On the Existence of a Complexity in Fixed Budget Bandit Identification

TL;DR

This paper investigates the fundamental limits of fixed budget bandit identification, revealing that for some tasks, no single algorithm can achieve the optimal error decay rate across all problem instances.

Contribution

It introduces the concept of complexity as a lower bound on error probability and shows it is determined by the best non-adaptive sampling strategy, highlighting limitations in fixed budget identification.

Findings

Existence of complexity depends on the problem

No universal optimal algorithm for Bernoulli best arm with two arms

Complexity is linked to non-adaptive sampling procedures

Abstract

In fixed budget bandit identification, an algorithm sequentially observes samples from several distributions up to a given final time. It then answers a query about the set of distributions. A good algorithm will have a small probability of error. While that probability decreases exponentially with the final time, the best attainable rate is not known precisely for most identification tasks. We show that if a fixed budget task admits a complexity, defined as a lower bound on the probability of error which is attained by the same algorithm on all bandit problems, then that complexity is determined by the best non-adaptive sampling procedure for that problem. We show that there is no such complexity for several fixed budget identification tasks including Bernoulli best arm identification with two arms: there is no single algorithm that attains everywhere the best possible rate.