Distribution-Dependent Rates for Multi-Distribution Learning

Rafael Hanashiro, Patrick Jaillet

Keywords: multi-distribution learning, multi-armed bandits

TL;DR: We devise adaptive and non-adaptive algorithms for the multi-distribution learning problem and provide distribution-dependent guarantees using tools from empirical process theory and drawing inspiration from pure exploration multi-armed bandits.

Abstract: To address the needs of modeling uncertainty in sensitive machine learning applications, the setup of distributionally robust optimization (DRO) seeks good performance uniformly across a variety of tasks. The recent multi-distribution learning (MDL) framework \cite{pmlr-v195-awasthi23a-open-prob} tackles this objective in a dynamic interaction with the environment, where the learner has sampling access to each target distribution. Drawing inspiration from the field of pure-exploration multi-armed bandits, we provide \textit{distribution-dependent} guarantees in the MDL regime, that scale with suboptimality gaps and result in superior dependence on the sample size when compared to the existing distribution-independent analyses. We investigate two non-adaptive strategies, uniform and non-uniform exploration, and present non-asymptotic regret bounds using novel tools from empirical process theory. Furthermore, we devise an adaptive optimistic algorithm, LCB-DR, that showcases enhanced dependence on the gaps, mirroring the contrast between uniform and optimistic allocation in the multi-armed bandit literature. We also conduct a small synthetic experiment illustrating the comparative strengths of each strategy.

Submission Number: 77