Information-Directed Selection for Top-Two Algorithms

TL;DR

This paper introduces an information-directed selection (IDS) method for top-two algorithms in multi-armed bandits, achieving asymptotic optimality for best-k arm identification and demonstrating superior empirical performance.

Contribution

It extends the top-two algorithm framework to best-k identification, introduces IDS for arm selection, and proves asymptotic optimality of top-two Thompson sampling.

Findings

IDS-based top-two algorithms outperform non-adaptive methods.

Proven asymptotic optimality for Gaussian best-arm identification.

Numerical results show significant performance improvements.

Abstract

We consider the best-k-arm identification problem for multi-armed bandits, where the objective is to select the exact set of k arms with the highest mean rewards by sequentially allocating measurement effort. We characterize the necessary and sufficient conditions for the optimal allocation using dual variables. Remarkably these optimality conditions lead to the extension of top-two algorithm design principle (Russo, 2020), initially proposed for best-arm identification. Furthermore, our optimality conditions induce a simple and effective selection rule dubbed information-directed selection (IDS) that selects one of the top-two candidates based on a measure of information gain. As a theoretical guarantee, we prove that integrated with IDS, top-two Thompson sampling is (asymptotically) optimal for Gaussian best-arm identification, solving a glaring open problem in the pure exploration literature (Russo, 2020). As a by-product, we show that for k > 1, top-two algorithms cannot achieve optimality even when the algorithm has access to the unknown "optimal" tuning parameter. Numerical experiments show the superior performance of the proposed top-two algorithms with IDS and considerable improvement compared with algorithms without adaptive selection.