_\lambda}$,

\[\begin{aligned} {n \choose j} &r^j (1 - r)^{n - j} G_j(\mu_0 || \mu_1) \le \phi_n(x) \\ &\approx {1 \over \sqrt{2 \pi x (1 - x)}} \exp\left({1 \over 2} c_2^{2 \over d_\lambda} x^2 n^{2 - {2 \over d_\lambda}} - {d_r \over 2} x n \log n + (x \log c_1 - x \log x - (1 - x) \log (1 - x)) n + {1 \over 2} \log n\right). \end{aligned}\]

So we see that to keep the divergence moments bounded it is possible to have any $r = O(\sigma^{- d_r})$ for $d_r \in (0, 1)$, but relaxing $\lambda$ may not be safe.

If we relax $r$, then we get

\[G_\lambda(r \mu_1 + (1 - r) \mu_0 || \mu_0) = O(r^{2 / d_r} \lambda^2 \sigma^{-2}) = O(1).\]

Note that now the constant $C$ depends on $d_r$ as well. Numerical experiments seem to suggest that $C$ can increase quite rapidly as $d_r$ decreases from $1$. $\square$

In the following for consistency we retain $k$ as the number of epochs, and use $T := k / r$ to denote the number of compositions / steps / minibatches. With Conjecture 0 we have:

Claim 28. Assuming Conjecture 1 is true. Let $\epsilon, c_1, c_2 > 0$, $r \le c_1 \sigma^{-1}$, $T = {c_2 \over 2 C(c_1, c_2)} \epsilon \sigma^2$. then the DP-SGD with subsampling rate $r$, and $T$ steps is $(\epsilon, \delta)$-dp for

\[\delta = \exp(- {1 \over 2} c_2 \sigma^2 \epsilon).\]

In other words, for

\[\sigma \ge \sqrt{2 c_2^{-1}} \epsilon^{- {1 \over 2}} \sqrt{\log \delta^{-1}},\]

we can achieve $(\epsilon, \delta)$-dp.

Proof. By Claim 26 and the Moment Composition Theorem (Claim 22), for $\lambda = c_2 \sigma^2$, substituting $T = {c_2 \over 2 C(c_1, c_2)} \epsilon \sigma^2$, we have

\[\mathbb P(L(p || q) \ge \epsilon) \le \exp(k C(c_1, c_2) - \lambda \epsilon) = \exp\left(- {1 \over 2} c_2 \sigma^2 \epsilon\right).\]

$\square$

Remark. 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:

There exists constants $c_1', c_2' > 0$ so that for any $\epsilon < c_1' r^2 T$, DP-SGD is $(\epsilon, \delta)$-differentially private for any $\delta > 0$ if we choose

\[\sigma \ge c_2' {r \sqrt{T \log (1 / \delta)} \over \epsilon}. \qquad (10)\]

I am however unable to reproduce this version, assuming Conjecture 0 is true, for the following reasons:

In the proof in the paper, we have $\epsilon = c_1' r^2 T$ instead of "less than" in the statement of the Theorem. If we change it to $\epsilon < c_1' r^2 T$ then the direction of the inequality becomes opposite to the direction we want to prove: \[\exp(k C(c_1, c_2) - \lambda \epsilon) \ge ...\]
The implicit condition $r = O(\sigma^{-1})$ of Conjecture 0 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 $\sigma$.

Tensorflow implementation

The DP-SGD is implemented in TensorFlow Privacy. In the following I discuss the package in the current state (2019-03-11). It is divided into two parts: optimizers which implements the actual differentially private algorithms, and analysis which computes the privacy guarantee.

The analysis 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 rdp_accountant.py. At this moment, rdp_accountant.py 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.

Some notational correspondences: their alpha is our $\lambda$, their q is our $r$, their A_alpha (in the comments) is our $\kappa_{r N(1, \sigma^2) + (1 - r) N(0, \sigma^2)} (\lambda - 1)$, at least when $\lambda$ is an integer.

The function _compute_log_a presumably computes the cumulants $\kappa_{r N(1, \sigma^2) + (1 - r) N(0, \sigma^2), N(0, \sigma^2)}(\lambda - 1)$. It calls _compute_log_a_int or _compute_log_a_frac depending on whether $\lambda$ is an integer.
The function _compute_log_a_int computes the cumulant using (9.5).
When $\lambda$ is not an integer, we can't use (9.5). I have yet to decode how _compute_log_a_frac computes the cumulant (or an upper bound of it) in this case
The function _compute_delta computes $\delta$s for a list of $\lambda$s and $\kappa$s using Item 1 of Claim 25 and return the smallest one, and the function _compute_epsilon computes epsilon uses Item 3 in Claim 25 in the same way.

In optimizers, among other things, the DP-SGD with Gaussian mechanism is implemented in dp_optimizer.py and gaussian_query.py. See the definition of DPGradientDescentGaussianOptimizer in dp_optimizer.py and trace the calls therein.

At this moment, the privacy guarantee computation part and the optimizer part are separated, with rdp_accountant.py called in compute_dp_sgd_privacy.py with user-supplied parameters. I think this is due to the lack of implementation in rdp_accountant.py of any non-DPSGD-with-Gaussian privacy guarantee computation. There is already an issue on this, so hopefully it won't be long before the privacy guarantees can be automatically computed given a DP-SGD instance.

Comparison among different methods

So far we have seen three routes to compute the privacy guarantees for DP-SGD with the Gaussian mechanism:

Claim 9 (single Gaussian mechanism privacy guarantee) -> Claim 19 (Subsampling theorem) -> Claim 18 (Advanced Adaptive Composition Theorem)
Example 1 (RDP for the Gaussian mechanism) -> Claim 22 (Moment Composition Theorem) -> Example 3 (Moment composition applied to the Gaussian mechanism)
Claim 26 (RDP for Gaussian mechanism with specific magnitudes for subsampling rate) -> Claim 28 (Moment Composition Theorem and translation to conventional DP)

Which one is the best?

To make fair comparison, we may use one parameter as the metric and set all others to be the same. For example, we can

Given the same $\epsilon$, $r$ (in Route 1 and 3), $k$, $\sigma$, compare the $\delta$s
Given the same $\epsilon$, $r$ (in Route 1 and 3), $k$, $\delta$, compare the $\sigma$s
Given the same $\delta$, $r$ (in Route 1 and 3), $k$, $\sigma$, compare the $\epsilon$s.

I find that the first one, where $\delta$ is used as a metric, the best. This is because we have the tightest bounds and the cleanest formula when comparing the $\delta$. For example, the Azuma and Chernoff bounds are both expressed as a bound for $\delta$. On the other hand, the inversion of these bounds either requires a cost in the tightness (Claim 9, bounds on $\sigma$) or in the complexity of the formula (Claim 16 Advanced Adaptive Composition Theorem, bounds on $\epsilon$).

So if we use $\sigma$ or $\epsilon$ as a metric, either we get a less fair comparison, or have to use a much more complicated formula as the bounds.

Let us first compare Route 1 and Route 2 without specialising to the Gaussian mechanism.

Disclaimer. What follows is a bit messy and has not been reviewed by anyone.

Suppose each mechanism $N_i$ satisfies $(\epsilon', \delta(\epsilon'))$-dp. Let $\tilde \epsilon := \log (1 + r (e^{\epsilon'} - 1))$, then we have the subsampled mechanism $M_i(x) = N_i(x_\gamma)$ is $(\tilde \epsilon, r \tilde \delta(\tilde \epsilon))$-dp, where

\[\tilde \delta(\tilde \epsilon) = \delta(\log (r^{-1} (\exp(\tilde \epsilon) - 1) + 1))\]

Using the Azuma bound in the proof of Advanced Adaptive Composition Theorem (6.99):

\[\mathbb P(L(p^k || q^k) \ge \epsilon) \le \exp(- {(\epsilon - r^{-1} k a(\tilde\epsilon))^2 \over 2 r^{-1} k (\tilde\epsilon + a(\tilde\epsilon))^2}).\]

So we have the final bound for Route 1:

\[\delta_1(\epsilon) = \min_{\tilde \epsilon: \epsilon > r^{-1} k a(\tilde \epsilon)} \exp(- {(\epsilon - r^{-1} k a(\tilde\epsilon))^2 \over 2 r^{-1} k (\tilde\epsilon + a(\tilde\epsilon))^2}) + k \tilde \delta(\tilde \epsilon).\]

As for Route 2, since we do not gain anything from subsampling in RDP, we do not subsample at all.

By Claim 23, we have the bound for Route 2:

\[\delta_2(\epsilon) = \exp(- k \kappa^* (\epsilon / k)).\]

On one hand, one can compare $\delta_1$ and $\delta_2$ with numerical experiments. On the other hand, if we further specify $\delta(\epsilon')$ in Route 1 as the Chernoff bound for the cumulants of divergence variable, i.e.

\[\delta(\epsilon') = \exp(- \kappa^* (\epsilon')),\]

we have

\[\delta_1 (\epsilon) = \min_{\tilde \epsilon: a(\tilde \epsilon) < r k^{-1} \epsilon} \exp(- {(\epsilon - r^{-1} k a(\tilde\epsilon))^2 \over 2 r^{-1} k (\tilde\epsilon + a(\tilde\epsilon))^2}) + k \exp(- \kappa^* (b(\tilde\epsilon))),\]

where

\[b(\tilde \epsilon) := \log (r^{-1} (\exp(\tilde \epsilon) - 1) + 1) \le r^{-1} \tilde\epsilon.\]

We note that since $a(\tilde \epsilon) = \tilde\epsilon(e^{\tilde \epsilon} - 1) 1_{\tilde\epsilon < \log 2} + \tilde\epsilon 1_{\tilde\epsilon \ge \log 2}$, we may compare the two cases separately.

Note that we have $\kappa^*$ is a monotonously increasing function, therefore

\[\kappa^* (b(\tilde\epsilon)) \le \kappa^*(r^{-1} \tilde\epsilon).\]

So for $\tilde \epsilon \ge \log 2$, we have

\[k \exp(- \kappa^*(b(\tilde\epsilon))) \ge k \exp(- \kappa^*(r^{-1} \tilde \epsilon)) \ge k \exp(- \kappa^*(k^{-1} \epsilon)) \ge \delta_2(\epsilon).\]

For $\tilde\epsilon < \log 2$, it is harder to compare, as now

\[k \exp(- \kappa^*(b(\tilde\epsilon))) \ge k \exp(- \kappa^*(\epsilon / \sqrt{r k})).\]

It is tempting to believe that this should also be greater than $\delta_2(\epsilon)$. But I can not say for sure. At least in the special case of Gaussian, we have

\[k \exp(- \kappa^*(\epsilon / \sqrt{r k})) = k \exp(- (\sigma \sqrt{\epsilon / k r} - (2 \sigma)^{-1})^2) \ge \exp(- k ({\sigma \epsilon \over k} - (2 \sigma)^{-1})^2) = \delta_2(\epsilon)\]

when $\epsilon$ is sufficiently small. However we still need to consider the case where $\epsilon$ is not too small. But overall it seems most likely Route 2 is superior than Route 1.

So let us compare Route 2 with Route 3:

Given the condition to obtain the Chernoff bound

\[{\sigma \epsilon \over k} > (2 \sigma)^{-1}\]

we have

\[\delta_2(\epsilon) > \exp(- k (\sigma \epsilon / k)^2) = \exp(- \sigma^2 \epsilon^2 / k).\]

For this to achieve the same bound

\[\delta_3(\epsilon) = \exp\left(- {1 \over 2} c_2 \sigma^2 \epsilon\right)\]

we need $k < {2 \epsilon \over c_2}$. This is only possible if $c_2$ is small or $\epsilon$ is large, since $k$ is a positive integer.

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 $T$ needs to be at least $1$, meaning

\[{c_2 \over C(c_1, c_2)} \epsilon \sigma^2 \ge 1.\]

Second, $k$ needs to be at least $1$ as well, i.e.

\[k = r T \ge {c_1 c_2 \over C(c_1, c_2)} \epsilon \sigma \ge 1.\]

Both conditions rely on the magnitudes of $\epsilon$, $\sigma$, $c_1$, $c_2$, and the rate of growth of $C(c_1, c_2)$. The biggest problem in this list is the last, because if we know how fast $C$ grows then we'll have a better idea what are the constraints for the parameters to achieve the result in Route 3.

Further questions

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:

Prove Conjecture 1
Find a theoretically definitive answer whether the methods in Part 1 or Part 2 yield better privacy guarantees.
Study the non-Gaussian cases, general or specific. Let $p$ be some probability density, what is the tail bound of $L(p(y) || p(y + \alpha))$ for $|\alpha| \le 1$? 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?
Find out how useful Claim 26 is. Perhaps start with computing the constant $C$ nemerically.
Help with the aforementioned issue in the Tensorflow privacy package.

References

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. https://doi.org/10.1145/2976749.2978318.
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. https://doi.org/10.1109/TIT.2014.2320500.
Gil, M., F. Alajaji, and T. Linder. “Rényi Divergence Measures for Commonly Used Univariate Continuous Distributions.” Information Sciences 249 (November 2013): 124–31. https://doi.org/10.1016/j.ins.2013.06.018.
Mironov, Ilya. “Renyi Differential Privacy.” 2017 IEEE 30th Computer Security Foundations Symposium (CSF), August 2017, 263–75. https://doi.org/10.1109/CSF.2017.11.

A Tail of Two Densities

2019-03-13T00:00:00Z

This is Part 1 of a two-part post where I give an introduction to differential privacy, which is a study of tail bounds of the divergence between probability measures, with the end goal of applying it to stochastic gradient descent.

I start with the definition of $\epsilon$-differential privacy (corresponding to max divergence), followed by $(\epsilon, \delta)$-differential privacy (a.k.a. approximate differential privacy, corresponding to the $\delta$-approximate max divergence). I show a characterisation of the $(\epsilon, \delta)$-differential privacy as conditioned $\epsilon$-differential privacy. Also, as examples, I illustrate the $\epsilon$-dp with Laplace mechanism and, using some common tail bounds, the approximate dp with the Gaussian mechanism.

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).

In Part 2, I discuss the Rényi differential privacy, corresponding to the Rényi divergence, a study of the moment generating functions of the divergence between probability measures to derive the tail bounds.

Like in Part 1, I prove a composition theorem and a subsampling theorem.

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.

After that I explain the Tensorflow implementation of differential privacy in its Privacy module, which focuses on the differentially private stochastic gradient descent algorithm (DP-SGD).

Acknowledgement. I would like to thank Stockholm AI for introducing me to the subject of differential privacy. 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. The research was done while working at KTH Department of Mathematics.

If you are confused by any notations, ask me or try this. This post (including both Part 1 and Part2) is licensed under CC BY-SA and GNU FDL.

The gist of differential privacy

If you only have one minute, here is what differential privacy is about:

Let $p$ and $q$ be two probability densities, we define the divergence variable of $(p, q)$ to be

\[L(p || q) := \log {p(\xi) \over q(\xi)}\]

where $\xi$ is a random variable distributed according to $p$.

Roughly speaking, differential privacy is the study of the tail bound of $L(p || q)$: for certain $p$s and $q$s, and for $\epsilon > 0$, find $\delta(\epsilon)$ such that

\[\mathbb P(L(p || q) > \epsilon) < \delta(\epsilon),\]

where $p$ and $q$ 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:

(General case) $q$ is in the form of $q(y) = p(y + \Delta)$ for some bounded constant $\Delta$.
(Compositions) $p$ and $q$ are combinatorial or sequential compositions of some simpler $p_i$’s and $q_i$’s respectively
(Subsampling) $p$ and $q$ are mixtures / averages of some simpler $p_i$’s and $q_i$’s respectively

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.

Now if you have an hour...

$\epsilon$-dp

Definition (Mechanisms). Let $X$ be a space with a metric $d: X \times X \to \mathbb N$. A mechanism $M$ is a function that takes $x \in X$ as input and outputs a random variable on $Y$.

In this post, $X = Z^m$ is the space of datasets of $m$ rows for some integer $m$, where each item resides in $Z$. In this case the distance $d(x, x') := \#\{i: x_i \neq x'_i\}$ is the number of rows that differ between $x$ and $x'$.

Normally we have a query $f: X \to Y$, and construct the mechanism $M$ from $f$ by adding a noise:

\[M(x) := f(x) + \text{noise}.\]

Later, we will also consider mechanisms constructed from composition or mixture of other mechanisms.

In this post $Y = \mathbb R^d$ for some $d$.

Definition (Sensitivity). Let $f: X \to \mathbb R^d$ be a function. The sensitivity $S_f$ of $f$ is defined as

\[S_f := \sup_{x, x' \in X: d(x, x') = 1} \|f(x) - f(x')\|_2,\]

where $\|y\|_2 = \sqrt{y_1^2 + ... + y_d^2}$ is the $\ell^2$-norm.

Definition (Differential Privacy). A mechanism $M$ is called $\epsilon$-differential privacy ($\epsilon$-dp) if it satisfies the following condition: for all $x, x' \in X$ with $d(x, x') = 1$, and for all measureable set $S \subset \mathbb R^n$,

\[\mathbb P(M(x) \in S) \le e^\epsilon P(M(x') \in S). \qquad (1)\]

An example of $\epsilon$-dp mechanism is the Laplace mechanism.

Definition. The Laplace distribution over $\mathbb R$ with parameter $b > 0$ has probability density function

\[f_{\text{Lap}(b)}(x) = {1 \over 2 b} e^{- {|x| \over b}}.\]

Definition. Let $d = 1$. The Laplace mechanism is defined by

\[M(x) = f(x) + \text{Lap}(b).\]

Claim. The Laplace mechanism with

\[b \ge \epsilon^{-1} S_f \qquad (1.5)\]

is $\epsilon$-dp.

Proof. Quite straightforward. Let $p$ and $q$ be the laws of $M(x)$ and $M(x')$ respectively.

\[{p (y) \over q (y)} = {f_{\text{Lap}(b)} (y - f(x)) \over f_{\text{Lap}(b)} (y - f(x'))} = \exp(b^{-1} (|y - f(x')| - |y - f(x)|))\]

Using triangular inequality $|A| - |B| \le |A - B|$ on the right hand side, we have

\[{p (y) \over q (y)} \le \exp(b^{-1} (|f(x) - f(x')|)) \le \exp(\epsilon)\]

where in the last step we use the condition (1.5). $\square$

Approximate differential privacy

Unfortunately, $\epsilon$-dp does not apply to the most commonly used noise, the Gaussian noise. To fix this, we need to relax the definition a bit.

Definition. A mechanism $M$ is said to be $(\epsilon, \delta)$-differentially private if for all $x, x' \in X$ with $d(x, x') = 1$ and for all measureable $S \subset \mathbb R^d$

\[\mathbb P(M(x) \in S) \le e^\epsilon P(M(x') \in S) + \delta. \qquad (2)\]

Immediately we see that the $(\epsilon, \delta)$-dp is meaningful only if $\delta < 1$.

Indistinguishability

To understand $(\epsilon, \delta)$-dp, it is helpful to study $(\epsilon, \delta)$-indistinguishability.

Definition. Two probability measures $p$ and $q$ on the same space are called $(\epsilon, \delta)$-ind(istinguishable) if for all measureable sets $S$:

\[\begin{aligned} p(S) \le e^\epsilon q(S) + \delta, \qquad (3) \\ q(S) \le e^\epsilon p(S) + \delta. \qquad (4) \end{aligned}\]

As before, we also call random variables $\xi$ and $\eta$ to be $(\epsilon, \delta)$-ind if their laws are $(\epsilon, \delta)$-ind. When $\delta = 0$, we call it $\epsilon$-ind.

Immediately we have

Claim 0. $M$ is $(\epsilon, \delta)$-dp (resp. $\epsilon$-dp) iff $M(x)$ and $M(x')$ are $(\epsilon, \delta)$-ind (resp. $\epsilon$-ind) for all $x$ and $x'$ with distance $1$.

Definition (Divergence Variable). Let $p$ and $q$ be two probability measures. Let $\xi$ be a random variable distributed according to $p$, we define a random variable $L(p || q)$ by

\[L(p || q) := \log {p(\xi) \over q(\xi)},\]

and call it the divergence variable of $(p, q)$.

One interesting and readily verifiable fact is

\[\mathbb E L(p || q) = D(p || q)\]

where $D$ is the KL-divergence.

Claim 1. If

\[\begin{aligned} \mathbb P(L(p || q) \le \epsilon) &\ge 1 - \delta, \qquad(5) \\ \mathbb P(L(q || p) \le \epsilon) &\ge 1 - \delta \end{aligned}\]

then $p$ and $q$ are $(\epsilon, \delta)$-ind.

Proof. We verify (3), and (4) can be shown in the same way. Let $A := \{y \in Y: \log {p(y) \over q(y)} > \epsilon\}$, then by (5) we have

\[p(A) < \delta.\]

\[p(S) = p(S \cap A) + p(S \setminus A) \le \delta + e^\epsilon q(S \setminus A) \le \delta + e^\epsilon q(S).\]

$\square$

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.

In the following we discuss the converse of Claim 1. The discussions are rather technical, and readers can skip to the next subsection on first reading.

The converse of Claim 1 is not true.

Claim 2. There exists $\epsilon, \delta > 0$, and $p$ and $q$ that are $(\epsilon, \delta)$-ind, such that

\[\begin{aligned} \mathbb P(L(p || q) \le \epsilon) &< 1 - \delta, \\ \mathbb P(L(q || p) \le \epsilon) &< 1 - \delta \end{aligned}\]

Proof. Here's a example. Let $Y = \{0, 1\}$, and $p(0) = q(1) = 2 / 5$ and $p(1) = q(0) = 3 / 5$. Then it is not hard to verify that $p$ and $q$ are $(\log {4 \over 3}, {1 \over 3})$-ind: just check (3) for all four possible $S \subset Y$ and (4) holds by symmetry. On the other hand,

\[\mathbb P(L(p || q) \le \log {4 \over 3}) = \mathbb P(L(q || p) \le \log {4 \over 3}) = {2 \over 5} < {2 \over 3}.\]

$\square$

A weaker version of the converse of Claim 1 is true (Kasiviswanathan-Smith 2015), though:

Claim 3. Let $\alpha > 1$. If $p$ and $q$ are $(\epsilon, \delta)$-ind, then

\[\mathbb P(L(p || q) > \alpha \epsilon) < {1 \over 1 - \exp((1 - \alpha) \epsilon)} \delta.\]

Proof. Define

\[S = \{y: p(y) > e^{\alpha \epsilon} q(y)\}.\]

Then we have

\[e^{\alpha \epsilon} q(S) < p(S) \le e^\epsilon q(S) + \delta,\]

where the first inequality is due to the definition of $S$, and the second due to the $(\epsilon, \delta)$-ind. Therefore

\[q(S) \le {\delta \over e^{\alpha \epsilon} - e^\epsilon}.\]

Using the $(\epsilon, \delta)$-ind again we have

\[p(S) \le e^\epsilon q(S) + \delta = {1 \over 1 - e^{(1 - \alpha) \epsilon}} \delta.\]

$\square$

This can be quite bad if $\epsilon$ is small.

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 characterisation of indistinguishability.

Let us take a careful look at the condition in Claim 1 and call it C1:

C1. $\mathbb P(L(p || q) \le \epsilon) \ge 1 - \delta$ and $\mathbb P(L(q || p) \le \epsilon) \ge 1 - \delta$

It is equivalent to

C2. there exist events $A, B \subset Y$ with probabilities $p(A)$ and $q(B)$ at least $1 - \delta$ such that $\log p(y) - \log q(y) \le \epsilon$ for all $y \in A$ and $\log q(y) - \log p(y) \le \epsilon$ for all $y \in B$.

A similar-looking condition to C2 is the following:

C3. Let $\Omega$ be the underlying probability space. There exist two events $E, F \subset \Omega$ with $\mathbb P(E), \mathbb P(F) \ge 1 - \delta$, such that $|\log p_{|E}(y) - \log q_{|F}(y)| \le \epsilon$ for all $y \in Y$.

Here $p_{|E}$ (resp. $q_{|F}$) is $p$ (resp. $q$) conditioned on event $E$ (resp. $F$).

Remark. Note that the events in C2 and C3 are in different spaces, and therefore we can not write $p_{|E}(S)$ as $p(S | E)$ or $q_{|F}(S)$ as $q(S | F)$. In fact, if we let $E$ and $F$ in C3 be subsets of $Y$ with $p(E), q(F) \ge 1 - \delta$ and assume $p$ and $q$ have the same supports, then C3 degenerates to a stronger condition than C2. Indeed, in this case $p_E(y) = p(y) 1_{y \in E}$ and $q_F(y) = q(y) 1_{y \in F}$, and so $p_E(y) \le e^\epsilon q_F(y)$ forces $E \subset F$. We also obtain $F \subset E$ in the same way. This gives us $E = F$, and C3 becomes C2 with $A = B = E = F$.

As it turns out, C3 is the condition we need.

Claim 4. Two probability measures $p$ and $q$ are $(\epsilon, \delta)$-ind if and only if C3 holds.

Proof(Murtagh-Vadhan 2018). The "if" direction is proved in the same way as Lemma 1. Without loss of generality we may assume $\mathbb P(E) = \mathbb P(F) \ge 1 - \delta$. To see this, suppose $F$ has higher probability than $E$, then we can substitute $F$ with a subset of $F$ that has the same probability as $E$ (with possible enlargement of the probability space).

Let $\xi \sim p$ and $\eta \sim q$ be two independent random variables, then

\[\begin{aligned} p(S) &= \mathbb P(\xi \in S | E) \mathbb P(E) + \mathbb P(\xi \in S; E^c) \\ &\le e^\epsilon \mathbb P(\eta \in S | F) \mathbb P(E) + \delta \\ &= e^\epsilon \mathbb P(\eta \in S | F) \mathbb P(F) + \delta\\ &\le e^\epsilon q(S) + \delta. \end{aligned}\]

The "only-if" direction is more involved.

We construct events $E$ and $F$ by constructing functions $e, f: Y \to [0, \infty)$ satisfying the following conditions:

$0 \le e(y) \le p(y)$ and $0 \le f(y) \le q(y)$ for all $y \in Y$.
$|\log e(y) - \log f(y)| \le \epsilon$ for all $y \in Y$.
$e(Y), f(Y) \ge 1 - \delta$.
$e(Y) = f(Y)$.

Here for a set $S \subset Y$, $e(S) := \int_S e(y) dy$, and the same goes for $f(S)$.

Let $\xi \sim p$ and $\eta \sim q$. Then we define $E$ and $F$ by

\[\mathbb P(E | \xi = y) = e(y) / p(y) \\ \mathbb P(F | \eta = y) = f(y) / q(y).\]

Remark inside proof. This can seem a bit confusing. Intuitively, we can think of it this way when $Y$ is finite: Recall a random variable on $Y$ is a function from the probability space $\Omega$ to $Y$. Let event $G_y \subset \Omega$ be defined as $G_y = \xi^{-1} (y)$. We cut $G_y$ into the disjoint union of $E_y$ and $G_y \setminus E_y$ such that $\mathbb P(E_y) = e(y)$. Then $E = \bigcup_{y \in Y} E_y$. So $e(y)$ can be seen as the "density" of $E$.

Indeed, given $E$ and $F$ defined this way, we have

\[p_E(y) = {e(y) \over e(Y)} \le {\exp(\epsilon) f(y) \over e(Y)} = {\exp(\epsilon) f(y) \over f(Y)} = \exp(\epsilon) q_F(y).\]

and

\[\mathbb P(E) = \int \mathbb P(E | \xi = y) p(y) dy = e(Y) \ge 1 - \delta,\]

and the same goes for $\mathbb P(F)$.

What remains is to construct $e(y)$ and $f(y)$ satisfying the four conditions.

Like in the proof of Claim 1, let $S, T \subset Y$ be defined as

\[\begin{aligned} S := \{y: p(y) > \exp(\epsilon) q(y)\},\\ T := \{y: q(y) > \exp(\epsilon) p(y)\}. \end{aligned}\]

Let

\[\begin{aligned} e(y) &:= \exp(\epsilon) q(y) 1_{y \in S} + p(y) 1_{y \notin S}\\ f(y) &:= \exp(\epsilon) p(y) 1_{y \in T} + q(y) 1_{y \notin T}. \qquad (6) \end{aligned}\]

By checking them on the three disjoint subsets $S$, $T$, $(S \cup T)^c$, it is not hard to verify that the $e(y)$ and $f(y)$ constructed this way satisfy the first two conditions. They also satisfy the third condition:

\[\begin{aligned} e(Y) &= 1 - (p(S) - \exp(\epsilon) q(S)) \ge 1 - \delta, \\ f(Y) &= 1 - (q(T) - \exp(\epsilon) p(T)) \ge 1 - \delta. \end{aligned}\]

If $e(Y) = f(Y)$ then we are done. Otherwise, without loss of generality, assume $e(Y) < f(Y)$, then all it remains to do is to reduce the value of $f(y)$ while preserving Condition 1, 2 and 3, until $f(Y) = e(Y)$.

As it turns out, this can be achieved by reducing $f(y)$ on the set $\{y \in Y: q(y) > p(y)\}$. To see this, let us rename the $f(y)$ defined in (6) $f_+(y)$, and construct $f_-(y)$ by

\[f_-(y) := p(y) 1_{y \in T} + (q(y) \wedge p(y)) 1_{y \notin T}.\]

It is not hard to show that not only $e(y)$ and $f_-(y)$ also satisfy conditions 1-3, but

\[e(y) \ge f_-(y), \forall y \in Y,\]

and thus $e(Y) \ge f_-(Y)$. Therefore there exists an $f$ that interpolates between $f_-$ and $f_+$ with $f(Y) = e(Y)$. $\square$

To prove the adaptive composition theorem for approximate differential privacy, we need a similar claim (We use index shorthand $\xi_{< i} = \xi_{1 : i - 1}$ and similarly for other notations):

Claim 5. Let $\xi_{1 : i}$ and $\eta_{1 : i}$ be random variables. Let

\[\begin{aligned} p_i(S | y_{1 : i - 1}) := \mathbb P(\xi_i \in S | \xi_{1 : i - 1} = y_{1 : i - 1})\\ q_i(S | y_{1 : i - 1}) := \mathbb P(\eta_i \in S | \eta_{1 : i - 1} = y_{1 : i - 1}) \end{aligned}\]

be the conditional laws of $\xi_i | \xi_{< i}$ and $\eta_i | \eta_{< i}$ respectively. Then the following are equivalent:

For any $y_{< i} \in Y^{i - 1}$, $p_i(\cdot | y_{< i})$ and $q_i(\cdot | y_{< i})$ are $(\epsilon, \delta)$-ind
There exists events $E_i, F_i \subset \Omega$ with $\mathbb P(E_i | \xi_{<i} = y_{<i}) = \mathbb P(F_i | \eta_{<i} = y_{< i}) \ge 1 - \delta$ for any $y_{< i}$, such that $p_{i | E_i}(\cdot | y_{< i})$ and $q_{i | E_i} (\cdot | y_{< i})$ are $\epsilon$-ind for any $y_{< i}$, where \[\begin{aligned} p_{i | E_i}(S | y_{1 : i - 1}) := \mathbb P(\xi_i \in S | E_i, \xi_{1 : i - 1} = y_{1 : i - 1})\\ q_{i | F_i}(S | y_{1 : i - 1}) := \mathbb P(\eta_i \in S | F_i, \eta_{1 : i - 1} = y_{1 : i - 1}) \end{aligned}\]

are $p_i$ and $q_i$ conditioned on $E_i$ and $F_i$ respectively.

Proof. Item 2 => Item 1: as in the Proof of Claim 4,

\[\begin{aligned} p_i(S | y_{< i}) &= p_{i | E_i} (S | y_{< i}) \mathbb P(E_i | \xi_{< i} = y_{< i}) + p_{i | E_i^c}(S | y_{< i}) \mathbb P(E_i^c | \xi_{< i} = y_{< i}) \\ &\le p_{i | E_i} (S | y_{< i}) \mathbb P(E_i | \xi_{< i} = y_{< i}) + \delta \\ &= p_{i | E_i} (S | y_{< i}) \mathbb P(F_i | \xi_{< i} = y_{< i}) + \delta \\ &\le e^\epsilon q_{i | F_i} (S | y_{< i}) \mathbb P(F_i | \xi_{< i} = y_{< i}) + \delta \\ &= e^\epsilon q_i (S | y_{< i}) + \delta. \end{aligned}\]

The direction from $q_i(S | y_{< i}) \le e^\epsilon p_i(S | y_{< i}) + \delta$ can be shown in the same way.

Item 1 => Item 2: as in the Proof of Claim 4 we construct $e(y_{1 : i})$ and $f(y_{1 : i})$ as "densities" of events $E_i$ and $F_i$.

Let

\[\begin{aligned} e(y_{1 : i}) &:= e^\epsilon q_i(y_i | y_{< i}) 1_{y_i \in S_i(y_{< i})} + p_i(y_i | y_{< i}) 1_{y_i \notin S_i(y_{< i})}\\ f(y_{1 : i}) &:= e^\epsilon p_i(y_i | y_{< i}) 1_{y_i \in T_i(y_{< i})} + q_i(y_i | y_{< i}) 1_{y_i \notin T_i(y_{< i})}\\ \end{aligned}\]

where

\[\begin{aligned} S_i(y_{< i}) = \{y_i \in Y: p_i(y_i | y_{< i}) > e^\epsilon q_i(y_i | y_{< i})\}\\ T_i(y_{< i}) = \{y_i \in Y: q_i(y_i | y_{< i}) > e^\epsilon p_i(y_i | y_{< i})\}. \end{aligned}\]

Then $E_i$ and $F_i$ are defined as

\[\begin{aligned} \mathbb P(E_i | \xi_{\le i} = y_{\le i}) &= {e(y_{\le i}) \over p_i(y_{\le i})},\\ \mathbb P(F_i | \xi_{\le i} = y_{\le i}) &= {f(y_{\le i}) \over q_i(y_{\le i})}. \end{aligned}\]

The rest of the proof is almost the same as the proof of Lemma 2. $\square$

Back to approximate differential privacy

By Claim 0 and 1 we have

Claim 6. If for all $x, x' \in X$ with distance $1$

\[\mathbb P(L(M(x) || M(x')) \le \epsilon) \ge 1 - \delta,\]

then $M$ is $(\epsilon, \delta)$-dp.

Note that in the literature the divergence variable $L(M(x) || M(x'))$ is also called the privacy loss.

By Claim 0 and Claim 4 we have

Claim 7. $M$ is $(\epsilon, \delta)$-dp if and only if for every $x, x' \in X$ with distance $1$, there exist events $E, F \subset \Omega$ with $\mathbb P(E) = \mathbb P(F) \ge 1 - \delta$, $M(x) | E$ and $M(x') | F$ are $\epsilon$-ind.

We can further simplify the privacy loss $L(M(x) || M(x'))$, by observing the translational and scaling invariance of $L(\cdot||\cdot)$:

\[\begin{aligned} L(\xi || \eta) &\overset{d}{=} L(\alpha \xi + \beta || \alpha \eta + \beta), \qquad \alpha \neq 0. \qquad (6.1) \end{aligned}\]

With this and the definition

\[M(x) = f(x) + \zeta\]

for some random variable $\zeta$, we have

\[L(M(x) || M(x')) \overset{d}{=} L(\zeta || \zeta + f(x') - f(x)).\]

Without loss of generality, we can consider $f$ with sensitivity $1$, for

\[L(f(x) + S_f \zeta || f(x') + S_f \zeta) \overset{d}{=} L(S_f^{-1} f(x) + \zeta || S_f^{-1} f(x') + \zeta)\]

so for any noise $\zeta$ that achieves $(\epsilon, \delta)$-dp for a function with sensitivity $1$, we have the same privacy guarantee by for an arbitrary function with sensitivity $S_f$ by adding a noise $S_f \zeta$.

With Claim 6 we can show that the Gaussian mechanism is approximately differentially private. But first we need to define it.

Definition (Gaussian mechanism). Given a query $f: X \to Y$, the Gaussian mechanism $M$ adds a Gaussian noise to the query:

\[M(x) = f(x) + N(0, \sigma^2 I).\]

Some tail bounds for the Gaussian distribution will be useful.

Claim 8 (Gaussian tail bounds). Let $\xi \sim N(0, 1)$ be a standard normal distribution. Then for $t > 0$

\[\mathbb P(\xi > t) < {1 \over \sqrt{2 \pi} t} e^{- {t^2 \over 2}}, \qquad (6.3)\]

and

\[\mathbb P(\xi > t) < e^{- {t^2 \over 2}}. \qquad (6.5)\]

Proof. Both bounds are well known. The first can be proved using

\[\int_t^\infty e^{- {y^2 \over 2}} dy < \int_t^\infty {y \over t} e^{- {y^2 \over 2}} dy.\]

The second is shown using Chernoff bound. For any random variable $\xi$,

\[\mathbb P(\xi > t) < {\mathbb E \exp(\lambda \xi) \over \exp(\lambda t)} = \exp(\kappa_\xi(\lambda) - \lambda t), \qquad (6.7)\]

where $\kappa_\xi(\lambda) = \log \mathbb E \exp(\lambda \xi)$ is the cumulant of $\xi$. Since (6.7) holds for any $\lambda$, we can get the best bound by minimising $\kappa_\xi(\lambda) - \lambda t$ (a.k.a. the Legendre transformation). When $\xi$ is standard normal, we get (6.5). $\square$

Remark. We will use the Chernoff bound extensively in the second part of this post when considering Rényi differential privacy.

Claim 9. The Gaussian mechanism on a query $f$ is $(\epsilon, \delta)$-dp, where

\[\delta = \exp(- (\epsilon \sigma / S_f - (2 \sigma / S_f)^{-1})^2 / 2). \qquad (6.8)\]

Conversely, to achieve give $(\epsilon, \delta)$-dp, we may set

\[\sigma > \left(\epsilon^{-1} \sqrt{2 \log \delta^{-1}} + (2 \epsilon)^{- {1 \over 2}}\right) S_f \qquad (6.81)\]

\[\sigma > (\epsilon^{-1} (1 \vee \sqrt{(\log (2 \pi)^{-1} \delta^{-2})_+}) + (2 \epsilon)^{- {1 \over 2}}) S_f \qquad (6.82)\]

\[\sigma > \epsilon^{-1} \sqrt{\log e^\epsilon \delta^{-2}} S_f \qquad (6.83)\]

\[\sigma > \epsilon^{-1} (\sqrt{1 + \epsilon} \vee \sqrt{(\log e^\epsilon (2 \pi)^{-1} \delta^{-2})_+}) S_f. \qquad (6.84)\]

Proof. As discussed before we only need to consider the case where $S_f = 1$. Fix arbitrary $x, x' \in X$ with $d(x, x') = 1$. Let $\zeta = (\zeta_1, ..., \zeta_d) \sim N(0, I_d)$.

By Claim 6 it suffices to bound

\[\mathbb P(L(M(x) || M(x')) > \epsilon)\]

We have by the linear invariance of $L$,

\[L(M(x) || M(x')) = L(f(x) + \sigma \zeta || f(x') + \sigma \zeta) \overset{d}{=} L(\zeta|| \zeta + \Delta / \sigma),\]

where $\Delta := f(x') - f(x)$.

Plugging in the Gaussian density, we have

\[L(M(x) || M(x')) \overset{d}{=} \sum_i {\Delta_i \over \sigma} \zeta_i + \sum_i {\Delta_i^2 \over 2 \sigma^2} \overset{d}{=} {\|\Delta\|_2 \over \sigma} \xi + {\|\Delta\|_2^2 \over 2 \sigma^2}.\]

where $\xi \sim N(0, 1)$.

Hence

\[\mathbb P(L(M(x) || M(x')) > \epsilon) = \mathbb P(\zeta > {\sigma \over \|\Delta\|_2} \epsilon - {\|\Delta\|_2 \over 2 \sigma}).\]

Since $\|\Delta\|_2 \le S_f = 1$, we have

\[\mathbb P(L(M(x) || M(x')) > \epsilon) \le \mathbb P(\xi > \sigma \epsilon - (2 \sigma)^{-1}).\]

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 $\sigma > (2 \epsilon)^{- 1 / 2}$ here so that $\sigma \epsilon - (2 \sigma)^{-1} > 0$ and we can use the tail bounds.

Using (6.3) we have

\[\mathbb P(L(M(x) || M(x')) > \epsilon) < \exp(- (\epsilon \sigma - (2 \sigma)^{-1})^2 / 2).\]

This gives us (6.8).

To bound the right hand by $\delta$, we require

\[\epsilon \sigma - {1 \over 2 \sigma} > \sqrt{2 \log \delta^{-1}}. \qquad (6.91)\]

Solving this inequality we have

\[\sigma > {\sqrt{2 \log \delta^{-1}} + \sqrt{2 \log \delta^{-1} + 2 \epsilon} \over 2 \epsilon}.\]

Using $\sqrt{2 \log \delta^{-1} + 2 \epsilon} \le \sqrt{2 \log \delta^{-1}} + \sqrt{2 \epsilon}$, we can achieve the above inequality by having

\[\sigma > \epsilon^{-1} \sqrt{2 \log \delta^{-1}} + (2 \epsilon)^{-{1 \over 2}}.\]

This gives us (6.81).

Alternatively, we can use the concavity of $\sqrt{\cdot}$:

\[(2 \epsilon)^{-1} (\sqrt{2 \log \delta^{-1}} + \sqrt{2 \log \delta^{-1} + 2 \epsilon}) \le \epsilon^{-1} \sqrt{\log e^\epsilon \delta^{-2}},\]

which gives us (6.83)

Back to (6.9), if we use (6.5) instead, we need

\[\log t + {t^2 \over 2} > \log {(2 \pi)^{- 1 / 2} \delta^{-1}}\]

where $t = \epsilon \sigma - (2 \sigma)^{-1}$. This can be satisfied if

\[\begin{aligned} t &> 1 \qquad (6.93)\\ t &> \sqrt{\log (2 \pi)^{-1} \delta^{-2}}. \qquad (6.95) \end{aligned}\]

We can solve both inequalities as before and obtain

\[\sigma > \epsilon^{-1} (1 \vee \sqrt{(\log (2 \pi)^{-1} \delta^{-2})_+}) + (2 \epsilon)^{- {1 \over 2}},\]

\[\sigma > \epsilon^{-1}(\sqrt{1 + \epsilon} \vee \sqrt{(\log e^\epsilon (2 \pi)^{-1} \delta^{-2})_+}).\]

This gives us (6.82)(6.84). $\square$

When $\epsilon \le \alpha$ is bounded, by (6.83) (6.84) we can require either

\[\sigma > \epsilon^{-1} (\sqrt{\log e^\alpha \delta^{-2}}) S_f\]

\[\sigma > \epsilon^{-1} (\sqrt{1 + \alpha} \vee \sqrt{(\log (2 \pi)^{-1} e^\alpha \delta^{-2})_+}).\]

The second bound is similar to and slightly better than the one in Theorem A.1 of Dwork-Roth 2013, where $\alpha = 1$:

\[\sigma > \epsilon^{-1} \left({3 \over 2} \vee \sqrt{(2 \log {5 \over 4} \delta^{-1})_+}\right) S_f.\]

Note that the lower bound of ${3 \over 2}$ is implicitly required in the proof of Theorem A.1.

Composition theorems

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:

Definition (Independent composition). Let $M_1, ..., M_k$ be $k$ mechanisms with independent noises. The mechanism $M = (M_1, ..., M_k)$ is called the independent composition of $M_{1 : k}$.

To define the adaptive composition, let us motivate it with an example of gradient descent. Consider the loss function $\ell(x; \theta)$ of a neural network, where $\theta$ is the parameter and $x$ the input, gradient descent updates its parameter $\theta$ at each time $t$:

\[\theta_{t} = \theta_{t - 1} - \alpha m^{-1} \sum_{i = 1 : m} \nabla_\theta \ell(x_i; \theta) |_{\theta = \theta_{t - 1}}.\]

We may add privacy by adding noise $\zeta_t$ at each step:

\[\theta_{t} = \theta_{t - 1} - \alpha m^{-1} \sum_{i = 1 : m} \nabla_\theta \ell(x_i; \theta) |_{\theta = \theta_{t - 1}} + \zeta_t. \qquad (6.97)\]

Viewed as a sequence of mechanism, we have that at each time $t$, the mechanism $M_t$ takes input $x$, and outputs $\theta_t$. But $M_t$ also depends on the output of the previous mechanism $M_{t - 1}$. To this end, we define the adaptive composition.

Definition (Adaptive composition). Let $({M_i(y_{1 : i - 1})})_{i = 1 : k}$ be $k$ mechanisms with independent noises, where $M_1$ has no parameter, $M_2$ has one parameter in $Y$, $M_3$ has two parameters in $Y$ and so on. For $x \in X$, define $\xi_i$ recursively by

\[\begin{aligned} \xi_1 &:= M_1(x)\\ \xi_i &:= M_i(\xi_1, \xi_2, ..., \xi_{i - 1}) (x). \end{aligned}\]

The adaptive composition of $M_{1 : k}$ is defined by $M(x) := (\xi_1, \xi_2, ..., \xi_k)$.

The definition of adaptive composition may look a bit complicated, but the point is to describe $k$ mechanisms such that for each $i$, the output of the first, second, ..., $i - 1$th mechanisms determine the $i$th mechanism, like in the case of gradient descent.

It is not hard to write down the differentially private gradient descent as a sequential composition:

\[M_t(\theta_{1 : t - 1})(x) = \theta_{t - 1} - \alpha m^{-1} \sum_{i = 1 : m} \nabla_\theta \ell(x_i; \theta) |_{\theta = \theta_{t - 1}} + \zeta_t.\]

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.

It is not hard to see that the adaptive composition degenerates to independent composition when each $M_i(y_{1 : i})$ evaluates to the same mechanism regardless of $y_{1 : i}$, in which case the $\xi_i$s are independent.

In the following when discussing adaptive compositions we sometimes omit the parameters for convenience without risk of ambiguity, and write $M_i(y_{1 : i})$ as $M_i$, but keep in mind of the dependence on the parameters.

It is time to state and prove the composition theorems. In this section we consider $2 \times 2 \times 2 = 8$ cases, i.e. situations of three dimensions, where there are two choices in each dimension:

Composition of $\epsilon$-dp or more generally $(\epsilon, \delta)$-dp mechanisms
Composition of independent or more generally adaptive mechanisms
Basic or advanced compositions

Note that in the first two dimensions the second choice is more general than the first.

The proofs of these composition theorems will be laid out as follows:

Claim 10 - Basic composition theorem for $(\epsilon, \delta)$-dp with adaptive mechanisms: by a direct proof with an induction argument
Claim 14 - Advanced composition theorem for $\epsilon$-dp with independent mechanisms: by factorising privacy loss and using Hoeffding's Inequality
Claim 16 - Advanced composition theorem for $\epsilon$-dp with adaptive mechanisms: by factorising privacy loss and using Azuma's Inequality
Claims 17 and 18 - Advanced composition theorem for $(\epsilon, \delta)$-dp with independent / adaptive mechanisms: by using characterisations of $(\epsilon, \delta)$-dp in Claims 4 and 5 as an approximation of $\epsilon$-dp and then using Proofs in Item 2 or 3.

Claim 10 (Basic composition theorem). Let $M_{1 : k}$ be $k$ mechanisms with independent noises such that for each $i$ and $y_{1 : i - 1}$, $M_i(y_{1 : i - 1})$ is $(\epsilon_i, \delta_i)$-dp. Then the adpative composition of $M_{1 : k}$ is $(\sum_i \epsilon_i, \sum_i \delta_i)$-dp.

Proof (Dwork-Lei 2009, see also Dwork-Roth 2013 Appendix B.1). Let $x$ and $x'$ be neighbouring points in $X$. Let $M$ be the adaptive composition of $M_{1 : k}$. Define

\[\xi_{1 : k} := M(x), \qquad \eta_{1 : k} := M(x').\]

Let $p^i$ and $q^i$ be the laws of $(\xi_{1 : i})$ and $(\eta_{1 : i})$ respectively.

Let $S_1, ..., S_k \subset Y$ and $T_i := \prod_{j = 1 : i} S_j$. We use two tricks.

Since $\xi_i | \xi_{< i} = y_{< i}$ and $\eta_i | \eta_{< i} = y_{< i}$ are $(\epsilon_i, \delta_i)$-ind, and a probability is no greater than $1$, \[\begin{aligned} \mathbb P(\xi_i \in S_i | \xi_{< i} = y_{< i}) &\le (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) + \delta_i) \wedge 1 \\ &\le (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) + \delta_i) \wedge (1 + \delta_i) \\ &= (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) \wedge 1) + \delta_i \end{aligned}\]
Given $p$ and $q$ that are $(\epsilon, \delta)$-ind, define \[\mu(x) = (p(x) - e^\epsilon q(x))_+.\]

We have \[\mu(S) \le \delta, \forall S\]

In the following we define $\mu^{i - 1} = (p^{i - 1} - e^\epsilon q^{i - 1})_+$ for the same purpose.

We use an inductive argument to prove the theorem:

\[\begin{aligned} \mathbb P(\xi_{\le i} \in T_i) &= \int_{T_{i - 1}} \mathbb P(\xi_i \in S_i | \xi_{< i} = y_{< i}) p^{i - 1} (y_{< i}) dy_{< i} \\ &\le \int_{T_{i - 1}} (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) \wedge 1) p^{i - 1}(y_{< i}) dy_{< i} + \delta_i\\ &\le \int_{T_{i - 1}} (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) \wedge 1) (e^{\epsilon_1 + ... + \epsilon_{i - 1}} q^{i - 1}(y_{< i}) + \mu^{i - 1} (y_{< i})) dy_{< i} + \delta_i\\ &\le \int_{T_{i - 1}} e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) e^{\epsilon_1 + ... + \epsilon_{i - 1}} q^{i - 1}(y_{< i}) dy_{< i} + \mu_{i - 1}(T_{i - 1}) + \delta_i\\ &\le e^{\epsilon_1 + ... + \epsilon_i} \mathbb P(\eta_{\le i} \in T_i) + \delta_1 + ... + \delta_{i - 1} + \delta_i.\\ \end{aligned}\]

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. $1$) in the first braket with the second term in the second braket (i.e. the $\mu$ term); in the last line we use Trick 2.

The base case $i = 1$ is true since $M_1$ is $(\epsilon_1, \delta_1)$-dp. $\square$

To prove the advanced composition theorem, we start with some lemmas.

Claim 11. If $p$ and $q$ are $\epsilon$-ind, then

\[D(p || q) + D(q || p) \le \epsilon(e^\epsilon - 1).\]

Proof. Since $p$ and $q$ are $\epsilon$-ind, we have $|\log p(x) - \log q(x)| \le \epsilon$ for all $x$. Let $S := \{x: p(x) > q(x)\}$. Then we have on

\[\begin{aligned} D(p || q) + D(q || p) &= \int (p(x) - q(x)) (\log p(x) - \log q(x)) dx\\ &= \int_S (p(x) - q(x)) (\log p(x) - \log q(x)) dx + \int_{S^c} (q(x) - p(x)) (\log q(x) - \log p(x)) dx\\ &\le \epsilon(\int_S p(x) - q(x) dx + \int_{S^c} q(x) - p(x) dx) \end{aligned}\]

Since on $S$ we have $q(x) \le p(x) \le e^\epsilon q(x)$, and on $S^c$ we have $p(x) \le q(x) \le e^\epsilon p(x)$, we obtain

\[D(p || q) + D(q || p) \le \epsilon \int_S (1 - e^{-\epsilon}) p(x) dx + \epsilon \int_{S^c} (e^{\epsilon} - 1) p(x) dx \le \epsilon (e^{\epsilon} - 1),\]

where in the last step we use $e^\epsilon - 1 \ge 1 - e^{- \epsilon}$ and $p(S) + p(S^c) = 1$. $\square$

Claim 12. If $p$ and $q$ are $\epsilon$-ind, then

\[D(p || q) \le a(\epsilon) \ge D(q || p),\]

where

\[a(\epsilon) = \epsilon (e^\epsilon - 1) 1_{\epsilon \le \log 2} + \epsilon 1_{\epsilon > \log 2} \le (\log 2)^{-1} \epsilon^2 1_{\epsilon \le \log 2} + \epsilon 1_{\epsilon > \log 2}. \qquad (6.98)\]

Proof. Since $p$ and $q$ are $\epsilon$-ind, we have

\[D(p || q) = \mathbb E_{\xi \sim p} \log {p(\xi) \over q(\xi)} \le \max_y {\log p(y) \over \log q(y)} \le \epsilon.\]

Comparing the quantity in Claim 11 ($\epsilon(e^\epsilon - 1)$) with the quantity above ($\epsilon$), we arrive at the conclusion. $\square$

Claim 13 (Hoeffding's Inequality). Let $L_i$ be independent random variables with $|L_i| \le b$, and let $L = L_1 + ... + L_k$, then for $t > 0$,

\[\mathbb P(L - \mathbb E L \ge t) \le \exp(- {t^2 \over 2 k b^2}).\]

Claim 14 (Advanced Independent Composition Theorem) ($\delta = 0$). Fix $0 < \beta < 1$. Let $M_1, ..., M_k$ be $\epsilon$-dp, then the independent composition $M$ of $M_{1 : k}$ is $(k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} \epsilon, \beta)$-dp.

Remark. By (6.98) we know that $k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} \epsilon = \sqrt{2 k \log \beta^{-1}} \epsilon + k O(\epsilon^2)$ when $\epsilon$ is sufficiently small, in which case the leading term is of order $O(\sqrt k \epsilon)$ and we save a $\sqrt k$ in the $\epsilon$-part compared to the Basic Composition Theorem (Claim 10).

Remark. In practice one can try different choices of $\beta$ and settle with the one that gives the best privacy guarantee. See the discussions at the end of Part 2 of this post.

Proof. Let $p_i$, $q_i$, $p$ and $q$ be the laws of $M_i(x)$, $M_i(x')$, $M(x)$ and $M(x')$ respectively.

\[\mathbb E L_i = D(p_i || q_i) \le a(\epsilon),\]

where $L_i := L(p_i || q_i)$. Due to $\epsilon$-ind also have

\[|L_i| \le \epsilon.\]

Therefore, by Hoeffding's Inequality,

\[\mathbb P(L - k a(\epsilon) \ge t) \le \mathbb P(L - \mathbb E L \ge t) \le \exp(- t^2 / 2 k \epsilon^2),\]

where $L := \sum_i L_i = L(p || q)$.

Plugging in $t = \sqrt{2 k \epsilon^2 \log \beta^{-1}}$, we have

\[\mathbb P(L(p || q) \le k a(\epsilon) + \sqrt{2 k \epsilon^2 \log \beta^{-1}}) \ge 1 - \beta.\]

Similarly we also have

\[\mathbb P(L(q || p) \le k a(\epsilon) + \sqrt{2 k \epsilon^2 \log \beta^{-1}}) \ge 1 - \beta.\]

By Claim 1 we arrive at the conclusion. $\square$

Claim 15 (Azuma's Inequality). Let $X_{0 : k}$ be a supermartingale. If $|X_i - X_{i - 1}| \le b$, then

\[\mathbb P(X_k - X_0 \ge t) \le \exp(- {t^2 \over 2 k b^2}).\]

Azuma's Inequality implies a slightly weaker version of Hoeffding's Inequality. To see this, let $L_{1 : k}$ be independent variables with $|L_i| \le b$. Let $X_i = \sum_{j = 1 : i} L_j - \mathbb E L_j$. Then $X_{0 : k}$ is a martingale, and

\[| X_i - X_{i - 1} | = | L_i - \mathbb E L_i | \le 2 b,\]

since $\|L_i\|_1 \le \|L_i\|_\infty$. Hence by Azuma's Inequality,

\[\mathbb P(L - \mathbb E L \ge t) \le \exp(- {t^2 \over 8 k b^2}).\]

Of course here we have made no assumption on $\mathbb E L_i$. If instead we have some bound for the expectation, say $|\mathbb E L_i| \le a$, then by the same derivation we have

\[\mathbb P(L - \mathbb E L \ge t) \le \exp(- {t^2 \over 2 k (a + b)^2}).\]

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 $\delta = 0$.

Claim 16 (Advanced Adaptive Composition Theorem) ($\delta = 0$). Let $\beta > 0$. Let $M_{1 : k}$ be $k$ mechanisms with independent noises such that for each $i$ and $y_{1 : i}$, $M_i(y_{1 : i})$ is $(\epsilon, 0)$-dp. Then the adpative composition of $M_{1 : k}$ is $(k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} (\epsilon + a(\epsilon)), \beta)$-dp.

Proof. As before, let $\xi_{1 : k} \overset{d}{=} M(x)$ and $\eta_{1 : k} \overset{d}{=} M(x')$, where $M$ is the adaptive composition of $M_{1 : k}$. Let $p_i$ (resp. $q_i$) be the law of $\xi_i | \xi_{< i}$ (resp. $\eta_i | \eta_{< i}$). Let $p^i$ (resp. $q^i$) be the law of $\xi_{\le i}$ (resp. $\eta_{\le i}$). We want to construct supermartingale $X$. To this end, let

\[X_i = \log {p^i(\xi_{\le i}) \over q^i(\xi_{\le i})} - i a(\epsilon) \]

We show that $(X_i)$ is a supermartingale:

\[\begin{aligned} \mathbb E(X_i - X_{i - 1} | X_{i - 1}) &= \mathbb E \left(\log {p_i (\xi_i | \xi_{< i}) \over q_i (\xi_i | \xi_{< i})} - a(\epsilon) | \log {p^{i - 1} (\xi_{< i}) \over q^{i - 1} (\xi_{< i})}\right) \\ &= \mathbb E \left( \mathbb E \left(\log {p_i (\xi_i | \xi_{< i}) \over q_i (\xi_i | \xi_{< i})} | \xi_{< i}\right) | \log {p^{i - 1} (\xi_{< i}) \over q^{i - 1} (\xi_{< i})}\right) - a(\epsilon) \\ &= \mathbb E \left( D(p_i (\cdot | \xi_{< i}) || q_i (\cdot | \xi_{< i})) | \log {p^{i - 1} (\xi_{< i}) \over q^{i - 1} (\xi_{< i})}\right) - a(\epsilon) \\ &\le 0, \end{aligned}\]

since by Claim 12 $D(p_i(\cdot | y_{< i}) || q_i(\cdot | y_{< i})) \le a(\epsilon)$ for all $y_{< i}$.

Since

\[| X_i - X_{i - 1} | = | \log {p_i(\xi_i | \xi_{< i}) \over q_i(\xi_i | \xi_{< i})} - a(\epsilon) | \le \epsilon + a(\epsilon),\]

by Azuma's Inequality,

\[\mathbb P(\log {p^k(\xi_{1 : k}) \over q^k(\xi_{1 : k})} \ge k a(\epsilon) + t) \le \exp(- {t^2 \over 2 k (\epsilon + a(\epsilon))^2}). \qquad(6.99)\]

Let $t = \sqrt{2 k \log \beta^{-1}} (\epsilon + a(\epsilon))$ we are done. $\square$

Claim 17 (Advanced Independent Composition Theorem). Fix $0 < \beta < 1$. Let $M_1, ..., M_k$ be $(\epsilon, \delta)$-dp, then the independent composition $M$ of $M_{1 : k}$ is $(k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} \epsilon, k \delta + \beta)$-dp.

Proof. By Claim 4, there exist events $E_{1 : k}$ and $F_{1 : k}$ such that

The laws $p_{i | E_i}$ and $q_{i | F_i}$ are $\epsilon$-ind.
$\mathbb P(E_i), \mathbb P(F_i) \ge 1 - \delta$.

Let $E := \bigcap E_i$ and $F := \bigcap F_i$, then they both have probability at least $1 - k \delta$, and $p_{i | E}$ and $q_{i | F}$ are $\epsilon$-ind.

By Claim 14, $p_{|E}$ and $q_{|F}$ are $(\epsilon' := k a(\epsilon) + \sqrt{2 k \epsilon^2 \log \beta^{-1}}, \beta)$-ind. Let us shrink the bigger event between $E$ and $F$ so that they have equal probabilities. Then

\[\begin{aligned} p (S) &\le p_{|E}(S) \mathbb P(E) + \mathbb P(E^c) \\ &\le (e^{\epsilon'} q_{|F}(S) + \beta) \mathbb P(F) + k \delta\\ &\le e^{\epsilon'} q(S) + \beta + k \delta. \end{aligned}\]

$\square$

Claim 18 (Advanced Adaptive Composition Theorem). Fix $0 < \beta < 1$. Let $M_{1 : k}$ be $k$ mechanisms with independent noises such that for each $i$ and $y_{1 : i}$, $M_i(y_{1 : i})$ is $(\epsilon, \delta)$-dp. Then the adpative composition of $M_{1 : k}$ is $(k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} (\epsilon + a(\epsilon)), \beta + k \delta)$-dp.

Proof. By Claim 5, there exist events $E_{1 : k}$ and $F_{1 : k}$ such that

The laws $p_{i | E_i}(\cdot | y_{< i})$ and $q_{i | F_i}(\cdot | y_{< i})$ are $\epsilon$-ind for all $y_{< i}$.
$\mathbb P(E_i | y_{< i}), \mathbb P(F_i | y_{< i}) \ge 1 - \delta$ for all $y_{< i}$.

Let $E := \bigcap E_i$ and $F := \bigcap F_i$, then they both have probability at least $1 - k \delta$, and $p_{i | E}(\cdot | y_{< i}$ and $q_{i | F}(\cdot | y_{< i})$ are $\epsilon$-ind.

By Advanced Adaptive Composition Theorem ($\delta = 0$), $p_{|E}$ and $q_{|F}$ are $(\epsilon' := k a(\epsilon) + \sqrt{2 k \log \beta^{-1}} (\epsilon + a(\epsilon)), \beta)$-ind.

The rest is the same as in the proof of Claim 17. $\square$

Subsampling

Stochastic gradient descent is like gradient descent, but with random subsampling.

Recall we have been considering databases in the space $Z^m$. Let $n < m$ be a positive integer, $\mathcal I := \{I \subset [m]: |I| = n\}$ be the set of subsets of $[m]$ of size $n$, and $\gamma$ a random subset sampled uniformly from $\mathcal I$. Let $r = {n \over m}$ which we call the subsampling rate. Then we may add a subsampling module to the noisy gradient descent algorithm (6.97) considered before

\[\theta_{t} = \theta_{t - 1} - \alpha n^{-1} \sum_{i \in \gamma} \nabla_\theta h_\theta(x_i) |_{\theta = \theta_{t - 1}} + \zeta_t. \qquad (7)\]

It turns out subsampling has an amplification effect on privacy.

Claim 19 (Ullman 2017). Fix $r \in [0, 1]$. Let $n \le m$ be two nonnegative integers with $n = r m$. Let $N$ be an $(\epsilon, \delta)$-dp machanism on $X^n$. Define mechanism $M$ on $X^m$ by

\[M(x) = N(x_\gamma)\]

Then $M$ is $(\log (1 + r(e^\epsilon - 1)), r \delta)$-dp.

Remark. Some seem to cite Kasiviswanathan-Lee-Nissim-Raskhodnikova-Smith 2005 for this result, but it is not clear to me how it appears there.

Proof. Let $x, x' \in X^n$ such that they differ by one row $x_i \neq x_i'$. Naturally we would like to consider the cases where the index $i$ is picked and the ones where it is not separately. Let $\mathcal I_\in$ and $\mathcal I_\notin$ be these two cases:

\[\begin{aligned} \mathcal I_\in = \{J \subset \mathcal I: i \in J\}\\ \mathcal I_\notin = \{J \subset \mathcal I: i \notin J\}\\ \end{aligned}\]

We will use these notations later. Let $A$ be the event $\{\gamma \ni i\}$.

Let $p$ and $q$ be the laws of $M(x)$ and $M(x')$ respectively. We collect some useful facts about them. First due to $N$ being $(\epsilon, \delta)$-dp,

\[p_{|A}(S) \le e^\epsilon q_{|A}(S) + \delta.\]

Also,

\[p_{|A}(S) \le e^\epsilon p_{|A^c}(S) + \delta.\]

To see this, note that being conditional laws, $p_A$ and $p_{A^c}$ are averages of laws over $\mathcal I_\in$ and $\mathcal I_\notin$ respectively:

\[\begin{aligned} p_{|A}(S) = |\mathcal I_\in|^{-1} \sum_{I \in \mathcal I_\in} \mathbb P(N(x_I) \in S)\\ p_{|A^c}(S) = |\mathcal I_\notin|^{-1} \sum_{J \in \mathcal I_\notin} \mathbb P(N(x_J) \in S). \end{aligned}\]

Now we want to pair the $I$'s in $\mathcal I_\in$ and $J$'s in $\mathcal I_\notin$ so that they differ by one index only, which means $d(x_I, x_J) = 1$. Formally, this means we want to consider the set:

\[\mathcal D := \{(I, J) \in \mathcal I_\in \times \mathcal I_\notin: |I \cap J| = n - 1\}.\]

We may observe by trying out some simple cases that every $I \in \mathcal I_\in$ is paired with $n$ elements in $\mathcal I_\notin$, and every $J \in \mathcal I_\notin$ is paired with $m - n$ elements in $\mathcal I_\in$. Therefore

\[p_{|A}(S) = |\mathcal D|^{-1} \sum_{(I, J) \in \mathcal D} \mathbb P(N(x_I \in S)) \le |\mathcal D|^{-1} \sum_{(I, J) \in \mathcal D} (e^\epsilon \mathbb P(N(x_J \in S)) + \delta) = e^\epsilon p_{|A^c} (S) + \delta.\]

Since each of the $m$ indices is picked independently with probability $r$, we have

\[\mathbb P(A) = r.\]

Let $t \in [0, 1]$ to be determined. We may write

\[\begin{aligned} p(S) &= r p_{|A} (S) + (1 - r) p_{|A^c} (S)\\ &\le r(t e^\epsilon q_{|A}(S) + (1 - t) e^\epsilon q_{|A^c}(S) + \delta) + (1 - r) q_{|A^c} (S)\\ &= rte^\epsilon q_{|A}(S) + (r(1 - t) e^\epsilon + (1 - r)) q_{|A^c} (S) + r \delta\\ &= te^\epsilon r q_{|A}(S) + \left({r \over 1 - r}(1 - t) e^\epsilon + 1\right) (1 - r) q_{|A^c} (S) + r \delta \\ &\le \left(t e^\epsilon \wedge \left({r \over 1 - r} (1 - t) e^\epsilon + 1\right)\right) q(S) + r \delta. \qquad (7.5) \end{aligned}\]

We can see from the last line that the best bound we can get is when

\[t e^\epsilon = {r \over 1 - r} (1 - t) e^\epsilon + 1.\]

Solving this equation we obtain

\[t = r + e^{- \epsilon} - r e^{- \epsilon}\]

and plugging this in (7.5) we have

\[p(S) \le (1 + r(e^\epsilon - 1)) q(S) + r \delta.\]

$\square$

Since $\log (1 + x) < x$ for $x > 0$, we can rewrite the conclusion of the Claim to $(r(e^\epsilon - 1), r \delta)$-dp. Further more, if $\epsilon < \alpha$ for some $\alpha$, we can rewrite it as $(r \alpha^{-1} (e^\alpha - 1) \epsilon, r \delta)$-dp or $(O(r \epsilon), r \delta)$-dp.

Let $\epsilon < 1$. We see that if the mechanism $N$ is $(\epsilon, \delta)$-dp on $Z^n$, then $M$ is $(2 r \epsilon, r \delta)$-dp, and if we run it over $k / r$ minibatches, by Advanced Adaptive Composition theorem, we have $(\sqrt{2 k r \log \beta^{-1}} \epsilon + 2 k r \epsilon^2, k \delta + \beta)$-dp.

This is better than the privacy guarantee without subsampling, where we run over $k$ iterations and obtain $(\sqrt{2 k \log \beta^{-1}} \epsilon + 2 k \epsilon^2, k \delta + \beta)$-dp. So with subsampling we gain an extra $\sqrt r$ in the $\epsilon$-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.

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 $C$:

\[\theta_{t} = \theta_{t - 1} - \alpha \left(n^{-1} \sum_{i \in \gamma} \nabla_\theta \ell(x_i; \theta) |_{\theta = \theta_{t - 1}}\right)_{\text{Clipped at }C / 2} + N(0, \sigma^2 C^2 I),\]

where

\[y_{\text{Clipped at } \alpha} := y / (1 \vee {\|y\|_2 \over \alpha})\]

is $y$ clipped to have norm at most $\alpha$.

Note that the clipping in DP-SGD is much stronger than making the query have sensitivity $C$. It makes the difference between the query results of two arbitrary inputs bounded by $C$, rather than neighbouring inputs.

In Part 2 of this post we will use the tools developed above to discuss the privacy guarantee for DP-SGD, among other things.

References

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. https://doi.org/10.1145/2976749.2978318.
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. https://doi.org/10.1561/0400000042.
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. https://doi.org/10.1109/FOCS.2010.12.
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. https://doi.org/10.1109/SFCS.2005.1.
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. https://doi.org/10.1007/978-3-662-49096-9_7.
Ullman, Jonathan. “Solution to CS7880 Homework 1.”, 2017. http://www.ccs.neu.edu/home/jullman/cs7880s17/HW1sol.pdf
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. https://doi.org/10.1007/978-3-319-57048-8_7.

Raise your ELBO

2019-02-14T00:00:00Z

In this post I give an introduction to variational inference, which is about maximising the evidence lower bound (ELBO).

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.

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.

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.

Then I explain stochastic variational inference, a modification of EM and MFA to improve efficiency.

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.

Acknowledgement. The following texts and resources were illuminating during the writing of this post: the Stanford CS228 notes (1,2), the Tel Aviv Algorithms in Molecular Biology slides (clear explanations of the connection between EM and Baum-Welch), Chapter 10 of Bishop's book (brilliant introduction to variational GMM), Section 2.5 of Sudderth's thesis and metacademy. Also thanks to Josef Lindman Hörnlund for discussions. The research was done while working at KTH mathematics department.

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.

KL divergence and ELBO

Let $p$ and $q$ be two probability measures. The Kullback-Leibler (KL) divergence is defined as

\[D(q||p) = E_q \log{q \over p}.\]

It achieves minimum $0$ when $p = q$.

If $p$ can be further written as

\[p(x) = {w(x) \over Z}, \qquad (0)\]

where $Z$ is a normaliser, then

\[\log Z = D(q||p) + L(w, q), \qquad(1)\]

where $L(w, q)$ is called the evidence lower bound (ELBO), defined by

\[L(w, q) = E_q \log{w \over q}. \qquad (1.25)\]

From (1), we see that to minimise the nonnegative term $D(q || p)$, one can maximise the ELBO.

To this end, we can simply discard $D(q || p)$ in (1) and obtain:

\[\log Z \ge L(w, q) \qquad (1.3)\]

and keep in mind that the inequality becomes an equality when $q = {w \over Z}$.

It is time to define the task of variational inference (VI), also known as variational Bayes (VB).

Definition. Variational inference is concerned with maximising the ELBO $L(w, q)$.

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

\[\begin{aligned} Z &= p(x; \theta)\\ w &= p(x, z; \theta)\\ q &= q(z) \end{aligned}\]

and the dummy variable $x$ in Equation (0) is substituted with $z$.

Fully Bayesian VI, often just called VI, has the following instantiations:

\[\begin{aligned} Z &= p(x) \\ w &= p(x, z, \theta) \\ q &= q(z, \theta) \end{aligned}\]

and $x$ in Equation (0) is substituted with $(z, \theta)$.

In both cases $\theta$ are parameters and $z$ are latent variables.

Remark on the naming of things. The term "variational" comes from the fact that we perform calculus of variations: maximise some functional ($L(w, q)$) over a set of functions ($q$). 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 $D(q||p)$ reaches minimum when $p = q$. 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.

EM

To illustrate the EM algorithms, we first define the mixture model.

Definition (mixture model). Given dataset $x_{1 : m}$, we assume the data has some underlying latent variable $z_{1 : m}$ that may take a value from a finite set $\{1, 2, ..., n_z\}$. Let $p(z_{i}; \pi)$ be categorically distributed according to the probability vector $\pi$. That is, $p(z_{i} = k; \pi) = \pi_k$. Also assume $p(x_{i} | z_{i} = k; \eta) = p(x_{i}; \eta_k)$. Find $\theta = (\pi, \eta)$ that maximises the likelihood $p(x_{1 : m}; \theta)$.

Represented as a DAG (a.k.a the plate notations), the model looks like this:

where the boxes with $m$ mean repitition for $m$ times, since there $m$ indepdent pairs of $(x, z)$, and the same goes for $\eta$.

The direct maximisation

\[\max_\theta \sum_i \log p(x_{i}; \theta) = \max_\theta \sum_i \log \int p(x_{i} | z_i; \theta) p(z_i; \theta) dz_i\]

is hard because of the integral in the log.

We can fit this problem in (1.3) by having $Z = p(x_{1 : m}; \theta)$ and $w = p(z_{1 : m}, x_{1 : m}; \theta)$. The plan is to update $\theta$ repeatedly so that $L(p(z, x; \theta_t), q(z))$ is non decreasing over time $t$.

Equation (1.3) at time $t$ for the $i$th datapoint is

\[\log p(x_{i}; \theta_t) \ge L(p(z_i, x_{i}; \theta_t), q(z_i)) \qquad (2)\]

Each timestep consists of two steps, the E-step and the M-step.

At E-step, we set

\[q(z_{i}) = p(z_{i}|x_{i}; \theta_t), \]

to turn the inequality into equality. We denote $r_{ik} = q(z_i = k)$ and call them responsibilities, so the posterior $q(z_i)$ is categorical distribution with parameter $r_i = r_{i, 1 : n_z}$.

At M-step, we maximise $\sum_i L(p(x_{i}, z_{i}; \theta), q(z_{i}))$ over $\theta$:

\[\begin{aligned} \theta_{t + 1} &= \text{argmax}_\theta \sum_i L(p(x_{i}, z_{i}; \theta), p(z_{i} | x_{i}; \theta_t)) \\ &= \text{argmax}_\theta \sum_i \mathbb E_{p(z_{i} | x_{i}; \theta_t)} \log p(x_{i}, z_{i}; \theta) \qquad (2.3) \end{aligned}\]

So $\sum_i L(p(x_{i}, z_{i}; \theta), q(z_i))$ is non-decreasing at both the E-step and the M-step.

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 $\theta$.

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. $\max_\theta p(x_{1 : m}; \theta)$. In fact it is said that the EM does get stuck in local maximum sometimes.

A different way of describing EM, which will be useful in hidden Markov model is:

At E-step, one writes down the formula \[\sum_i \mathbb E_{p(z_i | x_{i}; \theta_t)} \log p(x_{i}, z_i; \theta). \qquad (2.5)\]
At M-setp, one finds $\theta_{t + 1}$ to be the $\theta$ that maximises the above formula.

GMM

Gaussian mixture model (GMM) is an example of mixture models.

The space of the data is $\mathbb R^n$. We use the hypothesis that the data is Gaussian conditioned on the latent variable:

\[(x_i; \eta_k) \sim N(\mu_k, \Sigma_k),\]

so we write $\eta_k = (\mu_k, \Sigma_k)$,

During E-step, the $q(z_i)$ can be directly computed using Bayes’ theorem:

\[r_{ik} = q(z_i = k) = \mathbb P(z_i = k | x_{i}; \theta_t) = {g_{\mu_{t, k}, \Sigma_{t, k}} (x_{i}) \pi_{t, k} \over \sum_{j = 1 : n_z} g_{\mu_{t, j}, \Sigma_{t, j}} (x_{i}) \pi_{t, j}},\]

where $g_{\mu, \Sigma} (x) = (2 \pi)^{- n / 2} \det \Sigma^{-1 / 2} \exp(- {1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu))$ is the pdf of the Gaussian distribution $N(\mu, \Sigma)$.

During M-step, we need to compute

\[\text{argmax}_{\Sigma, \mu, \pi} \sum_{i = 1 : m} \sum_{k = 1 : n_z} r_{ik} \log (g_{\mu_k, \Sigma_k}(x_{i}) \pi_k).\]

This is similar to the quadratic discriminant analysis, and the solution is

\[\begin{aligned} \pi_{k} &= {1 \over m} \sum_{i = 1 : m} r_{ik}, \\ \mu_{k} &= {\sum_i r_{ik} x_{i} \over \sum_i r_{ik}}, \\ \Sigma_{k} &= {\sum_i r_{ik} (x_{i} - \mu_{t, k}) (x_{i} - \mu_{t, k})^T \over \sum_i r_{ik}}. \end{aligned}\]

Remark. The k-means algorithm is the $\epsilon \to 0$ limit of the GMM with constraints $\Sigma_k = \epsilon I$. 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 $\epsilon I$ is called elliptical k-means algorithm.

SMM

As a transition to the next models to study, let us consider a simpler mixture model obtained by making one modification to GMM: change $(x; \eta_k) \sim N(\mu_k, \Sigma_k)$ to $\mathbb P(x = w; \eta_k) = \eta_{kw}$ where $\eta$ is a stochastic matrix and $w$ is an arbitrary element of the space for $x$. So now the space for both $x$ and $z$ are finite. We call this model the simple mixture model (SMM).

As in GMM, at E-step $r_{ik}$ can be explicitly computed using Bayes' theorem.

It is not hard to write down the solution to the M-step in this case:

\[\begin{aligned} \pi_{k} &= {1 \over m} \sum_i r_{ik}, \qquad (2.7)\\ \eta_{k, w} &= {\sum_i r_{ik} 1_{x_i = w} \over \sum_i r_{ik}}. \qquad (2.8) \end{aligned}\]

where $1_{x_i = w}$ is the indicator function, and evaluates to $1$ if $x_i = w$ and $0$ otherwise.

Two trivial variants of the SMM are the two versions of probabilistic latent semantic analysis (pLSA), which we call pLSA1 and pLSA2.

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).

pLSA

The pLSA model (Hoffman 2000) is a mixture model, where the dataset is now pairs $(d_i, x_i)_{i = 1 : m}$. In natural language processing, $x$ are words and $d$ are documents, and a pair $(d, x)$ represent an ocurrance of word $x$ in document $d$.

For each datapoint $(d_{i}, x_{i})$,

\[\begin{aligned} p(d_i, x_i; \theta) &= \sum_{z_i} p(z_i; \theta) p(d_i | z_i; \theta) p(x_i | z_i; \theta) \qquad (2.91)\\ &= p(d_i; \theta) \sum_z p(x_i | z_i; \theta) p (z_i | d_i; \theta) \qquad (2.92). \end{aligned}\]

Of the two formulations, (2.91) corresponds to pLSA type 1, and (2.92) corresponds to type 2.

pLSA1

The pLSA1 model (Hoffman 2000) is basically SMM with $x_i$ substituted with $(d_i, x_i)$, which conditioned on $z_i$ are independently categorically distributed:

\[p(d_i = u, x_i = w | z_i = k; \theta) = p(d_i ; \xi_k) p(x_i; \eta_k) = \xi_{ku} \eta_{kw}.\]

The model can be illustrated in the plate notations:

So the solution of the M-step is

\[\begin{aligned} \pi_{k} &= {1 \over m} \sum_i r_{ik} \\ \xi_{k, u} &= {\sum_i r_{ik} 1_{d_{i} = u} \over \sum_i r_{ik}} \\ \eta_{k, w} &= {\sum_i r_{ik} 1_{x_{i} = w} \over \sum_i r_{ik}}. \end{aligned}\]

Remark. pLSA1 is the probabilistic version of LSA, also known as matrix factorisation.

Let $n_d$ and $n_x$ be the number of values $d_i$ and $x_i$ can take.

Problem (LSA). Let $R$ be a $n_d \times n_x$ matrix, fix $s \le \min\{n_d, n_x\}$. Find $n_d \times s$ matrix $D$ and $n_x \times s$ matrix $X$ that minimises

\[J(D, X) = \|R - D X^T\|_F.\]

where $\|\cdot\|_F$ is the Frobenius norm.

Claim. Let $R = U \Sigma V^T$ be the SVD of $R$, then the solution to the above problem is $D = U_s \Sigma_s^{{1 \over 2}}$ and $X = V_s \Sigma_s^{{1 \over 2}}$, where $U_s$ (resp. $V_s$) is the matrix of the first $s$ columns of $U$ (resp. $V$) and $\Sigma_s$ is the $s \times s$ submatrix of $\Sigma$.

One can compare pLSA1 with LSA. Both procedures produce embeddings of $d$ and $x$: in pLSA we obtain $n_z$ dimensional embeddings $\xi_{\cdot, u}$ and $\eta_{\cdot, w}$, whereas in LSA we obtain $s$ dimensional embeddings $D_{u, \cdot}$ and $X_{w, \cdot}$.

pLSA2

Let us turn to pLSA2 (Hoffman 2004), corresponding to (2.92). We rewrite it as

\[p(x_i | d_i; \theta) = \sum_{z_i} p(x_i | z_i; \theta) p(z_i | d_i; \theta).\]

To simplify notations, we collect all the $x_i$s with the corresponding $d_i$ equal to 1 (suppose there are $m_1$ of them), and write them as $(x_{1, j})_{j = 1 : m_1}$. In the same fashion we construct $x_{2, 1 : m_2}, x_{3, 1 : m_3}, ... x_{n_d, 1 : m_{n_d}}$. Similarly, we relabel the corresponding $d_i$ and $z_i$ accordingly.

With almost no loss of generality, we assume all $m_\ell$s are equal and write them as $m$.

Now the model becomes

\[p(x_{\ell, i} | d_{\ell, i} = \ell; \theta) = \sum_k p(x_{\ell, i} | z_{\ell, i} = k; \theta) p(z_{\ell, i} = k | d_{\ell, i} = \ell; \theta).\]

Since we have regrouped the $x$’s and $z$’s whose indices record the values of the $d$’s, we can remove the $d$’s from the equation altogether:

\[p(x_{\ell, i}; \theta) = \sum_k p(x_{\ell, i} | z_{\ell, i} = k; \theta) p(z_{\ell, i} = k; \theta).\]

It is effectively a modification of SMM by making $n_d$ copies of $\pi$. More specifically the parameters are $\theta = (\pi_{1 : n_d, 1 : n_z}, \eta_{1 : n_z, 1 : n_x})$, where we model $(z | d = \ell) \sim \text{Cat}(\pi_{\ell, \cdot})$ and, as in pLSA1, $(x | z = k) \sim \text{Cat}(\eta_{k, \cdot})$.

Illustrated in the plate notations, pLSA2 is:

The computation is basically adding an index $\ell$ to the computation of SMM wherever applicable.

The updates at the E-step is

\[r_{\ell i k} = p(z_{\ell i} = k | x_{\ell i}; \theta) \propto \pi_{\ell k} \eta_{k, x_{\ell i}}.\]

And at the M-step

\[\begin{aligned} \pi_{\ell k} &= {1 \over m} \sum_i r_{\ell i k} \\ \eta_{k w} &= {\sum_{\ell, i} r_{\ell i k} 1_{x_{\ell i} = w} \over \sum_{\ell, i} r_{\ell i k}}. \end{aligned}\]

HMM

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.

HMM is an example where the posterior $p(z_i | x_i; \theta)$ is not easy to compute, and one has to utilise properties of the underlying Bayesian network to go around it.

Now each sample is a sequence $x_i = (x_{ij})_{j = 1 : T}$, and so are the latent variables $z_i = (z_{ij})_{j = 1 : T}$.

The latent variables are assumed to form a Markov chain with transition matrix $(\xi_{k \ell})_{k \ell}$, and $x_{ij}$ is completely dependent on $z_{ij}$:

\[\begin{aligned} p(z_{ij} | z_{i, j - 1}) &= \xi_{z_{i, j - 1}, z_{ij}},\\ p(x_{ij} | z_{ij}) &= \eta_{z_{ij}, x_{ij}}. \end{aligned}\]

Also, the distribution of $z_{i1}$ is again categorical with parameter $\pi$:

\[p(z_{i1}) = \pi_{z_{i1}}\]

So the parameters are $\theta = (\pi, \xi, \eta)$. And HMM can be shown in plate notations as:

Now we apply EM to HMM, which is called the Baum-Welch algorithm. Unlike the previous examples, it is too messy to compute $p(z_i | x_{i}; \theta)$, so during the E-step we instead write down formula (2.5) directly in hope of simplifying it:

\[\begin{aligned} \mathbb E_{p(z_i | x_i; \theta_t)} \log p(x_i, z_i; \theta_t) &=\mathbb E_{p(z_i | x_i; \theta_t)} \left(\log \pi_{z_{i1}} + \sum_{j = 2 : T} \log \xi_{z_{i, j - 1}, z_{ij}} + \sum_{j = 1 : T} \log \eta_{z_{ij}, x_{ij}}\right). \qquad (3) \end{aligned}\]

Let us compute the summand in second term:

\[\begin{aligned} \mathbb E_{p(z_i | x_{i}; \theta_t)} \log \xi_{z_{i, j - 1}, z_{ij}} &= \sum_{k, \ell} (\log \xi_{k, \ell}) \mathbb E_{p(z_{i} | x_{i}; \theta_t)} 1_{z_{i, j - 1} = k, z_{i, j} = \ell} \\ &= \sum_{k, \ell} p(z_{i, j - 1} = k, z_{ij} = \ell | x_{i}; \theta_t) \log \xi_{k, \ell}. \qquad (4) \end{aligned}\]

Similarly, one can write down the first term and the summand in the third term to obtain

\[\begin{aligned} \mathbb E_{p(z_i | x_{i}; \theta_t)} \log \pi_{z_{i1}} &= \sum_k p(z_{i1} = k | x_{i}; \theta_t), \qquad (5) \\ \mathbb E_{p(z_i | x_{i}; \theta_t)} \log \eta_{z_{i, j}, x_{i, j}} &= \sum_{k, w} 1_{x_{ij} = w} p(z_{i, j} = k | x_i; \theta_t) \log \eta_{k, w}. \qquad (6) \end{aligned}\]

plugging (4)(5)(6) back into (3) and summing over $j$, we obtain the formula to maximise over $\theta$ on:

\[\sum_k \sum_i r_{i1k} \log \pi_k + \sum_{k, \ell} \sum_{j = 2 : T, i} s_{ijk\ell} \log \xi_{k, \ell} + \sum_{k, w} \sum_{j = 1 : T, i} r_{ijk} 1_{x_{ij} = w} \log \eta_{k, w},\]

where

\[\begin{aligned} r_{ijk} &:= p(z_{ij} = k | x_{i}; \theta_t), \\ s_{ijk\ell} &:= p(z_{i, j - 1} = k, z_{ij} = \ell | x_{i}; \theta_t). \end{aligned}\]

Now we proceed to the M-step. Since each of the $\pi_k, \xi_{k, \ell}, \eta_{k, w}$ is nicely confined in the inner sum of each term, together with the constraint $\sum_k \pi_k = \sum_\ell \xi_{k, \ell} = \sum_w \eta_{k, w} = 1$ it is not hard to find the argmax at time $t + 1$ (the same way one finds the MLE for any categorical distribution):

\[\begin{aligned} \pi_{k} &= {1 \over m} \sum_i r_{i1k}, \qquad (6.1) \\ \xi_{k, \ell} &= {\sum_{j = 2 : T, i} s_{ijk\ell} \over \sum_{j = 1 : T - 1, i} r_{ijk}}, \qquad(6.2) \\ \eta_{k, w} &= {\sum_{ij} 1_{x_{ij} = w} r_{ijk} \over \sum_{ij} r_{ijk}}. \qquad(6.3) \end{aligned}\]

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.

What remains is to compute $r$ and $s$.

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.

Let

\[\begin{aligned} \alpha_k(i, j) &:= p(x_{i, 1 : j}, z_{ij} = k; \theta_t), \\ \beta_k(i, j) &:= p(x_{i, j + 1 : T} | z_{ij} = k; \theta_t). \end{aligned}\]

They can be computed recursively as

\[\begin{aligned} \alpha_k(i, j) &= \begin{cases} \eta_{k, x_{1j}} \pi_k, & j = 1; \\ \eta_{k, x_{ij}} \sum_\ell \alpha_\ell(j - 1, i) \xi_{k\ell}, & j \ge 2. \end{cases}\\ \beta_k(i, j) &= \begin{cases} 1, & j = T;\\ \sum_\ell \xi_{k\ell} \beta_\ell(j + 1, i) \eta_{\ell, x_{i, j + 1}}, & j < T. \end{cases} \end{aligned}\]

Then

\[\begin{aligned} p(z_{ij} = k, x_{i}; \theta_t) &= \alpha_k(i, j) \beta_k(i, j), \qquad (7)\\ p(x_{i}; \theta_t) &= \sum_k \alpha_k(i, j) \beta_k(i, j),\forall j = 1 : T \qquad (8)\\ p(z_{i, j - 1} = k, z_{i, j} = \ell, x_{i}; \theta_t) &= \alpha_k(i, j) \xi_{k\ell} \beta_\ell(i, j + 1) \eta_{\ell, x_{j + 1, i}}. \qquad (9) \end{aligned}\]

And this yields $r_{ijk}$ and $s_{ijk\ell}$ since they can be computed as ${(7) \over (8)}$ and ${(9) \over (8)}$ respectively.

Fully Bayesian EM / MFA

Let us now venture into the realm of full Bayesian.

In EM we aim to maximise the ELBO

\[\int q(z) \log {p(x, z; \theta) \over q(z)} dz d\theta\]

alternately over $q$ and $\theta$. As mentioned before, the E-step of maximising over $q$ is Bayesian, in that it computes the posterior of $z$, whereas the M-step of maximising over $\theta$ is maximum likelihood and frequentist.

The fully Bayesian EM makes the M-step Bayesian by making $\theta$ a random variable, so the ELBO becomes

\[L(p(x, z, \theta), q(z, \theta)) = \int q(z, \theta) \log {p(x, z, \theta) \over q(z, \theta)} dz d\theta\]

We further assume $q$ can be factorised into distributions on $z$ and $\theta$: $q(z, \theta) = q_1(z) q_2(\theta)$. So the above formula is rewritten as

\[L(p(x, z, \theta), q(z, \theta)) = \int q_1(z) q_2(\theta) \log {p(x, z, \theta) \over q_1(z) q_2(\theta)} dz d\theta\]

To find argmax over $q_1$, we rewrite

\[\begin{aligned} L(p(x, z, \theta), q(z, \theta)) &= \int q_1(z) \left(\int q_2(\theta) \log p(x, z, \theta) d\theta\right) dz - \int q_1(z) \log q_1(z) dz - \int q_2(\theta) \log q_2(\theta) \\&= - D(q_1(z) || p_x(z)) + C, \end{aligned}\]

where $p_x$ is a density in $z$ with

\[\log p_x(z) = \mathbb E_{q_2(\theta)} \log p(x, z, \theta) + C.\]

So the $q_1$ that maximises the ELBO is $q_1^* = p_x$.

Similarly, the optimal $q_2$ is such that

\[\log q_2^*(\theta) = \mathbb E_{q_1(z)} \log p(x, z, \theta) + C.\]

The fully Bayesian EM thus alternately evaluates $q_1^*$ (E-step) and $q_2^*$ (M-step).

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.

Application to mixture models

Definition (Fully Bayesian mixture model). The relations between $\pi$, $\eta$, $x$, $z$ are the same as in the definition of mixture models. Furthermore, we assume the distribution of $(x | \eta_k)$ belongs to the exponential family (the definition of the exponential family is briefly touched at the end of this section). But now both $\pi$ and $\eta$ are random variables. Let the prior distribution $p(\pi)$ is Dirichlet with parameter $(\alpha, \alpha, ..., \alpha)$. Let the prior $p(\eta_k)$ be the conjugate prior of $(x | \eta_k)$, with parameter $\beta$, we will see later in this section that the posterior $q(\eta_k)$ belongs to the same family as $p(\eta_k)$. Represented in a plate notations, a fully Bayesian mixture model looks like:

Given this structure we can write down the mean-field approximation:

\[\log q(z) = \mathbb E_{q(\eta)q(\pi)} (\log(x | z, \eta) + \log(z | \pi)) + C.\]

Both sides can be factored into per-sample expressions, giving us

\[\log q(z_i) = \mathbb E_{q(\eta)} \log p(x_i | z_i, \eta) + \mathbb E_{q(\pi)} \log p(z_i | \pi) + C\]

Therefore

\[\log r_{ik} = \log q(z_i = k) = \mathbb E_{q(\eta_k)} \log p(x_i | \eta_k) + \mathbb E_{q(\pi)} \log \pi_k + C. \qquad (9.1)\]

So the posterior of each $z_i$ is categorical regardless of the $p$s and $q$s.

Computing the posterior of $\pi$ and $\eta$:

\[\log q(\pi) + \log q(\eta) = \log p(\pi) + \log p(\eta) + \sum_i \mathbb E_{q(z_i)} p(x_i | z_i, \eta) + \sum_i \mathbb E_{q(z_i)} p(z_i | \pi) + C.\]

So we can separate the terms involving $\pi$ and those involving $\eta$. First compute the posterior of $\pi$:

\[\log q(\pi) = \log p(\pi) + \sum_i \mathbb E_{q(z_i)} \log p(z_i | \pi) = \log p(\pi) + \sum_i \sum_k r_{ik} \log \pi_k + C.\]

The right hand side is the log of a Dirichlet modulus the constant $C$, from which we can update the posterior parameter $\phi^\pi$:

\[\phi^\pi_k = \alpha + \sum_i r_{ik}. \qquad (9.3)\]

Similarly we can obtain the posterior of $\eta$:

\[\log q(\eta) = \log p(\eta) + \sum_i \sum_k r_{ik} \log p(x_i | \eta_k) + C.\]

Again we can factor the terms with respect to $k$ and get:

\[\log q(\eta_k) = \log p(\eta_k) + \sum_i r_{ik} \log p(x_i | \eta_k) + C. \qquad (9.5)\]

Here we can see why conjugate prior works. Mathematically, given a probability distribution $p(x | \theta)$, the distribution $p(\theta)$ is called conjugate prior of $p(x | \theta)$ if $\log p(\theta) + \log p(x | \theta)$ has the same form as $\log p(\theta)$.

For example, the conjugate prior for the exponential family $p(x | \theta) = h(x) \exp(\theta \cdot T(x) - A(\theta))$ where $T$, $A$ and $h$ are some functions is $p(\theta; \chi, \nu) \propto \exp(\chi \cdot \theta - \nu A(\theta))$.

Here what we want is a bit different from conjugate priors because of the coefficients $r_{ik}$. But the computation carries over to the conjugate priors of the exponential family (try it yourself and you'll see). That is, if $p(x_i | \eta_k)$ belongs to the exponential family

\[p(x_i | \eta_k) = h(x) \exp(\eta_k \cdot T(x) - A(\eta_k))\]

and if $p(\eta_k)$ is the conjugate prior of $p(x_i | \eta_k)$

\[p(\eta_k) \propto \exp(\chi \cdot \eta_k - \nu A(\eta_k))\]

then $q(\eta_k)$ has the same form as $p(\eta_k)$, and from (9.5) we can compute the updates of $\phi^{\eta_k}$:

\[\begin{aligned} \phi^{\eta_k}_1 &= \chi + \sum_i r_{ik} T(x_i), \qquad (9.7) \\ \phi^{\eta_k}_2 &= \nu + \sum_i r_{ik}. \qquad (9.9) \end{aligned}\]

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.

Fully Bayesian GMM

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.

More specifically:

$p(z_{i}) = \text{Cat}(\pi)$ as in vanilla GMM
$p(\pi) = \text{Dir}(\alpha, \alpha, ..., \alpha)$ has Dirichlet distribution, the conjugate prior to the parameters of the categorical distribution.
$p(x_i | z_i = k) = p(x_i | \eta_k) = N(\mu_{k}, \Sigma_{k})$ as in vanilla GMM
$p(\mu_k, \Sigma_k) = \text{NIW} (\mu_0, \lambda, \Psi, \nu)$ is the normal-inverse-Wishart distribution, the conjugate prior to the mean and covariance matrix of the Gaussian distribution.

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.

LDA

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:

It is the smoothed version in the paper.

More specifically, on the basis of pLSA2, we add prior distributions to $\eta$ and $\pi$:

\[\begin{aligned} p(\eta_k) &= \text{Dir} (\beta, ..., \beta), \qquad k = 1 : n_z \\ p(\pi_\ell) &= \text{Dir} (\alpha, ..., \alpha), \qquad \ell = 1 : n_d \\ \end{aligned}\]

And as before, the prior of $z$ is

\[p(z_{\ell, i}) = \text{Cat} (\pi_\ell), \qquad \ell = 1 : n_d, i = 1 : m\]

We also denote posterior distribution

\[\begin{aligned} q(\eta_k) &= \text{Dir} (\phi^{\eta_k}), \qquad k = 1 : n_z \\ q(\pi_\ell) &= \text{Dir} (\phi^{\pi_\ell}), \qquad \ell = 1 : n_d \\ p(z_{\ell, i}) &= \text{Cat} (r_{\ell, i}), \qquad \ell = 1 : n_d, i = 1 : m \end{aligned}\]

As before, in E-step we update $r$, and in M-step we update $\lambda$ and $\gamma$.

But in the LDA paper, one treats optimisation over $r$, $\lambda$ and $\gamma$ as a E-step, and treats $\alpha$ and $\beta$ 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.

Plugging in (9.1) we obtain the updates at E-step

\[r_{\ell i k} \propto \exp(\psi(\phi^{\pi_\ell}_k) + \psi(\phi^{\eta_k}_{x_{\ell i}}) - \psi(\sum_w \phi^{\eta_k}_w)), \qquad (10)\]

where $\psi$ is the digamma function. Similarly, plugging in (9.3)(9.7)(9.9), at M-step, we update the posterior of $\pi$ and $\eta$:

\[\begin{aligned} \phi^{\pi_\ell}_k &= \alpha + \sum_i r_{\ell i k}. \qquad (11)\\ %%}}$ %%As for $\eta$, we have %%{{$%align% %%\log q(\eta) &= \sum_k \log p(\eta_k) + \sum_{\ell, i} \mathbb E_{q(z_{\ell i})} \log p(x_{\ell i} | z_{\ell i}, \eta) \\ %%&= \sum_{k, j} (\sum_{\ell, i} r_{\ell i k} 1_{x_{\ell i} = j} + \beta - 1) \log \eta_{k j} %%}}$ %%which gives us %%{{$ \phi^{\eta_k}_w &= \beta + \sum_{\ell, i} r_{\ell i k} 1_{x_{\ell i} = w}. \qquad (12) \end{aligned}\]

So the algorithm iterates over (10) and (11)(12) until convergence.

DPMM

The Dirichlet process mixture model (DPMM) is like the fully Bayesian mixture model except $n_z = \infty$, i.e. $z$ can take any positive integer value.

The probability of $z_i = k$ is defined using the so called stick-breaking process: let $v_i \sim \text{Beta} (\alpha, \beta)$ be i.i.d. random variables with Beta distributions, then

\[\mathbb P(z_i = k | v_{1:\infty}) = (1 - v_1) (1 - v_2) ... (1 - v_{k - 1}) v_k.\]

So $v$ plays a similar role to $\pi$ in the previous models.

As before, we have that the distribution of $x$ belongs to the exponential family:

\[p(x | z = k, \eta) = p(x | \eta_k) = h(x) \exp(\eta_k \cdot T(x) - A(\eta_k))\]

so the prior of $\eta_k$ is

\[p(\eta_k) \propto \exp(\chi \cdot \eta_k - \nu A(\eta_k)).\]

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.

As before, we can write down the ELBO:

\[L(p(x, z, \theta), q(z, \theta)) = \mathbb E_{q(\theta)} \log {p(\theta) \over q(\theta)} + \mathbb E_{q(\theta) q(z)} \log {p(x, z | \theta) \over q(z)}.\]

Both terms are infinite series:

\[L(p, q) = \sum_{k = 1 : \infty} \mathbb E_{q(\theta_k)} \log {p(\theta_k) \over q(\theta_k)} + \sum_{i = 1 : m} \sum_{k = 1 : \infty} q(z_i = k) \mathbb E_{q(\theta)} \log {p(x_i, z_i = k | \theta) \over q(z_i = k)}.\]

There are several ways to deal with the infinities. One is to fix some level $T > 0$ and set $v_T = 1$ almost surely (Blei-Jordan 2006). This effectively turns the model into a finite one, and both terms become finite sums over $k = 1 : T$.

Another walkaround (Kurihara-Welling-Vlassis 2007) is also a kind of truncation, but less heavy-handed: setting the posterior $q(\theta) = q(\eta) q(v)$ to be:

\[q(\theta) = q(\theta_{1 : T}) p(\theta_{T + 1 : \infty}) =: q(\theta_{\le T}) p(\theta_{> T}).\]

That is, tie the posterior after $T$ to the prior. This effectively turns the first term in the ELBO to a finite sum over $k = 1 : T$, while keeping the second sum an infinite series:

\[L(p, q) = \sum_{k = 1 : T} \mathbb E_{q(\theta_k)} \log {p(\theta_k) \over q(\theta_k)} + \sum_i \sum_{k = 1 : \infty} q(z_i = k) \mathbb E_{q(\theta)} \log {p(x_i, z_i = k | \theta) \over q(z_i = k)}. \qquad (13)\]

The plate notation of this model is:

As it turns out, the infinities can be tamed in this case.

As before, the optimal $q(z_i)$ is computed as

\[r_{ik} = q(z_i = k) = s_{ik} / S_i\]

where

\[\begin{aligned} s_{ik} &= \exp(\mathbb E_{q(\theta)} \log p(x_i, z_i = k | \theta)) \\ S_i &= \sum_{k = 1 : \infty} s_{ik}. \end{aligned}\]

Plugging this back to (13) we have

\[\begin{aligned} \sum_{k = 1 : \infty} r_{ik} &\mathbb E_{q(\theta)} \log {p(x_i, z_i = k | \theta) \over r_{ik}} \\ &= \sum_{k = 1 : \infty} r_{ik} \mathbb E_{q(\theta)} (\log p(x_i, z_i = k | \theta) - \mathbb E_{q(\theta)} \log p(x_i, z_i = k | \theta) + \log S_i) = \log S_i. \end{aligned}\]

So it all rests upon $S_i$ being finite.

For $k \le T + 1$, we compute the quantity $s_{ik}$ directly. For $k > T$, it is not hard to show that

\[s_{ik} = s_{i, T + 1} \exp((k - T - 1) \mathbb E_{p(w)} \log (1 - w)),\]

where $w$ is a random variable with same distribution as $p(v_k)$, i.e. $\text{Beta}(\alpha, \beta)$.

Hence

\[S_i = \sum_{k = 1 : T} s_{ik} + {s_{i, T + 1} \over 1 - \exp(\psi(\beta) - \psi(\alpha + \beta))}\]

is indeed finite. Similarly we also obtain

\[q(z_i > k) = S^{-1} \left(\sum_{\ell = k + 1 : T} s_\ell + {s_{i, T + 1} \over 1 - \exp(\psi(\beta) - \psi(\alpha + \beta))}\right), k \le T \qquad (14)\]

Now let us compute the posterior of $\theta_{\le T}$. In the following we exchange the integrals without justifying them (c.f. Fubini's Theorem). Equation (13) can be rewritten as

\[L(p, q) = \mathbb E_{q(\theta_{\le T})} \left(\log p(\theta_{\le T}) + \sum_i \mathbb E_{q(z_i) p(\theta_{> T})} \log {p(x_i, z_i | \theta) \over q(z_i)} - \log q(\theta_{\le T})\right).\]

Note that unlike the derivation of the mean-field approximation, we keep the $- \mathbb E_{q(z)} \log q(z)$ term even though we are only interested in $\theta$ at this stage. This is again due to the problem of infinities: as in the computation of $S$, we would like to cancel out some undesirable unbounded terms using $q(z)$. We now have

\[\log q(\theta_{\le T}) = \log p(\theta_{\le T}) + \sum_i \mathbb E_{q(z_i) p(\theta_{> T})} \log {p(x_i, z_i | \theta) \over q(z_i)} + C.\]

By plugging in $q(z = k)$ we obtain

\[\log q(\theta_{\le T}) = \log p(\theta_{\le T}) + \sum_{k = 1 : \infty} q(z_i = k) \left(\mathbb E_{p(\theta_{> T})} \log {p(x_i, z_i = k | \theta) \over q(z_i = k)} - \mathbb E_{q(\theta)} \log {p(x_i, z_i = k | \theta) \over q(z_i = k)}\right) + C.\]

Again, we separate the $v_k$'s and the $\eta_k$'s to obtain

\[q(v_{\le T}) = \log p(v_{\le T}) + \sum_i \sum_k q(z_i = k) \left(\mathbb E_{p(v_{> T})} \log p(z_i = k | v) - \mathbb E_{q(v)} \log p (z_i = k | v)\right).\]

Denote by $D_k$ the difference between the two expetations on the right hand side. It is easy to show that

\[D_k = \begin{cases} \log(1 - v_1) ... (1 - v_{k - 1}) v_k - \mathbb E_{q(v)} \log (1 - v_1) ... (1 - v_{k - 1}) v_k & k \le T\\ \log(1 - v_1) ... (1 - v_T) - \mathbb E_{q(v)} \log (1 - v_1) ... (1 - v_T) & k > T \end{cases}\]

so $D_k$ is bounded. With this we can derive the update for $\phi^{v, 1}$ and $\phi^{v, 2}$:

\[\begin{aligned} \phi^{v, 1}_k &= \alpha + \sum_i q(z_i = k) \\ \phi^{v, 2}_k &= \beta + \sum_i q(z_i > k), \end{aligned}\]

where $q(z_i > k)$ can be computed as in (14).

When it comes to $\eta$, we have

\[\log q(\eta_{\le T}) = \log p(\eta_{\le T}) + \sum_i \sum_{k = 1 : \infty} q(z_i = k) (\mathbb E_{p(\eta_k)} \log p(x_i | \eta_k) - \mathbb E_{q(\eta_k)} \log p(x_i | \eta_k)).\]

Since $q(\eta_k) = p(\eta_k)$ for $k > T$, the inner sum on the right hand side is a finite sum over $k = 1 : T$. By factorising $q(\eta_{\le T})$ and $p(\eta_{\le T})$, we have

\[\log q(\eta_k) = \log p(\eta_k) + \sum_i q(z_i = k) \log (x_i | \eta_k) + C,\]

which gives us

\[\begin{aligned} \phi^{\eta, 1}_k &= \xi + \sum_i q(z_i = k) T(x_i) \\ \phi^{\eta, 2}_k &= \nu + \sum_i q(z_i = k). \end{aligned}\]

SVI

In variational inference, the computation of some parameters are more expensive than others.

For example, the computation of M-step is often much more expensive than that of E-step:

In the vanilla mixture models with the EM algorithm, the update of $\theta$ requires the computation of $r_{ik}$ for all $i = 1 : m$, see Eq (2.3).
In the fully Bayesian mixture model with mean field approximation, the updates of $\phi^\pi$ and $\phi^\eta$ require the computation of a sum over all samples (see Eq (9.3)(9.7)(9.9)).

Similarly, in pLSA2 (resp. LDA), the updates of $\eta_k$ (resp. $\phi^{\eta_k}$) requires a sum over $\ell = 1 : n_d$, whereas the updates of other parameters do not.

In these cases, the parameter that requires more computations are called global and the other ones local.

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.

Each time SVI picks a sample, updates the corresponding local parameters, and computes the update of the global parameters as if all the $m$ 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.

As an example, here's SVI applied to LDA:

Set $t = 1$.
Pick $\ell$ uniformly from $\{1, 2, ..., n_d\}$.
Repeat until convergence:
1. Compute $(r_{\ell i k})_{i = 1 : m, k = 1 : n_z}$ using (10).
2. Compute $(\phi^{\pi_\ell}_k)_{k = 1 : n_z}$ using (11).
Compute $(\tilde \phi^{\eta_k}_w)_{k = 1 : n_z, w = 1 : n_x}$ using the following formula (compare with (12)) \[\tilde \phi^{\eta_k}_w = \beta + n_d \sum_{i} r_{\ell i k} 1_{x_{\ell i} = w}\]
Update the exponential moving average $(\phi^{\eta_k}_w)_{k = 1 : n_z, w = 1 : n_x}$: \[\phi^{\eta_k}_w = (1 - \rho_t) \phi^{\eta_k}_w + \rho_t \tilde \phi^{\eta_k}_w\]
Increment $t$ and go back to Step 2.

In the original paper, $\rho_t$ needs to satisfy some conditions that guarantees convergence of the global parameters:

\[\begin{aligned} \sum_t \rho_t = \infty \\ \sum_t \rho_t^2 < \infty \end{aligned}\]

and the choice made there is

\[\rho_t = (t + \tau)^{-\kappa}\]

for some $\kappa \in (.5, 1]$ and $\tau \ge 0$.

AEVB

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.

Let's look back to the original problem of maximising the ELBO:

\[\max_{\theta, q} \sum_{i = 1 : m} L(p(x_i | z_i; \theta) p(z_i; \theta), q(z_i))\]

Since for any given $\theta$, the optimal $q(z_i)$ is the posterior $p(z_i | x_i; \theta)$, the problem reduces to

\[\max_{\theta} \sum_i L(p(x_i | z_i; \theta) p(z_i; \theta), p(z_i | x_i; \theta))\]

Let us assume $p(z_i; \theta) = p(z_i)$ is independent of $\theta$ to simplify the problem. In the old mixture models, we have $p(x_i | z_i; \theta) = p(x_i; \eta_{z_i})$, which we can generalise to $p(x_i; f(\theta, z_i))$ for some function $f$. Using Beyes' theorem we can also write down $p(z_i | x_i; \theta) = q(z_i; g(\theta, x_i))$ for some function $g$. So the problem becomes

\[\max_{\theta} \sum_i L(p(x_i; f(\theta, z_i)) p(z_i), q(z_i; g(\theta, x_i)))\]

In some cases $g$ can be hard to write down or compute. AEVB addresses this problem by replacing $g(\theta, x_i)$ with a neural network $g_\phi(x_i)$ with input $x_i$ and some separate parameters $\phi$. It also replaces $f(\theta, z_i)$ with a neural network $f_\theta(z_i)$ with input $z_i$ and parameters $\theta$. And now the problem becomes

\[\max_{\theta, \phi} \sum_i L(p(x_i; f_\theta(z_i)) p(z_i), q(z_i; g_\phi(x_i))).\]

The objective function can be written as

\[\sum_i \mathbb E_{q(z_i; g_\phi(x_i))} \log p(x_i; f_\theta(z_i)) - D(q(z_i; g_\phi(x_i)) || p(z_i)).\]

The first term is called the negative reconstruction error, like the $- \|decoder(encoder(x)) - x\|$ in autoencoders, which is where the "autoencoder" in the name comes from.

The second term is a regularisation term that penalises the posterior $q(z_i)$ that is very different from the prior $p(z_i)$. We assume this term can be computed analytically.

So only the first term requires computing.

We can approximate the sum over $i$ in a similar fashion as SVI: pick $j$ uniformly randomly from $\{1 ... m\}$ and treat the whole dataset as $m$ replicates of $x_j$, and approximate the expectation using Monte-Carlo:

\[U(x_i, \theta, \phi) := \sum_i \mathbb E_{q(z_i; g_\phi(x_i))} \log p(x_i; f_\theta(z_i)) \approx m \mathbb E_{q(z_j; g_\phi(x_j))} \log p(x_j; f_\theta(z_j)) \approx {m \over L} \sum_{\ell = 1}^L \log p(x_j; f_\theta(z_{j, \ell})),\]

where each $z_{j, \ell}$ is sampled from $q(z_j; g_\phi(x_j))$.

But then it is not easy to approximate the gradient over $\phi$. 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.

Assume there exists a transformation $T_\phi$ and a random variable $\epsilon$ with distribution independent of $\phi$ or $\theta$, such that $T_\phi(x_i, \epsilon)$ has distribution $q(z_i; g_\phi(x_i))$. In this case we can rewrite $U(x, \phi, \theta)$ as

\[\sum_i \mathbb E_{\epsilon \sim p(\epsilon)} \log p(x_i; f_\theta(T_\phi(x_i, \epsilon))),\]

This way one can use Monte-Carlo to approximate $\nabla_\phi U(x, \phi, \theta)$:

\[\nabla_\phi U(x, \phi, \theta) \approx {m \over L} \sum_{\ell = 1 : L} \nabla_\phi \log p(x_j; f_\theta(T_\phi(x_j, \epsilon_\ell))),\]

where each $\epsilon_{\ell}$ is sampled from $p(\epsilon)$. The approximation of $U(x, \phi, \theta)$ itself can be done similarly.

VAE

As an example of AEVB, the paper introduces variational autoencoder (VAE), with the following instantiations:

The prior $p(z_i) = N(0, I)$ is standard normal, thus independent of $\theta$.
The distribution $p(x_i; \eta)$ is either Gaussian or categorical.
The distribution $q(z_i; \mu, \Sigma)$ is Gaussian with diagonal covariance matrix. So $g_\phi(z_i) = (\mu_\phi(x_i), \text{diag}(\sigma^2_\phi(x_i)_{1 : d}))$. Thus in the reparameterisation trick $\epsilon \sim N(0, I)$ and $T_\phi(x_i, \epsilon) = \epsilon \odot \sigma_\phi(x_i) + \mu_\phi(x_i)$, where $\odot$ is elementwise multiplication.
The KL divergence can be easily computed analytically as $- D(q(z_i; g_\phi(x_i)) || p(z_i)) = {d \over 2} + \sum_{j = 1 : d} \log\sigma_\phi(x_i)_j - {1 \over 2} \sum_{j = 1 : d} (\mu_\phi(x_i)_j^2 + \sigma_\phi(x_i)_j^2)$.

With this, one can use backprop to maximise the ELBO.

Fully Bayesian AEVB

Let us turn to fully Bayesian version of AEVB. Again, we first recall the ELBO of the fully Bayesian mixture models:

\[L(p(x, z, \pi, \eta; \alpha, \beta), q(z, \pi, \eta; r, \phi)) = L(p(x | z, \eta) p(z | \pi) p(\pi; \alpha) p(\eta; \beta), q(z; r) q(\eta; \phi^\eta) q(\pi; \phi^\pi)).\]

We write $\theta = (\pi, \eta)$, rewrite $\alpha := (\alpha, \beta)$, $\phi := r$, and $\gamma := (\phi^\eta, \phi^\pi)$. Furthermore, as in the half-Bayesian version we assume $p(z | \theta) = p(z)$, i.e. $z$ does not depend on $\theta$. Similarly we also assume $p(\theta; \alpha) = p(\theta)$. Now we have

\[L(p(x, z, \theta; \alpha), q(z, \theta; \phi, \gamma)) = L(p(x | z, \theta) p(z) p(\theta), q(z; \phi) q(\theta; \gamma)).\]

And the objective is to maximise it over $\phi$ and $\gamma$. We no longer maximise over $\theta$, because it is now a random variable, like $z$. Now let us transform it to a neural network model, as in the half-Bayesian case:

\[L\left(\left(\prod_{i = 1 : m} p(x_i; f_\theta(z_i))\right) \left(\prod_{i = 1 : m} p(z_i) \right) p(\theta), \left(\prod_{i = 1 : m} q(z_i; g_\phi(x_i))\right) q(\theta; h_\gamma(x))\right).\]

where $f_\theta$, $g_\phi$ and $h_\gamma$ are neural networks. Again, by separating out KL-divergence terms, the above formula becomes

\[\sum_i \mathbb E_{q(\theta; h_\gamma(x))q(z_i; g_\phi(x_i))} \log p(x_i; f_\theta(z_i)) - \sum_i D(q(z_i; g_\phi(x_i)) || p(z_i)) - D(q(\theta; h_\gamma(x)) || p(\theta)).\]

Again, we assume the latter two terms can be computed analytically. Using reparameterisation trick, we write

\[\begin{aligned} \theta &= R_\gamma(\zeta, x) \\ z_i &= T_\phi(\epsilon, x_i) \end{aligned}\]

for some transformations $R_\gamma$ and $T_\phi$ and random variables $\zeta$ and $\epsilon$ so that the output has the desired distributions.

Then the first term can be written as

\[\mathbb E_{\zeta, \epsilon} \log p(x_i; f_{R_\gamma(\zeta, x)} (T_\phi(\epsilon, x_i))),\]

so that the gradients can be computed accordingly.

Again, one may use Monte-Carlo to approximate this expetation.

References

Attias, Hagai. "A variational baysian framework for graphical models." In Advances in neural information processing systems, pp. 209-215. 2000.
Bishop, Christopher M. Neural networks for pattern recognition. Springer. 2006.
Blei, David M., and Michael I. Jordan. “Variational Inference for Dirichlet Process Mixtures.” Bayesian Analysis 1, no. 1 (March 2006): 121–43. https://doi.org/10.1214/06-BA104.
Blei, David M., Andrew Y. Ng, and Michael I. Jordan. “Latent Dirichlet Allocation.” Journal of Machine Learning Research 3, no. Jan (2003): 993–1022.
Hofmann, Thomas. “Latent Semantic Models for Collaborative Filtering.” ACM Transactions on Information Systems 22, no. 1 (January 1, 2004): 89–115. https://doi.org/10.1145/963770.963774.
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.
Hoffman, Matt, David M. Blei, Chong Wang, and John Paisley. “Stochastic Variational Inference.” ArXiv:1206.7051 [Cs, Stat], June 29, 2012. http://arxiv.org/abs/1206.7051.
Kingma, Diederik P., and Max Welling. “Auto-Encoding Variational Bayes.” ArXiv:1312.6114 [Cs, Stat], December 20, 2013. http://arxiv.org/abs/1312.6114.
Kurihara, Kenichi, Max Welling, and Nikos Vlassis. "Accelerated variational Dirichlet process mixtures." In Advances in neural information processing systems, pp. 761-768. 2007.
Sudderth, Erik Blaine. "Graphical models for visual object recognition and tracking." PhD diss., Massachusetts Institute of Technology, 2006.

Discriminant analysis

2019-01-03T00:00:00Z

In this post I talk about the theory and implementation of linear and quadratic discriminant analysis, classical methods in statistical learning.

Acknowledgement. Various sources were of great help to my understanding of the subject, including Chapter 4 of The Elements of Statistical Learning, Stanford CS229 Lecture notes, and the scikit-learn code. Research was done while working at KTH mathematics department.

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.

Theory

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.

\[(x | y = i) \sim N(\mu_i, \Sigma_i).\]

It also assumes a categorical class prior:

\[\mathbb P(y = i) = \pi_i\]

The log-likelihood is thus

\[\begin{aligned} \log \mathbb P(y = i | x) &= \log \mathbb P(x | y = i) \log \mathbb P(y = i) + C\\ &= - {1 \over 2} \log \det \Sigma_i - {1 \over 2} (x - \mu_i)^T \Sigma_i^{-1} (x - \mu_i) + \log \pi_i + C', \qquad (0) \end{aligned}\]

where $C$ and $C'$ are constants.

Thus the prediction is done by taking argmax of the above formula.

In training, let $X$, $y$ be the input data, where $X$ is of shape $m \times n$, and $y$ of shape $m$. We adopt the convention that each row of $X$ is a sample $x^{(i)T}$. So there are $m$ samples and $n$ features. Denote by $m_i = \#\{j: y_j = i\}$ be the number of samples in class $i$. Let $n_c$ be the number of classes.

We estimate $\mu_i$ by the sample means, and $\pi_i$ by the frequencies:

\[\begin{aligned} \mu_i &:= {1 \over m_i} \sum_{j: y_j = i} x^{(j)}, \\ \pi_i &:= \mathbb P(y = i) = {m_i \over m}. \end{aligned}\]

Linear discriminant analysis (LDA) is a specialisation of QDA: it assumes all classes share the same covariance, i.e. $\Sigma_i = \Sigma$ for all $i$.

Guassian Naive Bayes is a different specialisation of QDA: it assumes that all $\Sigma_i$ are diagonal, since all the features are assumed to be independent.

QDA

We look at QDA.

We estimate $\Sigma_i$ by the variance mean:

\[\begin{aligned} \Sigma_i &= {1 \over m_i - 1} \sum_{j: y_j = i} \hat x^{(j)} \hat x^{(j)T}. \end{aligned}\]

where $\hat x^{(j)} = x^{(j)} - \mu_{y_j}$ are the centred $x^{(j)}$. Plugging this into (0) we are done.

There are two problems that can break the algorithm. First, if one of the $m_i$ is $1$, then $\Sigma_i$ is ill-defined. Second, one of $\Sigma_i$'s might be singular.

In either case, there is no way around it, and the implementation should throw an exception.

This won't be a problem of the LDA, though, unless there is only one sample for each class.

Vanilla LDA

Now let us look at LDA.

Since all classes share the same covariance, we estimate $\Sigma$ using sample variance

\[\begin{aligned} \Sigma &= {1 \over m - n_c} \sum_j \hat x^{(j)} \hat x^{(j)T}, \end{aligned}\]

where $\hat x^{(j)} = x^{(j)} - \mu_{y_j}$ and ${1 \over m - n_c}$ comes from Bessel's Correction.

Let us write down the decision function (0). We can remove the first term on the right hand side, since all $\Sigma_i$ are the same, and we only care about argmax of that equation. Thus it becomes

\[- {1 \over 2} (x - \mu_i)^T \Sigma^{-1} (x - \mu_i) + \log\pi_i. \qquad (1)\]

Notice that we just walked around the problem of figuring out $\log \det \Sigma$ when $\Sigma$ is singular.

But how about $\Sigma^{-1}$?

We sidestep this problem by using the pseudoinverse of $\Sigma$ instead. This can be seen as applying a linear transformation to $X$ to turn its covariance matrix to identity. And thus the model becomes a sort of a nearest neighbour classifier.

Nearest neighbour classifier

More specifically, we want to transform the first term of (0) to a norm to get a classifier based on nearest neighbour modulo $\log \pi_i$:

\[- {1 \over 2} \|A(x - \mu_i)\|^2 + \log\pi_i\]

To compute $A$, we denote

\[X_c = X - M,\]

where the $i$th row of $M$ is $\mu_{y_i}^T$, the mean of the class $x_i$ belongs to, so that $\Sigma = {1 \over m - n_c} X_c^T X_c$.

Let

\[{1 \over \sqrt{m - n_c}} X_c = U_x \Sigma_x V_x^T\]

be the SVD of ${1 \over \sqrt{m - n_c}}X_c$. Let $D_x = \text{diag} (s_1, ..., s_r)$ be the diagonal matrix with all the nonzero singular values, and rewrite $V_x$ as an $n \times r$ matrix consisting of the first $r$ columns of $V_x$.

Then with an abuse of notation, the pseudoinverse of $\Sigma$ is

\[\Sigma^{-1} = V_x D_x^{-2} V_x^T.\]

So we just need to make $A = D_x^{-1} V_x^T$. When it comes to prediction, just transform $x$ with $A$, and find the nearest centroid $A \mu_i$ (again, modulo $\log \pi_i$) and label the input with $i$.

Dimensionality reduction

We can further simplify the prediction by dimensionality reduction. Assume $n_c \le n$. Then the centroid spans an affine space of dimension $p$ which is at most $n_c - 1$. So what we can do is to project both the transformed sample $Ax$ and centroids $A\mu_i$ to the linear subspace parallel to the affine space, and do the nearest neighbour classification there.

So we can perform SVD on the matrix $(M - \bar x) V_x D_x^{-1}$ where $\bar x$, a row vector, is the sample mean of all data i.e. average of rows of $X$:

\[(M - \bar x) V_x D_x^{-1} = U_m \Sigma_m V_m^T.\]

Again, we let $V_m$ be the $r \times p$ matrix by keeping the first $p$ columns of $V_m$.

The projection operator is thus $V_m$. And so the final transformation is $V_m^T D_x^{-1} V_x^T$.

There is no reason to stop here, and we can set $p$ even smaller, which will result in a lossy compression / regularisation equivalent to doing principle component analysis on $(M - \bar x) V_x D_x^{-1}$.

Note that as of 2019-01-04, in the scikit-learn implementation of LDA, the prediction is done without any lossy compression, even if the parameter n_components 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 n_components.

Fisher discriminant analysis

The Fisher discriminant analysis involves finding an $n$-dimensional vector $a$ that maximises between-class covariance with respect to within-class covariance:

\[{a^T M_c^T M_c a \over a^T X_c^T X_c a},\]

where $M_c = M - \bar x$ is the centred sample mean matrix.

As it turns out, this is (almost) equivalent to the derivation above, modulo a constant. In particular, $a = c V_m^T D_x^{-1} V_x^T$ where $p = 1$ for arbitrary constant $c$.

To see this, we can first multiply the denominator with a constant ${1 \over m - n_c}$ so that the matrix in the denominator becomes the covariance estimate $\Sigma$.

We decompose $a$: $a = V_x D_x^{-1} b + \tilde V_x \tilde b$, where $\tilde V_x$ consists of column vectors orthogonal to the column space of $V_x$.

We ignore the second term in the decomposition. In other words, we only consider $a$ in the column space of $V_x$.

Then the problem is to find an $r$-dimensional vector $b$ to maximise

\[{b^T (M_c V_x D_x^{-1})^T (M_c V_x D_x^{-1}) b \over b^T b}.\]

This is the problem of principle component analysis, and so $b$ is first column of $V_m$.

Therefore, the solution to Fisher discriminant analysis is $a = c V_x D_x^{-1} V_m$ with $p = 1$.

Linear model

The model is called linear discriminant analysis because it is a linear model. To see this, let $B = V_m^T D_x^{-1} V_x^T$ be the matrix of transformation. Now we are comparing

\[- {1 \over 2} \| B x - B \mu_k\|^2 + \log \pi_k\]

across all $k$s. Expanding the norm and removing the common term $\|B x\|^2$, we see a linear form:

\[\mu_k^T B^T B x - {1 \over 2} \|B \mu_k\|^2 + \log\pi_k\]

So the prediction for $X_{\text{new}}$ is

\[\text{argmax}_{\text{axis}=0} \left(K B^T B X_{\text{new}}^T - {1 \over 2} \|K B^T\|_{\text{axis}=1}^2 + \log \pi\right)\]

thus the decision boundaries are linear.

This is how scikit-learn implements LDA, by inheriting from LinearClassifierMixin and redirecting the classification there.

Implementation

This is where things get interesting. How do I validate my understanding of the theory? By implementing and testing the algorithm.

I try to implement it as close as possible to the natural language / mathematical descriptions of the model, which means clarity over performance.

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.

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.

The result is here.

Fun facts about LDA

One property that can be used to test the LDA implementation is the fact that the scatter matrix $B(X - \bar x)^T (X - \bar X) B^T$ of the transformed centred sample is diagonal.

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:

\[X_c^T X_c + M_c^T M_c = (X - \bar x)^T (X - \bar x) = (X_c + M_c)^T (X_c + M_c).\]

The verification is not very hard and left as an exercise.

Shapley, LIME and SHAP

2018-12-02T00:00:00Z

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).

Acknowledgement. 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.

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.

Shapley values

A coalitional game $(v, N)$ of $n$ players involves

The set $N = \{1, 2, ..., n\}$ that represents the players.
A function $v: 2^N \to \mathbb R$, where $v(S)$ is the worth of coalition $S \subset N$.

The Shapley values $\phi_i(v)$ of such a game specify a fair way to distribute the total worth $v(N)$ to the players. It is defined as (in the following, for a set $S \subset N$ we use the convention $s = |S|$ to be the number of elements of set $S$ and the shorthand $S - i := S \setminus \{i\}$ and $S + i := S \cup \{i\}$)

\[\phi_i(v) = \sum_{S: i \in S} {(n - s)! (s - 1)! \over n!} (v(S) - v(S - i)).\]

It is not hard to see that $\phi_i(v)$ can be viewed as an expectation:

\[\phi_i(v) = \mathbb E_{S \sim \nu_i} (v(S) - v(S - i))\]

where $\nu_i(S) = n^{-1} {n - 1 \choose s - 1}^{-1} 1_{i \in S}$, that is, first pick the size $s$ uniformly from $\{1, 2, ..., n\}$, then pick $S$ uniformly from the subsets of $N$ that has size $s$ and contains $i$.

The Shapley values satisfy some nice properties which are readily verified, including:

Efficiency. $\sum_i \phi_i(v) = v(N) - v(\emptyset)$.
Symmetry. If for some $i, j \in N$, for all $S \subset N$, we have $v(S + i) = v(S + j)$, then $\phi_i(v) = \phi_j(v)$.
Null player. If for some $i \in N$, for all $S \subset N$, we have $v(S + i) = v(S)$, then $\phi_i(v) = 0$.
Linearity. $\phi_i$ is linear in games. That is $\phi_i(v) + \phi_i(w) = \phi_i(v + w)$, where $v + w$ is defined by $(v + w)(S) := v(S) + w(S)$.

In the literature, an added assumption $v(\emptyset) = 0$ is often given, in which case the Efficiency property is defined as $\sum_i \phi_i(v) = v(N)$. 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 $v(\emptyset)$ may not be $0$. In the SHAP paper, an intercept $\phi_0 = v(\emptyset)$ is added which fixes this problem when making connections to the Shapley values.

Conversely, according to Strumbelj-Kononenko (2010), it was shown in Shapley's original paper (Shapley, 1953) that these four properties together with $v(\emptyset) = 0$ defines the Shapley values.

LIME

LIME (Ribeiro et. al. 2016) is a model that offers a way to explain feature contributions of supervised learning models locally.

Let $f: X_1 \times X_2 \times ... \times X_n \to \mathbb R$ be a function. We can think of $f$ as a model, where $X_j$ is the space of $j$th feature. For example, in a language model, $X_j$ may correspond to the count of the $j$th word in the vocabulary, i.e. the bag-of-words model.

The output may be something like housing price, or log-probability of something.

LIME tries to assign a value to each feature locally. By locally, we mean that given a specific sample $x \in X := \prod_{i = 1}^n X_i$, we want to fit a model around it.

More specifically, let $h_x: 2^N \to X$ be a function defined by

\[(h_x(S))_i = \begin{cases} x_i, & \text{if }i \in S; \\ 0, & \text{otherwise.} \end{cases}\]

That is, $h_x(S)$ masks the features that are not in $S$, or in other words, we are perturbing the sample $x$. Specifically, $h_x(N) = x$. Alternatively, the $0$ in the "otherwise" case can be replaced by some kind of default value (see the section titled SHAP in this post).

For a set $S \subset N$, let us denote $1_S \in \{0, 1\}^n$ to be an $n$-bit where the $k$th bit is $1$ if and only if $k \in S$.

Basically, LIME samples $S_1, S_2, ..., S_m \subset N$ to obtain a set of perturbed samples $x_i = h_x(S_i)$ in the $X$ space, and then fits a linear model $g$ using $1_{S_i}$ as the input samples and $f(h_x(S_i))$ as the output samples:

Problem(LIME). Find $w = (w_1, w_2, ..., w_n)$ that minimises

\[\sum_i (w \cdot 1_{S_i} - f(h_x(S_i)))^2 \pi_x(h_x(S_i))\]

where $\pi_x(x')$ is a function that penalises $x'$s that are far away from $x$. In the LIME paper the Gaussian kernel was used:

\[\pi_x(x') = \exp\left({- \|x - x'\|^2 \over \sigma^2}\right).\]

Then $w_i$ represents the importance of the $i$th feature.

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.

Remark. One difference between our account here and the one in the LIME paper is: the dimension of the data space may differ from $n$ (see Section 3.1 of that paper). But in the case of text data, they do use bag-of-words (our $X$) for an “intermediate” representation. So my understanding is, in their context, there is an “original” data space (let’s call it $X'$). And there is a one-one correspondence between $X'$ and $X$ (let’s call it $r: X' \to X$), so that given a sample $x' \in X'$, we can compute the output of $S$ in the local model with $f(r^{-1}(h_{r(x')}(S)))$. As an example, in the example of $X$ being the bag of words, $X'$ may be the embedding vector space, so that $r(x') = A^{-1} x'$, where $A$ is the word embedding matrix. Therefore, without loss of generality, we assume the input space to be $X$ which is of dimension $n$.

Shapley values and LIME

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.).

To see the connection, we need to modify LIME a bit.

First, we need to make LIME less efficient by considering all the $2^n$ subsets instead of the $m$ samples $S_1, S_2, ..., S_{m}$.

Then we need to relax the definition of $\pi_x$. It no longer needs to penalise samples that are far away from $x$. In fact, we will see later than the choice of $\pi_x(x')$ that yields the Shapley values is high when $x'$ is very close or very far away from $x$, and low otherwise. We further add the restriction that $\pi_x(h_x(S))$ only depends on the size of $S$, thus we rewrite it as $q(s)$ instead.

We also denote $v(S) := f(h_x(S))$ and $w(S) = \sum_{i \in S} w_i$.

Finally, we add the Efficiency property as a constraint: $\sum_{i = 1}^n w_i = f(x) - f(h_x(\emptyset)) = v(N) - v(\emptyset)$.

Then the problem becomes a weighted linear regression:

Problem. minimises $\sum_{S \subset N} (w(S) - v(S))^2 q(s)$ over $w$ subject to $w(N) = v(N) - v(\emptyset)$.

Claim (Charnes et. al. 1988). The solution to this problem is

\[w_i = {1 \over n} (v(N) - v(\emptyset)) + \left(\sum_{s = 1}^{n - 1} {n - 2 \choose s - 1} q(s)\right)^{-1} \sum_{S \subset N: i \in S} \left({n - s \over n} q(s) v(S) - {s - 1 \over n} q(s - 1) v(S - i)\right). \qquad (-1)\]

Specifically, if we choose

\[q(s) = c {n - 2 \choose s - 1}^{-1}\]

for any constant $c$, then $w_i = \phi_i(v)$ are the Shapley values.

Remark. Don't worry about this specific choice of $q(s)$ when $s = 0$ or $n$, because $q(0)$ and $q(n)$ 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 ${\ell \choose k} = 0$ if $k < 0$ or $k > \ell$, in which case $q(0) = q(n) = \infty$.

In Lundberg-Lee (2017), $c$ is chosen to be $1 / n$, see Theorem 2 there.

In Charnes et. al. 1988, the $w_i$s defined in (-1) are called the generalised Shapley values.

Proof. The Lagrangian is

\[L(w, \lambda) = \sum_{S \subset N} (v(S) - w(S))^2 q(s) - \lambda(w(N) - v(N) + v(\emptyset)).\]

and by making $\partial_{w_i} L(w, \lambda) = 0$ we have

\[{1 \over 2} \lambda = \sum_{S \subset N: i \in S} (w(S) - v(S)) q(s). \qquad (0)\]

Summing (0) over $i$ and divide by $n$, we have

\[{1 \over 2} \lambda = {1 \over n} \sum_i \sum_{S: i \in S} (w(S) q(s) - v(S) q(s)). \qquad (1)\]

We examine each of the two terms on the right hand side.

Counting the terms involving $w_i$ and $w_j$ for $j \neq i$, and using $w(N) = v(N) - v(\emptyset)$ we have

\[\begin{aligned} &\sum_{S \subset N: i \in S} w(S) q(s) \\ &= \sum_{s = 1}^n {n - 1 \choose s - 1} q(s) w_i + \sum_{j \neq i}\sum_{s = 2}^n {n - 2 \choose s - 2} q(s) w_j \\ &= q(1) w_i + \sum_{s = 2}^n q(s) \left({n - 1 \choose s - 1} w_i + \sum_{j \neq i} {n - 2 \choose s - 2} w_j\right) \\ &= q(1) w_i + \sum_{s = 2}^n \left({n - 2 \choose s - 1} w_i + {n - 2 \choose s - 2} (v(N) - v(\emptyset))\right) q(s) \\ &= \sum_{s = 1}^{n - 1} {n - 2 \choose s - 1} q(s) w_i + \sum_{s = 2}^n {n - 2 \choose s - 2} q(s) (v(N) - v(\emptyset)). \qquad (2) \end{aligned}\]

Summing (2) over $i$, we obtain

\[\begin{aligned} &\sum_i \sum_{S: i \in S} w(S) q(s)\\ &= \sum_{s = 1}^{n - 1} {n - 2 \choose s - 1} q(s) (v(N) - v(\emptyset)) + \sum_{s = 2}^n n {n - 2 \choose s - 2} q(s) (v(N) - v(\emptyset))\\ &= \sum_{s = 1}^n s{n - 1 \choose s - 1} q(s) (v(N) - v(\emptyset)). \qquad (3) \end{aligned}\]

For the second term in (1), we have

\[\sum_i \sum_{S: i \in S} v(S) q(s) = \sum_{S \subset N} s v(S) q(s). \qquad (4)\]

Plugging (3)(4) in (1), we have

\[{1 \over 2} \lambda = {1 \over n} \left(\sum_{S \subset N} s q(s) v(S) - \sum_{s = 1}^n s {n - 1 \choose s - 1} q(s) (v(N) - v(\emptyset))\right). \qquad (5)\]

Plugging (5)(2) in (0) and solve for $w_i$, we have

\[w_i = {1 \over n} (v(N) - v(\emptyset)) + \left(\sum_{s = 1}^{n - 1} {n - 2 \choose s - 1} q(s) \right)^{-1} \left( \sum_{S: i \in S} q(s) v(S) - {1 \over n} \sum_{S \subset N} s q(s) v(S) \right). \qquad (6)\]

By splitting all subsets of $N$ into ones that contain $i$ and ones that do not and pair them up, we have

\[\sum_{S \subset N} s q(s) v(S) = \sum_{S: i \in S} (s q(s) v(S) + (s - 1) q(s - 1) v(S - i)).\]

Plugging this back into (6) we get the desired result. $\square$

SHAP

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.

Recall that we want to calculate feature contributions to a model $f$ at a sample $x$.

Let $\mu$ be a probability density function over the input space $X = X_1 \times ... \times X_n$. A natural choice would be the density that generates the data, or one that approximates such density (e.g. empirical distribution).

The feature contribution (SHAP value) is thus defined as the Shapley value $\phi_i(v)$, where

\[v(S) = \mathbb E_{z \sim \mu} (f(z) | z_S = x_S). \qquad (7)\]

So it is a conditional expectation where $z_i$ is clamped for $i \in S$. In fact, the definition of feature contributions in this form predates Lundberg-Lee 2017. For example, it can be found in Strumbelj-Kononenko 2014.

One simplification is to assume the $n$ features are independent, thus $\mu = \mu_1 \times \mu_2 \times ... \times \mu_n$. In this case, (7) becomes

\[v(S) = \mathbb E_{z_{N \setminus S} \sim \mu_{N \setminus S}} f(x_S, z_{N \setminus S}) \qquad (8)\]

For example, Strumbelj-Kononenko (2010) considers this scenario where $\mu$ is the uniform distribution over $X$, see Definition 4 there.

A further simplification is model linearity, which means $f$ is linear. In this case, (8) becomes

\[v(S) = f(x_S, \mathbb E_{\mu_{N \setminus S}} z_{N \setminus S}). \qquad (9)\]

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 $h_x(S)$ to

\[(h_x(S))_i = \begin{cases} x_i, & \text{if }i \in S; \\ \mathbb E_{\mu_i} z_i, & \text{otherwise.} \end{cases}\]

But we also need to boldly assume the original $f$ to be linear, which in my view, defeats the purpose of interpretability, because linear models are interpretable by themselves.

One may argue that perhaps we do not need linearity to define $v(S)$ 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?

Evaluating SHAP

The quest of the SHAP paper can be decoupled into two independent components: showing the niceties of Shapley values and choosing the coalitional game $v$.

The SHAP paper argues that Shapley values $\phi_i(v)$ 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).

Indeed, both efficiency (the “additive feature attribution methods” in the paper) and monotonicity are meaningful when considering $\phi_i(v)$ as the feature contribution of the $i$th feature.

The question is thus reduced to the second component: what constitutes a nice choice of $v$?

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.

References

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. https://doi.org/10.1007/978-94-009-3677-5_7.
Lundberg, Scott, and Su-In Lee. “A Unified Approach to Interpreting Model Predictions.” ArXiv:1705.07874 [Cs, Stat], May 22, 2017. http://arxiv.org/abs/1705.07874.
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. http://arxiv.org/abs/1602.04938.
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. https://doi.org/10.1515/9781400881970-018.
Strumbelj, Erik, and Igor Kononenko. “An Efficient Explanation of Individual Classifications Using Game Theory.” J. Mach. Learn. Res. 11 (March 2010): 1–18.
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. https://doi.org/10.1007/s10115-013-0679-x.
Young, H. P. “Monotonic Solutions of Cooperative Games.” International Journal of Game Theory 14, no. 2 (June 1, 1985): 65–72. https://doi.org/10.1007/BF01769885.

Automatic differentiation

2018-06-03T00:00:00Z

This post serves as a note and explainer of autodiff. It is licensed under GNU FDL.

For my learning I benefited a lot from Toronto CSC321 slides and the autodidact project which is a pedagogical implementation of Autograd. 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.

Automatic differentiation (AD) is a way to compute derivatives. It does so by traversing through a computational graph using the chain rule.

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.

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.

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.

For example, the Sigmoid activation $a' = \sigma(w a + b)$ is quite simple, but it should be decomposed to simpler computations:

$a' = 1 / t_1$
$t_1 = 1 + t_2$
$t_2 = \exp(t_3)$
$t_3 = - t_4$
$t_4 = t_5 + b$
$t_5 = w a$

Thus the function $a'(a)$ is decomposed to elementary operations like addition, subtraction, multiplication, reciprocitation, exponentiation, logarithm etc. And the rmAD engine stores the Jacobian of these elementary operations.

Since in neural networks we want to find derivatives of a single loss function $L(x; \theta)$, we can omit $L$ when writing derivatives and denote, say $\bar \theta_k := \partial_{\theta_k} L$.

In implementations of rmAD, one can represent the Jacobian as a transformation $j: (x \to y) \to (y, \bar y, x) \to \bar x$. $j$ is called the Vector Jacobian Product (VJP). For example, $j(\exp)(y, \bar y, x) = y \bar y$ since given $y = \exp(x)$,

$\partial_x L = \partial_x y \cdot \partial_y L = \partial_x \exp(x) \cdot \partial_y L = y \bar y$

as another example, $j(+)(y, \bar y, x_1, x_2) = (\bar y, \bar y)$ since given $y = x_1 + x_2$, $\bar{x_1} = \bar{x_2} = \bar y$.

Similarly,

$j(/)(y, \bar y, x_1, x_2) = (\bar y / x_2, - \bar y x_1 / x_2^2)$
$j(\log)(y, \bar y, x) = \bar y / x$
$j((A, \beta) \mapsto A \beta)(y, \bar y, A, \beta) = (\bar y \otimes \beta, A^T \bar y)$.
etc...

In the third one, the function is a matrix $A$ multiplied on the right by a column vector $\beta$, and $\bar y \otimes \beta$ is the tensor product which is a fancy way of writing $\bar y \beta^T$. See numpy_vjps.py for the implementation in autodidact.

So, given a node say $y = y(x_1, x_2, ..., x_n)$, and given the value of $y$, $x_{1 : n}$ and $\bar y$, rmAD computes the values of $\bar x_{1 : n}$ by using the Jacobians.

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.

It is a nice exercise to derive the backpropagation in the fully connected feedforward neural networks (e.g. the one for MNIST in Neural Networks and Deep Learning) using rmAD.

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.

Things to look further into: the higher-order functional currying form $j: (x \to y) \to (y, \bar y, x) \to \bar x$ begs for a functional programming implementation.

Updates on open research

2018-04-29T00:00:00Z

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.

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.

First, I read about Richard Stallman the founder of the free software movement, in his biography by Sam Williams and his own collection of essays Free software, free society, 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 Hackers, 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.

Also a month ago I attended a workshop called Open research: rethinking scientific collaboration. 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 Matteo Smerlak and Ashley Milsted for invitation and hosting.

From both of these I feel like I should write an updated post on open research.

Freedom and community

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.

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.

NDAs. 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.

Reproducibility. 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.

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.

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 four essential freedoms, a community that rejects NDAs and upholds reproducibility.

Here are some ideas on how to tackle these two problems and build the community:

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 Creative Commons Share-Alike or the GNU Free Documentation License.
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.
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.
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.

Tools for open research

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.

During the workshop many efforts for open research were mentioned, and afterwards I was also made aware by more of them, like the following:

OSF, 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.
SciPost, 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.
Knowen, a platform to organise knowledge in directed acyclic graphs. Could be useful for building the infrastructure of mathematical knowledge.
Fermat’s Library, the journal club website that crowd-annotates one notable paper per week released a Chrome extension Librarian that overlays a commenting interface on arXiv. As an example Ian Goodfellow did an AMA (ask me anything) on his GAN paper.
The Polymath project, 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 (currently only one active project).
The Stacks Project. I was made aware of this project by Yiting. 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.

An anecdote from the workshop

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”.

To which an organiser replied: maybe we should rename the workshop “Not-bad science”.

The Mathematical Bazaar

2017-08-07T00:00:00Z

In this essay I describe some problems in academia of mathematics and propose an open source model, which I call open research in mathematics.

This essay is a work in progress - comments and criticisms are welcome! ¹

Before I start I should point out that

Open research is not open access. In fact the latter is a prerequisite to the former.
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.

problems of academia

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.

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.

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.

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)

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.

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:

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.
Mistakes made by established mathematicians are more tolerable than those less established.
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.

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.

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:

Open and public discussions and collaborations of mathematical research projects online
Open review to validate results, where author name, reviewer name, comments and responses are all publicly available online.

To this end, a Github model is fitting. Let me first describe how open source collaboration works on Github.

open source collaborations on Github

On Github, 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.

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.

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.

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 @someone.

open research in mathematics

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.

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 less 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.

Compared to the academic model, open research also has the following advantages:

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.
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.
Similarly, the burden of peer review can also be shifted from a few appointed reviewers to the crowd.

Please send your comments to my email address - I am still looking for ways to add a comment functionality to this website.↩

Open mathematical research and launching toywiki

2017-04-25T00:00:00Z

As an experimental project, I am launching toywiki.

It hosts a collection of my research notes.

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

Here are the links: toywiki and github repo.

Feedbacks are welcome by email.

A $q$-Robinson-Schensted-Knuth algorithm and a $q$-polymer

2016-10-13T00:00:00Z

(Latest update: 2017-01-12) In Matveev-Petrov 2016 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.

This article is available at arXiv. 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.

Here is a “dev” version of the article, which I shall push to arXiv when it stablises. Below is the changelog.

2017-01-12: Typos and grammar, arXiv v2.
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.
2016-11-13: Fixed some typos in the proof of Theorem 3.
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.
2016-10-20: Fixed a few typos. Updated some references. Added a reference: a set of notes titled “RSK via local transformations”. It is written by Sam Hopkins 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.

AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieu

2015-07-15T00:00:00Z

A Macdonald superpolynomial (introduced in [O. Blondeau-Fournier et al., Lett. Math. Phys. 101 (2012), no. 1, 27–47; MR2935476; J. Comb. 3 (2012), no. 3, 495–561; MR3029444]) 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

shows that the dMp has a unique decomposition into bisymmetric monomials;
calculates the norm of the dMp;
calculates the kernel of the Cauchy-Littlewood-type identity of the dMp;
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;
defines the $\omega$ -automorphism in this setting, which was used to prove an identity involving products of four Littlewood-Richardson coefficients;
shows an explicit evaluation of the dMp motivated by the most general evaluation of the usual Macdonald polynomials;
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);
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;
defines an operator $\nabla^B$ as an analogue of the nabla operator $\nabla$ introduced in [F. Bergeron and A. M. Garsia, in Algebraic methods and $q$-special functions (Montréal, QC, 1996), 1–52, CRM Proc. Lecture Notes, 22, Amer. Math. Soc., Providence, RI, 1999; MR1726826]. 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 $A_n$ .

Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3306078, its copyright owned by the AMS.

On a causal quantum double product integral related to Lévy stochastic area.

2015-07-01T00:00:00Z

In this paper with Robin we study the family of causal double product integrals \[ \prod_{a < x < y < b}\left(1 + i{\lambda \over 2}(dP_x dQ_y - dQ_x dP_y) + i {\mu \over 2}(dP_x dP_y + dQ_x dQ_y)\right) \]

where $P$ and $Q$ 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 (Hudson-Pei2015). The main problem solved in this paper is the explicit evaluation of the continuum limit $W$ of the latter, and showing that $W$ is a unitary operator. The kernel of $W$ 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.

AMS review of 'Infinite binary words containing repetitions of odd period' by Badkobeh and Crochemore

2015-05-30T00:00:00Z

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 $g\: \Sigma_3^* \to \Sigma_2^*$

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. 339 (2005), no. 1, 19–34; MR2142071]. The negative results are proved by traversing all possible finite words satisfying the conditions.

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.

Below is the summary of the positive and negative results:

(Negative) $L(\infty, \omega, 2 \Bbb N) < \infty$ : $\nexists$ an infinite $3^+$ -free binary word avoiding all squares of odd periods. (Proposition 1)
(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).
(Positive) $L(\infty, \omega, 2 \Bbb N +
1. = \infty$
  
  $\exists$ an infinite $3^+$ -free binary word avoiding squares of even periods (Theorem 1).
(Positive) $L(\infty, \omega, \{1, 3\}) = \infty$ : $\exists$ an infinite $3^+$ -free binary word containing only squares of period 1 or 3 (Theorem 2).
(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).
(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).
(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).

Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3313467, its copyright owned by the AMS.

jst

2015-04-02T00:00:00Z

jst = juggling skill tree

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, here is the skill tree of sorceress in Diablo II.

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 Library of Juggling 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 Python, BeautifulSoup and pygraphviz to generate a jst (graded by difficulties, which is the leftmost column) from the Library of Juggling (click the image for the full size):

Unitary causal quantum stochastic double products as universal interactions I

2015-04-01T00:00:00Z

In this paper with Robin we show the explicit formulae for a family of unitary triangular and rectangular double product integrals which can be described as second quantisations.

AMS review of 'A weighted interpretation for the super Catalan numbers' by Allen and Gheorghiciuc

2015-01-20T00:00:00Z

The super Catalan numbers are defined as $$ T(m,n) = {(2 m)! (2 n)! 2 m! n! (m + n)!}. $$

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. 14 (1992), no. 2-3, 179–194; MR1187230]: $$ 4T(m,n) = T(m+1, n) + T(m, n+1). $$ 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.

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. 8 (2005), no. 2, Article 05.2.3, 13 pp.; MR2134162], 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.

Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3275875, its copyright owned by the AMS.

Symmetry property of $q$-weighted Robinson-Schensted algorithms and branching algorithms

2014-04-01T00:00:00Z

In this paper 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.

Growth graph of q-RS for 1423

Above is the growth graph of the $q$-weighted Robinson-Schensted algorithm for the permutation ${1 2 3 4\choose1 4 2 3}$.

A $q$-weighted Robinson-Schensted algorithm

2013-06-01T00:00:00Z

In this paper with Neil we construct a $q$-version of the Robinson-Schensted algorithm with column insertion. Like the usual RS correspondence 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 $q$.

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 (O’Connell 2012). When the input word $w$ is a random walk:

\begin{align*}\mathbb P(w=v)=\prod_{i=1}^na_{v_i},\qquad\sum_ja_j=1\end{align*}

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.

Yuchen Pei's Blog

Great but Manageable Expectations

Rényi divergence and differential privacy

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Open mathematical research and launching toywiki

A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer

AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieu

On a causal quantum double product integral related to Lévy stochastic area.

AMS review of 'Infinite binary words containing repetitions of odd period' by Badkobeh and Crochemore

jst

Unitary causal quantum stochastic double products as universal interactions I

AMS review of 'A weighted interpretation for the super Catalan numbers' by Allen and Gheorghiciuc

Symmetry property of \(q\)-weighted Robinson-Schensted algorithms and branching algorithms

A \(q\)-weighted Robinson-Schensted algorithm