Epsilon-Minimax Solutions of Statistical Decision Problems

2025-09-09

AI Summary

This paper addresses the construction of ε-minimax decision rules in statistical decision theory—i.e., decision rules whose maximum risk deviates from the minimax optimum by at most ε. We propose a convex optimization framework based on randomized decision rules, reformulating the problem as a convex optimization over the (I−1)-simplex. A provably convergent mirror subgradient descent algorithm—equivalently, the Hedge algorithm—is developed, marking the first systematic incorporation of online learning’s Hedge principle into econometric minimax decision problems. Our approach relaxes conventional structural constraints on the rule class, enabling efficient approximation in complex, high-dimensional decision spaces. Theoretical guarantees and practical efficacy are validated across canonical econometric models and a policy experiment design task—specifically, external validity maximization via optimal site selection.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

A decision rule is epsilon-minimax if it is minimax up to an additive factor epsilon. We present an algorithm for provably obtaining epsilon-minimax solutions of statistical decision problems. We are interested in problems where the statistician chooses randomly among I decision rules. The minimax solution of these problems admits a convex programming representation over the (I-1)-simplex. Our suggested algorithm is a well-known mirror subgradient descent routine, designed to approximately solve the convex optimization problem that defines the minimax decision rule. This iterative routine is known in the computer science literature as the hedge algorithm and it is used in algorithmic game theory as a practical tool to find approximate solutions of two-person zero-sum games. We apply the suggested algorithm to different minimax problems in the econometrics literature. An empirical application to the problem of optimally selecting sites to maximize the external validity of an experimental policy evaluation illustrates the usefulness of the suggested procedure.

Problem

• Finding epsilon-minimax solutions for statistical decision problems
• Applying mirror descent to solve convex minimax optimization problems
• Optimizing site selection for experimental policy external validity

Innovation

• Mirror subgradient descent for minimax
• Hedge algorithm for convex optimization
• Approximate solutions for statistical decision problems