The Max $K$-Armed Bandit: PAC Lower Bounds and Efficient Algorithms

TL;DR

This paper studies the Max K-Armed Bandit problem, establishing fundamental lower bounds on sample complexity under PAC constraints and proposing algorithms that nearly achieve these bounds, with analysis of robustness and performance comparisons.

Contribution

It provides the first PAC lower bounds for the Max K-Armed Bandit problem and introduces an efficient algorithm that nearly matches these bounds.

Findings

Lower bounds on sample complexity under PAC framework.

An algorithm that nearly attains the lower bounds.

Performance comparison between distinguishable and indistinguishable arms.

Abstract

We consider the Max $K$-Armed Bandit problem, where a learning agent is faced with several stochastic arms, each a source of i.i.d. rewards of unknown distribution. At each time step the agent chooses an arm, and observes the reward of the obtained sample. Each sample is considered here as a separate item with the reward designating its value, and the goal is to find an item with the highest possible value. Our basic assumption is a known lower bound on the {\em tail function} of the reward distributions. Under the PAC framework, we provide a lower bound on the sample complexity of any $(\epsilon,\delta)$-correct algorithm, and propose an algorithm that attains this bound up to logarithmic factors. We analyze the robustness of the proposed algorithm and in addition, we compare the performance of this algorithm to the variant in which the arms are not distinguishable by the agent and are chosen randomly at each stage. Interestingly, when the maximal rewards of the arms happen to be similar, the latter approach may provide better performance.