Effective Dimension in Bandit Problems under Censorship

TL;DR

This paper introduces the concept of effective dimension to analyze the performance of bandit algorithms under censorship, providing a framework that captures the statistical complexity and adapts classical results to censored environments.

Contribution

It develops a broad class of censorship models analyzed through effective dimension, extending classical bandit analysis and introducing a generalized elliptical potential inequality.

Findings

Effective dimension captures the complexity of censored bandit problems.

A transient phase allows correction of initial censorship misspecification.

Results extend classical bandit regret bounds to censored settings.

Abstract

In this paper, we study both multi-armed and contextual bandit problems in censored environments. Our goal is to estimate the performance loss due to censorship in the context of classical algorithms designed for uncensored environments. Our main contributions include the introduction of a broad class of censorship models and their analysis in terms of the effective dimension of the problem -- a natural measure of its underlying statistical complexity and main driver of the regret bound. In particular, the effective dimension allows us to maintain the structure of the original problem at first order, while embedding it in a bigger space, and thus naturally leads to results analogous to uncensored settings. Our analysis involves a continuous generalization of the Elliptical Potential Inequality, which we believe is of independent interest. We also discover an interesting property of decision-making under censorship: a transient phase during which initial misspecification of censorship is self-corrected at an extra cost, followed by a stationary phase that reflects the inherent slowdown of learning governed by the effective dimension. Our results are useful for applications of sequential decision-making models where the feedback received depends on strategic uncertainty (e.g., agents' willingness to follow a recommendation) and/or random uncertainty (e.g., loss or delay in arrival of information).