Sample complexity of partition identification using multi-armed bandits

TL;DR

This paper investigates the sample complexity of identifying the correct partition of a vector of distributions in multi-armed bandit problems, providing bounds and algorithms for various geometric partition structures.

Contribution

It introduces a unified framework for partition identification in bandits, deriving lower bounds and proposing algorithms that asymptotically match these bounds across different geometric settings.

Findings

Derived lower bounds on sample complexity based on problem geometry

Proposed algorithms match lower bounds asymptotically

Applicable to diverse applications like finance

Abstract

Given a vector of probability distributions, or arms, each of which can be sampled independently, we consider the problem of identifying the partition to which this vector belongs from a finitely partitioned universe of such vector of distributions. We study this as a pure exploration problem in multi armed bandit settings and develop sample complexity bounds on the total mean number of samples required for identifying the correct partition with high probability. This framework subsumes well studied problems such as finding the best arm or the best few arms. We consider distributions belonging to the single parameter exponential family and primarily consider partitions where the vector of means of arms lie either in a given set or its complement. The sets considered correspond to distributions where there exists a mean above a specified threshold, where the set is a half space and where either the set or its complement is a polytope, or more generally, a convex set. In these settings, we characterize the lower bounds on mean number of samples for each arm highlighting their dependence on the problem geometry. Further, inspired by the lower bounds, we propose algorithms that can match these bounds asymptotically with decreasing probability of error. Applications of this framework may be diverse. We briefly discuss one associated with finance.