# Locally Private Hypothesis Selection

arXiv: https://arxiv.org/abs/2002.09465 · HTML: https://mlsanity.com/paper/2002.09465/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2020.

## Built on by (most cited first, 17 of 17)

- [Private High-Dimensional Hypothesis Testing](https://mlsanity.com/paper/2203.01537/) (2022)
- [Communication-constrained hypothesis testing: Optimality, robustness, and reverse data processing inequalities](https://mlsanity.com/paper/2206.02765/) (2022)
- [Simple Binary Hypothesis Testing under Local Differential Privacy and Communication Constraints](https://mlsanity.com/paper/2301.03566/) (2023)
- [Statistically Near-Optimal Hypothesis Selection](https://mlsanity.com/paper/2108.07880/) (2021)
- [The Role of Interactivity in Structured Estimation](https://mlsanity.com/paper/2203.06870/) (2022)
- [DP-HyPO: An Adaptive Private Hyperparameter Optimization Framework](https://mlsanity.com/paper/2306.05734/) (2023)
- [Data Structures for Density Estimation](https://mlsanity.com/paper/2306.11312/) (2023)
- [Distribution Learnability and Robustness](https://mlsanity.com/paper/2406.17814/) (2024)
- [Sample-Optimal Locally Private Hypothesis Selection and the Provable Benefits of Interactivity](https://mlsanity.com/paper/2312.05645/) (2023)
- [On the Price of Privacy for Language Identification and Generation](https://mlsanity.com/paper/2604.07238/) (2026)
- [Statistical-Computational Trade-offs for Density Estimation](https://mlsanity.com/paper/2410.23087/) (2024)
- [Optimal Algorithms for Augmented Testing of Discrete Distributions](https://mlsanity.com/paper/2412.00974/) (2024)
- [Open Problem: Is Interaction Necessary for Order-Optimal 1-bit Mean Estimation?](https://mlsanity.com/paper/2607.02896/) (2026)
- [Robust Online Classification: From Estimation to Denoising](https://mlsanity.com/paper/2309.01698/) (2023)
- [Online Distribution Learning with Local Private Constraints](https://mlsanity.com/paper/2402.00315/) (2024)
- [Nearly-Linear Time Private Hypothesis Selection with the Optimal Approximation Factor](https://mlsanity.com/paper/2506.01162/) (2025)
- [Query-Efficient Locally Private Hypothesis Selection via the Scheffe Graph](https://mlsanity.com/paper/2509.16180/) (2025)

## Abstract (arXiv)

We initiate the study of hypothesis selection under local differential privacy. Given samples from an unknown probability distribution p and a set of k probability distributions Q, we aim to output, under the constraints of varepsilon-local differential privacy, a distribution from Q whose total variation distance to p is comparable to the best such distribution. This is a generalization of the classic problem of k-wise simple hypothesis testing, which corresponds to when p ∈ Q, and we wish to identify p. Absent privacy constraints, this problem requires O(log k) samples from p, and it was recently shown that the same complexity is achievable under (central) differential privacy. However, the naive approach to this problem under local differential privacy would require tilde O(k2) samples. We first show that the constraint of local differential privacy incurs an exponential increase in cost: any algorithm for this problem requires at least Omega(k) samples. Second, for the special case of k-wise simple hypothesis testing, we provide a non-interactive algorithm which nearly matches this bound, requiring tilde O(k) samples. Finally, we provide sequentially interactive algorithms for the general case, requiring tilde O(k) samples and only O(log log k) rounds of interactivity. Our algorithms are achieved through a reduction to maximum selection with adversarial comparators, a problem of independent interest for which we initiate study in the parallel setting. For this problem, we provide a family of algorithms for each number of allowed rounds of interaction t, as well as lower bounds showing that they are near-optimal for every t. Notably, our algorithms result in exponential improvements on the round complexity of previous methods.