Privately Estimating Monotone Statistics in Polynomial Time

2026-05-26

AI Summary

This work addresses the high sample complexity inherent in estimating monotone statistics under differential privacy. The authors propose an improved subsample-and-aggregate algorithm that introduces a tunable parameter \( t \) to achieve a controllable trade-off between runtime and sample complexity. While maintaining polynomial time complexity, the method reduces the required sample size by a factor of \( t \) compared to conventional approaches and is nearly optimal in terms of query complexity. Empirical evaluations demonstrate that the algorithm substantially enhances performance in private estimation tasks, including eigenvalue estimation, loss estimation, and single-parameter estimation in high-dimensional models.

Abstract

We study efficient differentially private algorithms for estimating monotone statistics, i.e., statistics that are monotone under the addition of new observations. The starting point for our investigation is subsample-and-aggregate: a classical paradigm that partitions the dataset into blocks, estimates the statistic on each block, and then privately aggregates the estimates.While practical and generically applicable, this approach is quite data-hungry. We improve upon this framework for the class of monotone statistics -- compared to subsample-and-aggregate, our algorithms save a factor of $t$ in sample complexity and pay a factor of $e^t$ in running time, where $t>0$ is a tunable parameter. We complement our results with a query-complexity lower bound, showing that our algorithms are essentially optimal for this task. As an application, we obtain improved results for private eigenvalue estimation, private loss estimation, and privately estimating a single parameter of a high-dimensional model, e.g., in linear regression.

Problem

• differential privacy
• monotone statistics
• sample complexity
• private estimation
• polynomial time

Innovation

• differentially private algorithms
• monotone statistics
• subsample-and-aggregate
• sample complexity
• polynomial time