Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universal Asymptotics for Jensen--Shannon Divergence under Shuffling

Alex Shvets

TL;DR

The paper analyzes the Jensen--Shannon divergence between transcript distributions in the shuffle model under neighboring datasets, establishing a universal two-term asymptotic expansion as the number of users $n$ grows. By representing the log-likelihood ratio through an additive statistic $U_n$ and expanding a pointwise functional $D(1+U_n)$, the authors derive $\mathrm{JSD}(T_n\|T'_n) = \frac{\chi^2(W_1\|W_0)}{8n} + \frac{1}{n^2}\big[\frac{7}{64}\chi^4(W_1\|W_0) - \frac{1}{16}\mu_3(W)\big] + O(n^{-3})$ under a mild positivity assumption. They further specialize the result to binary and $k$-ary randomized responses and extend to multi-message protocols by independent repetition, where the leading term scales with $\big(1+\chi^2\big)^m-1$. The work provides explicit remainder control and yields precise privacy-utility guarantees for shuffle-based privacy amplification, with clear guidance for protocol design. The universality across fixed local channels makes the results broadly applicable to a range of practical privacy settings.

Abstract

We study the Jensen--Shannon divergence (JSD) between transcript distributions induced by neighboring datasets in the shuffle model when each user applies a fixed local randomizer and a trusted shuffler releases the output histogram. Under a mild positivity assumption, we prove an explicit two-term asymptotic expansion where the leading term is chi-squared divergence divided by 8n. Binary randomized response and k-ary randomized response follow as corollaries. For multi-message protocols based on independent repetition, the leading coefficient becomes (1 + chi-squared)^m - 1. A fully explicit remainder control is provided in the appendix.

Universal Asymptotics for Jensen--Shannon Divergence under Shuffling

TL;DR

The paper analyzes the Jensen--Shannon divergence between transcript distributions in the shuffle model under neighboring datasets, establishing a universal two-term asymptotic expansion as the number of users grows. By representing the log-likelihood ratio through an additive statistic and expanding a pointwise functional , the authors derive under a mild positivity assumption. They further specialize the result to binary and -ary randomized responses and extend to multi-message protocols by independent repetition, where the leading term scales with . The work provides explicit remainder control and yields precise privacy-utility guarantees for shuffle-based privacy amplification, with clear guidance for protocol design. The universality across fixed local channels makes the results broadly applicable to a range of practical privacy settings.

Abstract

We study the Jensen--Shannon divergence (JSD) between transcript distributions induced by neighboring datasets in the shuffle model when each user applies a fixed local randomizer and a trusted shuffler releases the output histogram. Under a mild positivity assumption, we prove an explicit two-term asymptotic expansion where the leading term is chi-squared divergence divided by 8n. Binary randomized response and k-ary randomized response follow as corollaries. For multi-message protocols based on independent repetition, the leading coefficient becomes (1 + chi-squared)^m - 1. A fully explicit remainder control is provided in the appendix.
Paper Structure (22 sections, 15 theorems, 79 equations, 4 figures)

This paper contains 22 sections, 15 theorems, 79 equations, 4 figures.

Table of Contents

  1. Model and divergences
  2. Shuffle model and neighboring datasets
  3. Positivity assumption
  4. Jensen--Shannon divergence and $\chi^2$
  5. Exact shuffle likelihood ratio
  6. Linearity in the histogram
  7. I.i.d. representation under $T_n$
  8. JSD as an expectation of a pointwise functional
  9. Taylor expansion of the pointwise functional
  10. Moments of $U_n$
  11. Main universal expansion
  12. Special cases
  13. Binary randomized response
  14. $k$-ary randomized response
  15. Multi-message shuffle protocols by independent repetition
  16. ...and 7 more sections

Key Result

Lemma 2.1

For every histogram $N = (N_y)_{y \in \mathcal{Y}}$ with $\sum_y N_y = n$,

Figures (4)

  • Figure 1: Binary randomized response: $\chi^2(W_1\|W_0)$ as a function of the flip probability $\delta$.
  • Figure 2: Multi-message shuffling (binary RR, $\delta=0.3$): the leading coefficient grows as $(1+\chi^2)^m-1$ with $\chi^2=0.7619047619\approx 0.7619$.
  • Figure 3: Binary randomized response with $\delta=0.3$: $n\,\mathrm{JSD}(T_n\|T'_n)$ converges to $\chi^2(W_1\|W_0)/8$.
  • Figure 4: Binary randomized response with $\delta=0.3$: the compensated quantity $n^2\!\left(\mathrm{JSD}(T_n\|T'_n)-\chi^2/(8n)\right)$ converges to $(3/64)\,\chi^4$.

Theorems & Definitions (37)

  • Definition 1.1: Neighboring datasets
  • Remark 1.2: Scope of universality and direction of the neighboring change
  • Definition 1.3: Minimum output mass under $W_0$
  • Remark 1.5
  • Definition 1.6: Jensen--Shannon divergence
  • Definition 1.7: $\chi^2$-divergence
  • Lemma 2.1: Exact likelihood ratio
  • proof
  • Lemma 3.1: Pointwise JSD functional
  • proof
  • ...and 27 more