Nearly-Linear Time Private Hypothesis Selection with the Optimal Approximation Factor

• URL: http://arxiv.org/abs/2506.01162v1
• Date: Sun, 01 Jun 2025 20:46:46 GMT
• Title: Nearly-Linear Time Private Hypothesis Selection with the Optimal Approximation Factor
• Authors: Maryam Aliakbarpour, Zhan Shi, Ria Stevens, Vincent X. Wang,
• Abstract summary: Estimating the density of a distribution from its samples is a fundamental problem in statistics.<n> Hypothesis selection addresses the setting where, in addition to a sample set, we are given $n$ candidate distributions.<n>We propose a differentially private algorithm in the central model that runs in nearly-linear time with respect to the number of hypotheses.
• Score: 7.069470347531414
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Estimating the density of a distribution from its samples is a fundamental problem in statistics. Hypothesis selection addresses the setting where, in addition to a sample set, we are given $n$ candidate distributions -- referred to as hypotheses -- and the goal is to determine which one best describes the underlying data distribution. This problem is known to be solvable very efficiently, requiring roughly $O(\log n)$ samples and running in $\tilde{O}(n)$ time. The quality of the output is measured via the total variation distance to the unknown distribution, and the approximation factor of the algorithm determines how large this distance is compared to the optimal distance achieved by the best candidate hypothesis. It is known that $\alpha = 3$ is the optimal approximation factor for this problem. We study hypothesis selection under the constraint of differential privacy. We propose a differentially private algorithm in the central model that runs in nearly-linear time with respect to the number of hypotheses, achieves the optimal approximation factor, and incurs only a modest increase in sample complexity, which remains polylogarithmic in $n$. This resolves an open question posed by [Bun, Kamath, Steinke, Wu, NeurIPS 2019]. Prior to our work, existing upper bounds required quadratic time.

Related papers

• Instance-Optimality for Private KL Distribution Estimation [41.35506763248454]
  We study the fundamental problem of estimating an unknown discrete distribution $p$ over $d$ symbols, given $n$ i.i.d. samples from the distribution.<n>We propose algorithms that achieve instance-optimality up to constant factors, with and without a differential privacy constraint.
  arXiv  Detail & Related papers  (2025-05-29T16:27:57Z)
• Statistical-Computational Trade-offs for Density Estimation [60.81548752871115]
  We show that for a broad class of data structures their bounds cannot be significantly improved. This is a novel emphstatistical-computational trade-off for density estimation.
  arXiv  Detail & Related papers  (2024-10-30T15:03:33Z)
• Instance-Optimal Private Density Estimation in the Wasserstein Distance [37.58527481568219]
  Estimating the density of a distribution from samples is a fundamental problem in statistics. We study differentially private density estimation in the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-06-27T22:51:06Z)
• Optimal Multi-Distribution Learning [88.3008613028333]
  Multi-distribution learning seeks to learn a shared model that minimizes the worst-case risk across $k$ distinct data distributions. We propose a novel algorithm that yields an varepsilon-optimal randomized hypothesis with a sample complexity on the order of (d+k)/varepsilon2.
  arXiv  Detail & Related papers  (2023-12-08T16:06:29Z)
• Estimating Optimal Policy Value in General Linear Contextual Bandits [50.008542459050155]
  In many bandit problems, the maximal reward achievable by a policy is often unknown in advance. We consider the problem of estimating the optimal policy value in the sublinear data regime before the optimal policy is even learnable. We present a more practical, computationally efficient algorithm that estimates a problem-dependent upper bound on $V*$.
  arXiv  Detail & Related papers  (2023-02-19T01:09:24Z)
• Simple Binary Hypothesis Testing under Local Differential Privacy and Communication Constraints [8.261182037130407]
  We study simple binary hypothesis testing under both local differential privacy (LDP) and communication constraints. We qualify our results as either minimax optimal or instance optimal.
  arXiv  Detail & Related papers  (2023-01-09T18:36:49Z)
• Privately Estimating a Gaussian: Efficient, Robust and Optimal [6.901744415870126]
  We give efficient algorithms for privately estimating a Gaussian distribution in both pure and approximate differential privacy (DP) models. In the pure DP setting, we give an efficient algorithm that estimates an unknown $d$-dimensional Gaussian distribution up to an arbitrary tiny total variation error. For the special case of mean estimation, our algorithm achieves the optimal sample complexity of $widetilde O(d)$, improving on a $widetilde O(d1.5)$ bound from prior work.
  arXiv  Detail & Related papers  (2022-12-15T18:27:39Z)
• Distributionally Robust Optimization with Markovian Data [8.126833795693699]
  We study a program where the probability distribution of the uncertain problem parameters is unknown. We propose a data-driven distributionally to estimate the problem's objective function and optimal solution.
  arXiv  Detail & Related papers  (2021-06-12T10:59:02Z)
• Private Stochastic Non-Convex Optimization: Adaptive Algorithms and Tighter Generalization Bounds [72.63031036770425]
  We propose differentially private (DP) algorithms for bound non-dimensional optimization. We demonstrate two popular deep learning methods on the empirical advantages over standard gradient methods.
  arXiv  Detail & Related papers  (2020-06-24T06:01:24Z)
• Breaking the Sample Size Barrier in Model-Based Reinforcement Learning with a Generative Model [50.38446482252857]
  This paper is concerned with the sample efficiency of reinforcement learning, assuming access to a generative model (or simulator) We first consider $gamma$-discounted infinite-horizon Markov decision processes (MDPs) with state space $mathcalS$ and action space $mathcalA$. We prove that a plain model-based planning algorithm suffices to achieve minimax-optimal sample complexity given any target accuracy level.
  arXiv  Detail & Related papers  (2020-05-26T17:53:18Z)
• Locally Private Hypothesis Selection [96.06118559817057]
  We output a distribution from $mathcalQ$ whose total variation distance to $p$ is comparable to the best such distribution. We show that the constraint of local differential privacy incurs an exponential increase in cost. Our algorithms result in exponential improvements on the round complexity of previous methods.
  arXiv  Detail & Related papers  (2020-02-21T18:30:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.