Minimax Optimal Algorithms for Fixed-Budget Best Arm Identification

TL;DR

This paper characterizes the optimal exponential decay rate of misidentification probability in fixed-budget best arm identification, introducing two rates and algorithms that outperform existing methods.

Contribution

It introduces two new rates, $R^{ ext{go}}$ and $R^{ ext{go}}_ extfty$, and proposes algorithms achieving these rates, advancing understanding of the problem's fundamental limits.

Findings

The rate $R^{ ext{go}}$ outperforms existing algorithms.

The rate $R^{ ext{go}}_ extinfty$ is achievable with the DOT algorithm.

The paper provides a minimax optimal characterization of the problem.

Abstract

We consider the fixed-budget best arm identification problem where the goal is to find the arm of the largest mean with a fixed number of samples. It is known that the probability of misidentifying the best arm is exponentially small to the number of rounds. However, limited characterizations have been discussed on the rate (exponent) of this value. In this paper, we characterize the minimax optimal rate as a result of an optimization over all possible parameters. We introduce two rates, $R^{\mathrm{go}}$ and $R^{\mathrm{go}}_{\infty}$, corresponding to lower bounds on the probability of misidentification, each of which is associated with a proposed algorithm. The rate $R^{\mathrm{go}}$ is associated with $R^{\mathrm{go}}$-tracking, which can be efficiently implemented by a neural network and is shown to outperform existing algorithms. However, this rate requires a nontrivial condition to be achievable. To address this issue, we introduce the second rate $R^{\mathrm{go}}_\infty$. We show that this rate is indeed achievable by introducing a conceptual algorithm called delayed optimal tracking (DOT).