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:

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

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

1. it is also \((\epsilon, \exp((\lambda - 1) (\rho - \epsilon)))\)-dp for any \(\epsilon \ge \rho\).
2. Alternatively, \(M\) is \((\epsilon, - \exp(\kappa_M^*(\epsilon)))\)-dp for any \(\epsilon > 0\).
3. 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.

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 \(M\) be the Gaussian mechanism with subsampling rate \(r\), and \(p\) and \(q\) be the laws of \(M(x)\) and \(M(x')\) respectively, where \(d(x, x') = 1\). Then

\[D_\lambda (p || q) \le D_\lambda (r \mu_1 + (1 - r) \mu_0 || \mu_0)\]

where \(\mu_i = N(i, \sigma^2)\).

Remark. Conjecture 1 is heuristically reasonable. To see this, let us 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 for \(I \in \mathcal I_\in\),

\[D_\lambda(p_I || q_I) \le D_\lambda(\mu_0 || \mu_1),\]

and for \(I \in \mathcal I_\notin\),

\[D_\lambda(p_I || q_I) = 0 = D_\lambda(\mu_0 || \mu_0).\]

Since we are taking an average over \(\mathcal I\), of which \(r |\mathcal I|\) are in \(\mathcal I_\in\) and \((1 - r) |\mathcal I|\) are in \(\mathcal I_\notin\), (9.3) says "the inequalities carry over averaging".

A more general version of Conjecture 1 has been proven false. The counter example for the general version does not apply here, so it is still possible Conjecture 1 is true.

Let \(p_\in\) (resp. \(q_\in\)) be the average of \(p_I\) (resp. \(q_I\)) over \(I \in \mathcal I_\in\), and \(p_\notin\) (resp. \(q_\notin\)) be the average of \(p_I\) (resp. \(q_I\)) over \(I \in \mathcal I_\notin\).

Immediately we have \(p_\notin = q_\notin\), hence

\[D_\lambda(p_\notin || q_\notin) = 0 = D_\lambda(\mu_0 || \mu_0). \qquad(9.7)\]

By Claim 25, we have

\[D_\lambda(p_\in || q_\in) \le D_\lambda (\mu_1 || \mu_0). \qquad(9.9) \]

So one way to prove Conjecture 1 is perhaps prove a more specialised comparison theorem than the false conjecture:

Given (9.7) and (9.9), show that

\[D_\lambda(r p_\in + (1 - r) p_\notin || r q_\in + (1 - r) q_\notin) \le D_\lambda(r \mu_1 + (1 - r) \mu_0 || \mu_0).\]

[End of Remark]

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. By Conjecture 1, it suffices to prove the following:

If \(r \le c_1 \sigma^{-1}\) and \(\lambda \le c_2 \sigma^2\) for some positive constant \(c_1\) and \(c_2\), 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 Claim 26.

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 Claim 26 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 1 is true, for the following reasons:

1. 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: \[\exp(k C(c_1, c_2) - \lambda \epsilon) \ge ...\]
2. The condition \(r = O(\sigma^{-1})\) of Claim 26 whose result is used in the proof of this theorem is not mentioned in the statement of the proof. The implication is that (10) becomes an ill-formed condition as the right hand side also depends on \(\sigma\).

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.

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

1. Claim 9 (single Gaussian mechanism privacy guarantee) -> Claim 19 (Subsampling theorem) -> Claim 18 (Advanced Adaptive Composition Theorem)
2. Example 1 (RDP for the Gaussian mechanism) -> Claim 22 (Moment Composition Theorem) -> Example 3 (Moment composition applied to the Gaussian mechanism)
3. 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

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

Warning. What follows is a bit messy.

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.

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:

1. Prove Conjecture 1
2. Find a theoretically definitive answer whether the methods in Part 1 or Part 2 yield better privacy guarantees.
3. 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?
4. Find out how useful Claim 26 is. Perhaps start with computing the constant \(C\) nemerically.
5. Help with the aforementioned issue in the Tensorflow privacy package.

• 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.

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 variable¹ 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:

1. (General case) \(q\) is in the form of \(q(y) = p(y + \Delta)\) for some bounded constant \(\Delta\).
2. (Compositions) \(p\) and \(q\) are combinatorial or sequential compositions of some simpler \(p_i\)'s and \(q_i\)'s respectively
3. (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…

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 some space \(Z\). 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 \(S \subset \mathbb R^n\),

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

Practically speaking, this means given the results from perturbed query on two known databases that differs by one row, it is hard to determine which result is from which database.

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). \(\square\)

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 \(\delta < 1\).

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:

1. Composition of \(\epsilon\)-dp or more generally \((\epsilon, \delta)\)-dp mechanisms
2. Composition of independent or more generally adaptive mechanisms
3. 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:

1. Claim 10 - Basic composition theorem for \((\epsilon, \delta)\)-dp with adaptive mechanisms: by a direct proof with an induction argument
2. Claim 14 - Advanced composition theorem for \(\epsilon\)-dp with independent mechanisms: by factorising privacy loss and using Hoeffding's Inequality
3. Claim 16 - Advanced composition theorem for \(\epsilon\)-dp with adaptive mechanisms: by factorising privacy loss and using Azuma's Inequality
4. 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.

1. 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 \(1\), \[\begin{aligned} \mathbb P(\xi_i \in S_i | \xi_{< i} = y_{< i}) &\le (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) + \delta_i) \wedge 1 \\ &\le (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) + \delta_i) \wedge (1 + \delta_i) \\ &= (e^{\epsilon_i} \mathbb P(\eta_i \in S_i | \eta_{< i} = y_{< i}) \wedge 1) + \delta_i \end{aligned}\]

2. 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

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

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.

Remark. This theorem appeared in Dwork-Rothblum-Vadhan 2010, but I could not find a proof there. A proof can be found in Dwork-Roth 2013 (See Theorem 3.20 there). Here I prove it in a similar way, except that instead of the use of an intermediate random variable there, I use the conditional probability results from Claim 5, the approach mentioned in Vadhan 2017.

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

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

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 mechanism on \(Z^n\). Define mechanism \(M\) on \(Z^m\) 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 \(x, x' \in Z^n\) 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.

• 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.

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.

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.

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.

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:

1. Set \(t = 1\).
2. Pick \(\ell\) uniformly from \(\{1, 2, ..., n_d\}\).
3. 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).
4. Compute \((\tilde \phi^{\eta_k}_w)_{k = 1 : n_z, w = 1 : n_x}\) using the following formula (compare with (12)) \[\tilde \phi^{\eta_k}_w = \beta + n_d \sum_{i} r_{\ell i k} 1_{x_{\ell i} = w}\]
5. Update the exponential moving average \((\phi^{\eta_k}_w)_{k = 1 : n_z, w = 1 : n_x}\): \[\phi^{\eta_k}_w = (1 - \rho_t) \phi^{\eta_k}_w + \rho_t \tilde \phi^{\eta_k}_w\]
6. 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 \(\tau \ge 0\).

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.

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• 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.

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.

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.

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 (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 \(n\).

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

\[{1 \over 2} \lambda = {1 \over n} \left(\sum_{s = 1}^n s {n - 1 \choose s - 1} q(s) (v(N) - v(\emptyset)) - \sum_{S \subset N} s q(s) v(S) \right). \qquad (5)\]

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. \(\square\)

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?

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.

• 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.

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:

1. 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.
2. 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.
3. 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.
4. 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.

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.

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".

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.

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.

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.

• The Cathedral and the Bazaar by Eric Raymond
• Doing sience online by Michael Nielson
• Is massively collaborative mathematics possible? by Timothy Gowers