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

2025-11-20

AI Summary

This paper addresses the statistical testing problem of a composite null hypothesis—comprising M candidate distributions—against a simple alternative hypothesis—a single distribution—with the objective of jointly identifying an approximately least-favorable null distribution and an approximately optimal test. We propose a convex optimization framework based on Stochastic Mirror Descent (SMD), the first systematic application of this algorithm to such composite hypothesis testing problems. We establish finite-step convergence guarantees and provide explicit design guidelines for the initial point, step size, and iteration count. The resulting test enjoys rigorous statistical guarantees—including controlled Type-I error under the worst-case null—and demonstrably outperforms existing heuristic approaches. Crucially, it achieves a favorable trade-off between robustness—via minimax optimality under distributional uncertainty—and computational efficiency—through scalable first-order updates.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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, Müller, and Watson (2015), whose theoretical performance guarantees are unknown.

Problem

• Finding approximate least-favorable distributions for hypothesis testing
• Developing nearly optimal tests via stochastic mirror descent
• Providing finite-iteration guarantees for convex optimization solutions

Innovation

• Stochastic mirror descent for convex optimization
• Approximate least-favorable distribution generation
• Nearly optimal hypothesis testing construction