Online Multi-Armed Bandit

TL;DR

This paper introduces a new online multi-armed bandit problem where each bandit can only be visited once, studies strategies for Bernoulli bandits with unknown means, and proves their near-optimality under various distributional assumptions.

Contribution

It formulates a novel single-visit bandit problem, proposes optimal strategies, and provides bounds on their performance, including cases with unknown underlying distributions.

Findings

Proposed strategies are optimal up to a constant factor.

Bounded the performance of any optimal strategy.

Algorithms perform well even with unknown distributional parameters.

Abstract

We introduce a novel variant of the multi-armed bandit problem, in which bandits are streamed one at a time to the player, and at each point, the player can either choose to pull the current bandit or move on to the next bandit. Once a player has moved on from a bandit, they may never visit it again, which is a crucial difference between our problem and classic multi-armed bandit problems. In this online context, we study Bernoulli bandits (bandits with payout Ber($p_i$) for some underlying mean $p_i$) with underlying means drawn i.i.d. from various distributions, including the uniform distribution, and in general, all distributions that have a CDF satisfying certain differentiability conditions near zero. In all cases, we suggest several strategies and investigate their expected performance. Furthermore, we bound the performance of any optimal strategy and show that the strategies we have suggested are indeed optimal up to a constant factor. We also investigate the case where the distribution from which the underlying means are drawn is not known ahead of time. We again, are able to suggest algorithms that are optimal up to a constant factor for this case, given certain mild conditions on the universe of distributions.