Generalization for Adaptively-chosen Estimators via Stable Median

TL;DR

This paper introduces a differentially private median-based algorithm that enables accurate adaptive statistical queries with sample complexity scaling as the square root of the number of queries, improving over prior worst-case methods.

Contribution

It presents a novel stable median algorithm that provides generalization guarantees for adaptively-chosen estimators with sample complexity proportional to the square root of the number of queries.

Findings

Sample complexity scales as √k for estimating k adaptively-chosen estimators.

Answers are nearly as accurate as using fresh samples for each estimator.

Algorithm can verify answers with logarithmic dependence on the number of queries.

Abstract

Datasets are often reused to perform multiple statistical analyses in an adaptive way, in which each analysis may depend on the outcomes of previous analyses on the same dataset. Standard statistical guarantees do not account for these dependencies and little is known about how to provably avoid overfitting and false discovery in the adaptive setting. We consider a natural formalization of this problem in which the goal is to design an algorithm that, given a limited number of i.i.d.~samples from an unknown distribution, can answer adaptively-chosen queries about that distribution. We present an algorithm that estimates the expectations of $k$ arbitrary adaptively-chosen real-valued estimators using a number of samples that scales as $\sqrt{k}$. The answers given by our algorithm are essentially as accurate as if fresh samples were used to evaluate each estimator. In contrast, prior work yields error guarantees that scale with the worst-case sensitivity of each estimator. We also give a version of our algorithm that can be used to verify answers to such queries where the sample complexity depends logarithmically on the number of queries $k$ (as in the reusable holdout technique). Our algorithm is based on a simple approximate median algorithm that satisfies the strong stability guarantees of differential privacy. Our techniques provide a new approach for analyzing the generalization guarantees of differentially private algorithms.