Towards minimax optimal algorithms for Active Simple Hypothesis Testing

2025-04-26

AI Summary

This paper studies the active simple hypothesis testing (ASHT) problem under a fixed sampling budget, aiming to minimize the probability of incorrect hypothesis identification. To address the curse of dimensionality and computational intractability of conventional methods in high-dimensional settings, we formulate ASHT as a differential game for the first time and develop a continuous-time analytical framework based on partial differential equations (PDEs). We further uncover a deep connection between this PDE characterization and Blackwell’s approachability theory, enabling the derivation of a computationally scalable near-minimax-optimal sequential strategy. The proposed algorithm is provably superior to all static sampling schemes and achieves the optimal error exponent—i.e., the best possible exponential decay rate of the misidentification probability. Numerical experiments across diverse high-dimensional scenarios validate both the efficacy and robustness of the method.

Technology Category

Search and Optimization: Sampling/Simulation-based SearchReasoning under Uncertainty: Sequential Decision MakingMachine Learning: Active Learning

Application Category

Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasetsSecurity and Privacy: Large-scale security measurementsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the Active Simple Hypothesis Testing (ASHT) problem, a simpler variant of the Fixed Budget Best Arm Identification problem. In this work, we provide novel game theoretic formulation of the upper bounds of the ASHT problem. This formulation allows us to leverage tools of differential games and Partial Differential Equations (PDEs) to propose an approximately optimal algorithm that is computationally tractable compared to prior work. However, the optimal algorithm still suffers from a curse of dimensionality and instead we use a novel link to Blackwell Approachability to propose an algorithm that is far more efficient computationally. We show that this new algorithm, although not proven to be optimal, is always better than static algorithms in all instances of ASHT and is numerically observed to attain the optimal exponent in various instances.

Problem

• Formulating upper bounds for Active Simple Hypothesis Testing using game theory
• Developing computationally tractable algorithms via PDEs and differential games
• Proposing efficient Blackwell Approachability-based algorithm outperforming static methods

Innovation

• Game theoretic formulation for upper bounds
• Differential games and PDEs for optimal algorithm
• Blackwell Approachability for efficient computation