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authorYuchen Pei <me@ypei.me>2019-03-15 19:58:21 +0100
committerYuchen Pei <me@ypei.me>2019-03-15 19:58:21 +0100
commit2fa0512a84daa2b5d1ef77f70a58b216e49ab851 (patch)
treee15f5310b2ab5d37cd754ed7d9d5161b31453f2e /posts
parent64ad21bb25e93a901e346b557a5cd98c0dd2586c (diff)
gaussian higher dim renyi
Diffstat (limited to 'posts')
-rw-r--r--posts/2019-03-14-great-but-manageable-expectations.md8
1 files changed, 4 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 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 $\lambda > 1$.