Approximate Least-Favorable Distributions and Nearly Optimal Tests via Stochastic Mirror Descent

TL;DR

This paper introduces a stochastic mirror descent method to efficiently find nearly optimal hypothesis tests and least-favorable distributions in complex testing scenarios, providing theoretical guarantees and practical implementation guidance.

Contribution

It develops a novel stochastic mirror descent algorithm for hypothesis testing, offering provable convergence to approximate least-favorable distributions and nearly optimal tests.

Findings

Algorithm converges after finitely many iterations

Provides concrete implementation recommendations

Extends and modifies previous algorithms with unknown guarantees

Abstract

We consider a class of hypothesis testing problems where the null hypothesis postulates $M$ distributions for the observed data, and there is only one possible distribution under the alternative. We show that one can use a stochastic mirror descent routine for convex optimization to provably obtain - after finitely many iterations - both an approximate least-favorable distribution and a nearly optimal test, in a sense we make precise. Our theoretical results yield concrete recommendations about the algorithm's implementation, including its initial condition, its step size, and the number of iterations. Importantly, our suggested algorithm can be viewed as a slight variation of the algorithm suggested by Elliott, M\"uller, and Watson (2015), whose theoretical performance guarantees are unknown.