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arxiv: 2605.28952 · v2 · pith:SE3OYTN5new · submitted 2026-05-27 · 💻 cs.CR · cs.DS· cs.IT· cs.LG· math.IT· math.ST· stat.TH

Optimal Rates for Differentially Private Hypothesis Testing with E-values

Ben Jacobsen , Tomas Gonzalez , Gavin Brown , Kassem Fawaz , Aaditya Ramdas This is my paper

Pith reviewed 2026-06-29 11:21 UTC · model grok-4.3

classification 💻 cs.CR cs.DScs.ITcs.LGmath.ITmath.STstat.TH
keywords differential privacye-valueshypothesis testingoptimal ratese-powersequential testinge-processprivate inference
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The pith

An explicit algorithm achieves the exact optimal e-power rate for ε-differentially private hypothesis testing of P against Q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper determines the highest e-power achievable when e-values for testing whether data comes from P or from Q must also obey ε-differential privacy. E-values support anytime-valid inference that remains valid even when the analyst decides when to stop, which is useful for adaptive analysis of sensitive data. The authors derive the precise rate at which e-power grows with sample size under the privacy constraint and construct an algorithm attaining that rate. They further supply matching upper and lower bounds on the stopping times of any private e-process in the sequential setting. A reader would care because the result gives the minimal data requirement for valid private tests and shows how to meet it.

Core claim

Given two distributions P and Q, the maximum achievable e-power for testing X ~ P^n against X ~ Q^n with e-values that satisfy ε-differential privacy is characterized by an optimal rate, and there exists an algorithm that matches this rate exactly. In the sequential setting, when observations arrive one-by-one and the analyst chooses when to halt, matching upper and lower bounds are given on the stopping times of any private e-process.

What carries the argument

The ε-differentially private e-value construction that attains the characterized optimal e-power rate for distinguishing P from Q.

If this is right

  • The minimal number of samples required to reach a target e-power level under ε-privacy is now known exactly.
  • Sequential private testing procedures have stopping times that cannot be improved beyond the matching bounds without violating privacy or validity.
  • The supplied algorithm requires fewer samples than DP-SPRT across tested sequential problems and privacy levels.
  • The same rate and bounds apply uniformly to any pair of distributions P and Q.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The optimality result may extend to composite or nonparametric hypotheses if similar privacy-preserving constructions can be found.
  • The bounds could inform privacy-utility trade-offs in other e-value applications such as multiple testing or change-point detection.
  • In deployment, the explicit rate gives a concrete way to set sample sizes for private anytime-valid tests in domains requiring both validity and privacy.

Load-bearing premise

The e-values are constructed from n i.i.d. samples drawn from either P or Q while satisfying the ε-differential privacy definition.

What would settle it

Finding an ε-differentially private e-value whose e-power grows faster than the characterized optimal rate for some fixed P, Q and ε, or a private e-process whose stopping time falls below the derived lower bound, would falsify the optimality claim.

Figures

Figures reproduced from arXiv: 2605.28952 by Aaditya Ramdas, Ben Jacobsen, Gavin Brown, Kassem Fawaz, Tomas Gonzalez.

Figure 1
Figure 1. Figure 1: Left: Illustrates our definition of Qe for ε = 1 and two densities P and Q over [0, 1]. The shaded region in A and the shaded region in B both have area TV(Q, e Q). Right: Compares the true likelihood ratio dQ/dP to the non-private bounded e-variable E∗ , which can be interpreted as dQ/dP e . Simply clipping dQ/dP to a fixed range could introduce either positive or negative bias; the distribution-dependent… view at source ↗
Figure 2
Figure 2. Figure 2: Empirical CDFs of stopping times for sequential tests over 100 trials under [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
read the original abstract

E-values have attracted considerable interest in recent years as flexible tools for enabling anytime-valid and adaptive data analysis. Hypothesis testing is at the core of many of these applications, which can often involve private or sensitive data. In this work, we answer a simple but important question: given two distributions $\mathbb{P}$ and $\mathbb{Q}$, what is the maximum achievable e-power when testing $X\sim \mathbb{P}^n$ against $X\sim\mathbb{Q}^n$ with e-values that satisfy $\varepsilon$-differential privacy? We characterize the optimal rate for this problem and provide an algorithm which matches it exactly. In the sequential setting, when observations arrive one-by-one and the analyst chooses when to halt, we give matching upper and lower bounds on the stopping times of any private e-process. Numerical experiments confirm the practicality of our algorithms, which require less data than the recently proposed DP-SPRT across a range of sequential testing problems and privacy levels.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper characterizes the optimal e-power rate achievable by ε-differentially private e-values for simple hypothesis testing between two fixed distributions P and Q on the basis of n i.i.d. samples. It supplies an explicit algorithm attaining this rate exactly. In the sequential setting it derives matching upper and lower bounds on the stopping times of any ε-DP e-process. Numerical experiments are reported showing that the proposed procedures require fewer samples than DP-SPRT across several privacy levels.

Significance. If the claimed characterization and matching bounds hold, the work supplies the first tight information-theoretic limits for private e-value testing together with explicit, practical mechanisms. The combination of batch optimality, sequential stopping-time bounds, and empirical outperformance of prior methods constitutes a substantive contribution to the intersection of differential privacy and anytime-valid inference.

minor comments (2)
  1. [Abstract] The abstract states that the algorithm 'matches it exactly'; the main text should explicitly state whether this is with respect to the leading constant or only the asymptotic rate (e.g., §3 or §4).
  2. [Section 3] Notation for the privacy parameter is introduced as ε but the dependence on ε is not displayed in the displayed rate expressions; adding the explicit functional dependence would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our work, as well as their recommendation to accept the manuscript. We are pleased that the contributions on optimal e-power rates, the explicit algorithm, sequential stopping-time bounds, and empirical comparisons were viewed as substantive.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper characterizes optimal e-power rates for ε-DP hypothesis testing between P^n and Q^n by deriving matching upper and lower bounds from standard change-of-measure arguments and privacy-loss accounting applied to the external definitions of e-power (expectation under the alternative) and ε-differential privacy. The sequential stopping-time bounds follow similarly from these primitives without reducing any claimed optimum to a fitted quantity or self-citation chain internal to the paper. No load-bearing step equates a result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definitions of e-values (non-negative random variables with expectation ≤1 under the null) and ε-differential privacy; no free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (2)
  • standard math E-values are non-negative random variables whose expectation is at most 1 under the null hypothesis
    Standard definition from the e-value literature invoked for hypothesis testing.
  • standard math The mechanism producing the e-value must satisfy ε-differential privacy
    Core constraint used to define the feasible set of e-values whose e-power is maximized.

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discussion (0)

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