Fast and Regret Optimal Best Arm Identification: Fundamental Limits and Low-Complexity Algorithms

TL;DR

This paper introduces a new algorithm for the multi-armed bandit problem that simultaneously achieves quick identification of the best arm and maximizes reward, with proven optimality and efficient stopping times.

Contribution

It proposes the ROBAI framework and the EOCP algorithm, achieving asymptotic optimal regret and near-optimal sample complexity for dual objectives in bandit problems.

Findings

EOCP achieves asymptotic optimal regret in Gaussian and general bandits.

EOCP commits to the optimal arm in O(log T) rounds with pre-determined stopping time.

Classic UCB algorithms may over-explore, leading to unnecessary regret.

Abstract

This paper considers a stochastic Multi-Armed Bandit (MAB) problem with dual objectives: (i) quick identification and commitment to the optimal arm, and (ii) reward maximization throughout a sequence of $T$ consecutive rounds. Though each objective has been individually well-studied, i.e., best arm identification for (i) and regret minimization for (ii), the simultaneous realization of both objectives remains an open problem, despite its practical importance. This paper introduces \emph{Regret Optimal Best Arm Identification} (ROBAI) which aims to achieve these dual objectives. To solve ROBAI with both pre-determined stopping time and adaptive stopping time requirements, we present an algorithm called EOCP and its variants respectively, which not only achieve asymptotic optimal regret in both Gaussian and general bandits, but also commit to the optimal arm in $\mathcal{O}(\log T)$ rounds with pre-determined stopping time and $\mathcal{O}(\log^2 T)$ rounds with adaptive stopping time. We further characterize lower bounds on the commitment time (equivalent to the sample complexity) of ROBAI, showing that EOCP and its variants are sample optimal with pre-determined stopping time, and almost sample optimal with adaptive stopping time. Numerical results confirm our theoretical analysis and reveal an interesting "over-exploration" phenomenon carried by classic UCB algorithms, such that EOCP has smaller regret even though it stops exploration much earlier than UCB, i.e., $\mathcal{O}(\log T)$ versus $\mathcal{O}(T)$, which suggests over-exploration is unnecessary and potentially harmful to system performance.