On the Optimal Sample Complexity for Best Arm Identification

TL;DR

This paper advances understanding of the sample complexity in best arm identification for stochastic bandits, introducing improved algorithms and lower bounds, especially for the two-arm case, and proposing a conjecture on optimality.

Contribution

It presents a new algorithm for BEST-1-ARM that surpasses previous bounds, and establishes a novel lower bound for the two-arm sign problem, extending classical results and linking them to BEST-1-ARM.

Findings

New upper bound algorithm for BEST-1-ARM

A simplified, extended lower bound for the sign problem

Reduction from Sign problem to BEST-1-ARM for lower bounds

Abstract

We study the best arm identification (BEST-1-ARM) problem, which is defined as follows. We are given $n$ stochastic bandit arms. The $i$th arm has a reward distribution $D_i$ with an unknown mean $\mu_{i}$. Upon each play of the $i$th arm, we can get a reward, sampled i.i.d. from $D_i$. We would like to identify the arm with the largest mean with probability at least $1-\delta$, using as few samples as possible. We provide a nontrivial algorithm for BEST-1-ARM, which improves upon several prior upper bounds on the same problem. We also study an important special case where there are only two arms, which we call the sign problem. We provide a new lower bound of sign, simplifying and significantly extending a classical result by Farrell in 1964, with a completely new proof. Using the new lower bound for sign, we obtain the first lower bound for BEST-1-ARM that goes beyond the classic Mannor-Tsitsiklis lower bound, by an interesting reduction from Sign to BEST-1-ARM. We propose an interesting conjecture concerning the optimal sample complexity of BEST-1-ARM from the perspective of instance-wise optimality.