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Bounds on the privacy amplification of arbitrary channels via the contraction of $f_α$-divergence

Leonhard Grosse, Sara Saeidian, Tobias J. Oechtering, Mikael Skoglund

TL;DR

The paper investigates privacy amplification under arbitrary discrete channels by analyzing the contraction of $f_α$-divergence, which translates to Rényi-divergence via a monotone transform. It develops contraction bounds under restricted input priors, including an improved Pinsker-type inequality and a general SDPI framework that captures cross-channel effects. By applying these results to RLDP, the authors show that post-processing by sparse channels can significantly amplify privacy even when the channel lacks a finite LDP guarantee, with potential convergence to perfect privacy as the input alphabet grows. This has practical implications for enhancing privacy in systems that already satisfy local DP by leveraging post-processing with carefully structured channels. Overall, the work provides structural conditions, tight inequality bounds, and actionable guidance for designing post-processing steps to strengthen privacy guarantees in discrete data pipelines.

Abstract

We examine the privacy amplification of channels that do not necessarily satisfy any LDP guarantee by analyzing their contraction behavior in terms of $f_α$-divergence, an $f$-divergence related to Rényi-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 $f_α$-divergence based on the joint range technique by Harremoës and Vajda. The presented bound is tight whenever the value of the total variation distance is larger than 1/$α$. By applying these inequalities in a cross-channel setting, we arrive at strong data processing inequalities for $f_α$-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.

Bounds on the privacy amplification of arbitrary channels via the contraction of $f_α$-divergence

TL;DR

The paper investigates privacy amplification under arbitrary discrete channels by analyzing the contraction of -divergence, which translates to Rényi-divergence via a monotone transform. It develops contraction bounds under restricted input priors, including an improved Pinsker-type inequality and a general SDPI framework that captures cross-channel effects. By applying these results to RLDP, the authors show that post-processing by sparse channels can significantly amplify privacy even when the channel lacks a finite LDP guarantee, with potential convergence to perfect privacy as the input alphabet grows. This has practical implications for enhancing privacy in systems that already satisfy local DP by leveraging post-processing with carefully structured channels. Overall, the work provides structural conditions, tight inequality bounds, and actionable guidance for designing post-processing steps to strengthen privacy guarantees in discrete data pipelines.

Abstract

We examine the privacy amplification of channels that do not necessarily satisfy any LDP guarantee by analyzing their contraction behavior in terms of -divergence, an -divergence related to Rényi-divergence via a monotonic transformation. We present bounds on contraction for restricted sets of prior distributions via -divergence inequalities and present an improved Pinsker's inequality for -divergence based on the joint range technique by Harremoës and Vajda. The presented bound is tight whenever the value of the total variation distance is larger than 1/. By applying these inequalities in a cross-channel setting, we arrive at strong data processing inequalities for -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.
Paper Structure (13 sections, 4 theorems, 64 equations, 2 figures)

This paper contains 13 sections, 4 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Preliminaries
  5. Rényi-divergence, $f_\alpha$-divergene and $E_\gamma$-divergence
  6. Strong Data Processing Inequalities
  7. Conditions for $\eta_\alpha(P_{Y|X},\mathcal{P}_{\mathcal{X}})=1$
  8. Connection to the Confusion Graph of $P_{Y|X}$
  9. Pinsker-type Inequalities for $f_\alpha$-divergence
  10. Strong Data Processing with Pinsker
  11. Privacy Amplification Bounds for RLDP
  12. Proof of Theorem \ref{['thm:pinsker']}
  13. Proof of Proposition \ref{['prop:etaisone']}

Key Result

Proposition 3

Consider a discrete channel $P_{Y|X}$. Whenever we have $\eta_{\alpha}(P_{Y|X},\mathcal{P}_{\mathcal{X}}) = 1$.

Figures (2)

  • Figure 1: Joint range of $D_{f_{\alpha}}(P||Q)$ and $\text{TV}(P||Q)$ for the case $\alpha=4$ as well as the bound presented in Theorem \ref{['thm:pinsker']}.
  • Figure 2: RLDP guarantees and the corresponding bounds in Example \ref{['ex:RLDPex']}.

Theorems & Definitions (13)

  • Definition 1: $f_\alpha$-divergence
  • Definition 2: $E_{\gamma}$-divergence, see, e.g., 7552457
  • Proposition 3
  • Example 4: Block-diagonal matrices
  • Example 5: $k$-singular channels
  • Theorem 6: Reverse Pinkser's Inequality for $f_\alpha$-divergence, special case of 8630660
  • Remark 7
  • Theorem 8: Pinsker's inequality for $f_\alpha$-divergences
  • Remark 9
  • Remark 10
  • ...and 3 more