Fixed-Confidence Best Arm Identification with Decreasing Variance

TL;DR

This paper addresses best-arm identification in stochastic bandits with decreasing reward variances, proposing two novel policies that balance sampling and waiting, outperforming existing methods.

Contribution

It introduces two new policies tailored for decreasing variance scenarios, with analytical guarantees and improved performance over state-of-the-art algorithms.

Findings

Proposed policies outperform classical methods in simulations.

Analytical guarantees established for both policies.

Effective handling of decreasing variance in reward distributions.

Abstract

We focus on the problem of best-arm identification in a stochastic multi-arm bandit with temporally decreasing variances for the arms' rewards. We model arm rewards as Gaussian random variables with fixed means and variances that decrease with time. The cost incurred by the learner is modeled as a weighted sum of the time needed by the learner to identify the best arm, and the number of samples of arms collected by the learner before termination. Under this cost function, there is an incentive for the learner to not sample arms in all rounds, especially in the initial rounds. On the other hand, not sampling increases the termination time of the learner, which also increases cost. This trade-off necessitates new sampling strategies. We propose two policies. The first policy has an initial wait period with no sampling followed by continuous sampling. The second policy samples periodically and uses a weighted average of the rewards observed to identify the best arm. We provide analytical guarantees on the performance of both policies and supplement our theoretical results with simulations which show that our polices outperform the state-of-the-art policies for the classical best arm identification problem.