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Optimal Rates for Differentially Private Hypothesis Testing with E-values

NeurIPS 2026 · Federated learning and privacy › Differential privacy · approach: Local-DP and private estimation · 0 citations (OpenAlex)

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Characterizes the optimal asymptotic e-power of ε-DP e-values for simple hypothesis testing (batch) via an intermediate distribution, gives a matching private e-variable construction, and a universal lower bound and matching batching-based algorithm for DP e-processes in the sequential setting.

Problem

Maximum achievable e-power (log-growth) of e-values that are ε-differentially private when testing P^n against Q^n, in batch and sequential (stopping-time) settings.

Approach

Shows the optimal private e-power rate equals inf over Q' of KL(Q'||P)+ε·TV(Q',Q), attained by a clipped likelihood ratio; privatizes it with Laplace noise on a sum of log-clipped likelihood ratios using a merging function and a Bernoulli-free compensator, and batches samples in the sequential setting with a calibrated schedule.

Key results

• Two-sided private e-process stops earlier than DP-SPRT across the tested Bernoulli settings and privacy levels (100 trials).
• Batch optimal rate under ε-DP equals inf_Q' KL(Q'||P)+ε TV(Q',Q), matched exactly by the constructed private e-variable.
• Sequential lower bound: E^Q[N] ≥ [(1-β)log((1-β)/α)+β log(β/(1-α))] / R(Q||P) for any ε-DP level-α sequential test.

Surprising findings

• Unbounded private e-variables are always suboptimal: a bounded transformation of any unbounded ε-DP e-variable strictly improves log-optimality.

Limitations

• Results stated for pure ε-DP; approximate DP optimal rate is unknown.
• Optimal e-variable E* requires integrals over full support of P and Q, which may be intractable in high dimensions.
• Constant-factor and lower-order-term gaps: distribution-independent statistic matches rate only up to constants and for ε=Θ(1); open question for ε=o(1).

Experimental setup

Benchmarks: Bernoulli(0.3) vs Bernoulli(q) sequential testing

Metrics: stopping time empirical CDF stopping time N at alpha=beta=1/40

Baselines: DP-SPRT

Builds on: DP-SPRT AboveThreshold Donsker-Varadhan Wald's equation

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