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| -rw-r--r-- | posts/2019-03-13-a-tail-of-two-densities.md | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/posts/2019-03-13-a-tail-of-two-densities.md b/posts/2019-03-13-a-tail-of-two-densities.md index ba62777..e71651a 100644 --- a/posts/2019-03-13-a-tail-of-two-densities.md +++ b/posts/2019-03-13-a-tail-of-two-densities.md @@ -1094,8 +1094,8 @@ It turns out subsampling has an amplification effect on privacy. **Claim 19 (Ullman 2017)**. Fix $r \in [0, 1]$. Let $n \le m$ be two nonnegative integers with -$n = r m$. Let $N$ be an $(\epsilon, \delta)$-dp machanism on $X^n$. -Define mechanism $M$ on $X^m$ by +$n = r m$. Let $N$ be an $(\epsilon, \delta)$-dp machanism on $Z^n$. +Define mechanism $M$ on $Z^m$ by $$M(x) = N(x_\gamma)$$ @@ -1105,7 +1105,7 @@ Then $M$ is $(\log (1 + r(e^\epsilon - 1)), r \delta)$-dp. Kasiviswanathan-Lee-Nissim-Raskhodnikova-Smith 2005 for this result, but it is not clear to me how it appears there. -**Proof**. Let $x, x' \in X^n$ such that they differ by one +**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: |