Bounds on the privacy amplification of arbitrary channels via the contraction of $f_\alpha$-divergence

TL;DR

This paper establishes bounds on privacy amplification for arbitrary channels using $f_eta$-divergence contraction, providing tighter inequalities and demonstrating significant privacy gains even with sparse channels in local differential privacy settings.

Contribution

It introduces new bounds on $f_eta$-divergence contraction for arbitrary channels, including an improved Pinsker's inequality, and applies these to privacy amplification analysis.

Findings

Tight bounds on $f_eta$-divergence contraction for channels.

Enhanced Pinsker's inequality for $f_eta$-divergence.

Sparse channels can achieve substantial privacy amplification.

Abstract

We examine the privacy amplification of channels that do not necessarily satisfy any LDP guarantee by analyzing their contraction behavior in terms of $f_\alpha$-divergence, an $f$-divergence related to R\'enyi-divergence via a monotonic transformation. We present bounds on contraction for restricted sets of prior distributions via $f$-divergence inequalities and present an improved Pinsker's inequality for $f_\alpha$-divergence based on the joint range technique by Harremo\"es and Vajda. The presented bound is tight whenever the value of the total variation distance is larger than 1/$\alpha$. By applying these inequalities in a cross-channel setting, we arrive at strong data processing inequalities for $f_\alpha$-divergence that can be adapted to use-case specific restrictions of input distributions and channel. The application of these results to privacy amplification shows that even very sparse channels can lead to significant privacy amplification when used as a post-processing step after local differentially private mechanisms.