Combinatorial Pure Exploration with Bottleneck Reward Function

TL;DR

This paper investigates the combinatorial pure exploration problem with bottleneck rewards, proposing new algorithms for fixed-confidence and fixed-budget settings that are statistically optimal and empirically superior.

Contribution

It introduces novel algorithms tailored for CPE-B, addressing unique bottleneck challenges and achieving optimal sample complexity and error guarantees.

Findings

Proposed algorithms with optimal sample complexity for FC setting.

State-of-the-art error guarantees for FB setting.

Empirical validation showing superiority over baselines.

Abstract

In this paper, we study the Combinatorial Pure Exploration problem with the Bottleneck reward function (CPE-B) under the fixed-confidence (FC) and fixed-budget (FB) settings. In CPE-B, given a set of base arms and a collection of subsets of base arms (super arms) following a certain combinatorial constraint, a learner sequentially plays a base arm and observes its random reward, with the objective of finding the optimal super arm with the maximum bottleneck value, defined as the minimum expected reward of the base arms contained in the super arm. CPE-B captures a variety of practical scenarios such as network routing in communication networks, and its \emph{unique challenges} fall on how to utilize the bottleneck property to save samples and achieve the statistical optimality. None of the existing CPE studies (most of them assume linear rewards) can be adapted to solve such challenges, and thus we develop brand-new techniques to handle them. For the FC setting, we propose novel algorithms with optimal sample complexity for a broad family of instances and establish a matching lower bound to demonstrate the optimality (within a logarithmic factor). For the FB setting, we design an algorithm which achieves the state-of-the-art error probability guarantee and is the first to run efficiently on fixed-budget path instances, compared to existing CPE algorithms. Our experimental results on the top-$k$, path and matching instances validate the empirical superiority of the proposed algorithms over their baselines.