Combinatorial Pure Exploration with Continuous and Separable Reward Functions and Its Applications (Extended Version)

TL;DR

This paper introduces a new algorithm for the combinatorial pure exploration problem with continuous, separable reward functions in stochastic bandits, providing bounds on sample complexity and handling non-linear rewards.

Contribution

It proposes an adaptive learning algorithm for CPE-CS, introduces the consistent optimality hardness measure, and establishes upper and lower bounds on sample complexity.

Findings

The algorithm achieves near-optimal sample complexity bounds.

The hardness measure effectively captures problem difficulty.

The method handles non-linear reward functions successfully.

Abstract

We study the Combinatorial Pure Exploration problem with Continuous and Separable reward functions (CPE-CS) in the stochastic multi-armed bandit setting. In a CPE-CS instance, we are given several stochastic arms with unknown distributions, as well as a collection of possible decisions. Each decision has a reward according to the distributions of arms. The goal is to identify the decision with the maximum reward, using as few arm samples as possible. The problem generalizes the combinatorial pure exploration problem with linear rewards, which has attracted significant attention in recent years. In this paper, we propose an adaptive learning algorithm for the CPE-CS problem, and analyze its sample complexity. In particular, we introduce a new hardness measure called the consistent optimality hardness, and give both the upper and lower bounds of sample complexity. Moreover, we give examples to demonstrate that our solution has the capacity to deal with non-linear reward functions.