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Diffstat (limited to 'posts')
| -rw-r--r-- | posts/2019-03-14-great-but-manageable-expectations.md | 14 |
1 files changed, 10 insertions, 4 deletions
diff --git a/posts/2019-03-14-great-but-manageable-expectations.md b/posts/2019-03-14-great-but-manageable-expectations.md index d622fd4..d5ca684 100644 --- a/posts/2019-03-14-great-but-manageable-expectations.md +++ b/posts/2019-03-14-great-but-manageable-expectations.md @@ -341,7 +341,7 @@ then we have $(\lambda, O(r^2 \lambda / \sigma^2))$-rdp. Wait, why is there a conjecture? Well, I have tried but not been able to prove the following, which is a hidden assumption in the original proof: -**Conjecture 1**. Let $p_i$, $q_i$, $\mu_i$, $\nu_i$ be +**Conjecture 1** \[Probably [FALSE](https://math.stackexchange.com/a/3152296/149540), to be removed\]. Let $p_i$, $q_i$, $\mu_i$, $\nu_i$ be probability densities on the same space for $i = 1 : n$. If $D_\lambda(p_i || q_i) \le D_\lambda(\mu_i || \nu_i)$ for all $i$, then @@ -351,9 +351,12 @@ Basically, it is saying \"if for each $i$, $p_i$ and $q_i$ are closer to each other than $\mu_i$ and $\nu_i$, then so are their averages over $i$\". So it is heuristically reasonable. +But it is probably [**FALSE**](https://math.stackexchange.com/a/3152296/149540). +<!--- This conjecture is equivalent to its special case when $n = 2$ by an induction argument (replacing one pair of densities at a time). +--> Recall the definition of $G_\lambda$ under the definition of Rényi differential privacy. The following Claim will be useful. @@ -381,7 +384,9 @@ I will break the proof into two parts: $c_1$, $c_2$ and the function $C(c_1, c_2)$ are important to the practicality and usefulness of Conjecture 0. -Part 1 can be derived using Conjecture 1. +Part 1 can be derived using Conjecture 1, but since Conjecture 1 is probably false, +let us rename Part 1 itself _Conjecture 2_, which needs to be verified by other means. +<!--- We use the notations $p_I$ and $q_I$ to be $q$ and $p$ conditioned on the subsampling index $I$, just like in the proof of the subsampling theorems (Claim 19 and 24). Then @@ -400,6 +405,7 @@ $|\mathcal I_\in| = r |\mathcal I|$, by Conjecture 1, we have Part 1. it suffices to prove a weaker version of it, by specialising on mixture of Gaussians, in order to have a Claim 26 without any conjectural assumptions. I have in fact posted the Conjecture on [Stackexchange](https://math.stackexchange.com/questions/3147963/an-inequality-related-to-the-renyi-divergence). +--> Now let us verify Part 2. @@ -486,7 +492,7 @@ In the following for consistency we retain $k$ as the number of epochs, and use $T := k / r$ to denote the number of compositions / steps / minibatches. With Conjecture 0 we have: -**Claim 28**. Assuming Conjecture 1 is true. Let $\epsilon, c_1, c_2 > 0$, +**Claim 28**. Assuming Conjecture 2 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 @@ -733,7 +739,7 @@ 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 +1. Prove Conjecture 2 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 |