From 2fa0512a84daa2b5d1ef77f70a58b216e49ab851 Mon Sep 17 00:00:00 2001 From: Yuchen Pei Date: Fri, 15 Mar 2019 19:58:21 +0100 Subject: gaussian higher dim renyi --- posts/2019-03-14-great-but-manageable-expectations.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) (limited to 'posts') diff --git a/posts/2019-03-14-great-but-manageable-expectations.md b/posts/2019-03-14-great-but-manageable-expectations.md index 2ec280a..8b4187c 100644 --- a/posts/2019-03-14-great-but-manageable-expectations.md +++ b/posts/2019-03-14-great-but-manageable-expectations.md @@ -84,18 +84,18 @@ $$D_\lambda(p || q) = (\lambda - 1)^{-1} \kappa_{p, q}(\lambda - 1).$$ In the following, whenever you see $t$, think of it as $\lambda - 1$. -**Example 1 (RDP for Gaussian +**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 -$$L(N(\mu_1, \sigma^2) || N(\mu_2, \sigma^2)) \overset{d}{=} L(N(0, 1) || N((\mu_2 - \mu_1) / \sigma, 1)).$$ +$$L(N(\mu_1, \sigma^2 I) || N(\mu_2, \sigma^2 I)) \overset{d}{=} L(N(0, I) || N((\mu_2 - \mu_1) / \sigma, I)).$$ With this we get -$$D_\lambda(N(\mu_1, \sigma^2) || N(\mu_2, \sigma^2)) = {\lambda (\mu_2 - \mu_1)^2 \over 2 \sigma^2} = D_\lambda(N(\mu_2, \sigma^2) || N(\mu_1, \sigma^2)).$$ +$$D_\lambda(N(\mu_1, \sigma^2 I) || N(\mu_2, \sigma^2 I)) = {\lambda \|\mu_2 - \mu_1\|_2^2 \over 2 \sigma^2} = D_\lambda(N(\mu_2, \sigma^2 I) || N(\mu_1, \sigma^2 I)).$$ -Also due to the scaling invariance of $L$, we only need to consider $f$ +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 $\lambda > 1$. -- cgit v1.2.3