DP-SPRT: Differentially Private Sequential Probability Ratio Tests

We revisit Wald’s celebrated Sequential Probability Ratio Test for sequential tests of two simple hypotheses, under privacy constraints. We propose DP-SPRT, a wrapper that can be calibrated to achieve desired error probabilities and privacy constraints, addressing a significant gap in previous work. DP-SPRT relies on a private mechanism that processes a sequence of queries and stops after privately determining when the query results fall outside a predefined interval. This OutsideInterval mechanism improves upon naive composition of existing techniques like AboveThreshold, achieving a factor-of-2 privacy improvement and thus potentially benefiting other continual monitoring procedures. We prove generic upper bounds on the error and sample complexity of DP-SPRT that can accommodate various noise distributions based on the practitioner’s privacy needs. We exemplify them in two settings: Laplace noise (pure Differential Privacy) and Gaussian noise (Rényi differential privacy). In the former setting, by providing a lower bound on the sample complexity of any $\varepsilon$-DP test with prescribed type I and type II errors, we show that DP-SPRT is near optimal when both errors are small and the two hypotheses are close. Moreover, we conduct an experimental study revealing its good practical performance.

💡 Research Summary

This paper addresses the problem of performing sequential hypothesis testing under differential privacy (DP) constraints. Classical Wald’s Sequential Probability Ratio Test (SPRT) is optimal for testing two simple hypotheses when there are no privacy concerns: it achieves the smallest expected sample size among all tests that satisfy prescribed Type‑I (α) and Type‑II (β) error probabilities. However, in sensitive domains such as clinical trials or finance, the decision to stop data collection can itself leak private information about the most recent observations. Existing DP hypothesis‑testing work focuses on fixed‑sample settings and does not directly apply to the adaptive nature of sequential testing.

The authors propose DP‑SPRT, a general framework that wraps any noise‑adding mechanism (Laplace for pure ε‑DP, Gaussian for Rényi DP) around the SPRT statistic while preserving the desired error guarantees. The core technical contribution is the “OutsideInterval” mechanism, a novel DP primitive that simultaneously monitors whether a query value lies below a lower threshold, above an upper threshold, or inside the interval between them. For each query f_i, the mechanism adds a shared noise Z to both thresholds and an independent noise Y_i to the query result. By reusing the same Z for both comparisons, the privacy cost is essentially halved compared to naïvely composing two AboveThreshold instances. Theorem 1 shows that if Z and Y_i satisfy ε_Z‑DP (sensitivity Δ) and ε_Y‑DP (sensitivity 2Δ) respectively, the whole procedure is (ε_Z+ε_Y)‑DP; a similar bound holds for Rényi DP with a factor‑2 improvement in the privacy parameters.

The paper conducts a detailed sample‑complexity analysis for Bernoulli observations, which have bounded sensitivity and thus fit naturally into the DP setting. With Laplace noise, thresholds are calibrated exactly to γ_0=β and γ_1=1/α, and the noise scale is set to Δ/ε. The expected stopping time satisfies
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