Effective Dimension in Bandit Problems under Censorship
This work addresses sequential decision-making applications with strategic or random uncertainty, such as recommendation systems, but is incremental as it extends classical bandit methods to censored settings.
The paper tackles bandit problems in censored environments by analyzing performance loss and introducing effective dimension as a measure of statistical complexity, showing that it maintains problem structure and leads to regret bounds analogous to uncensored settings, with results including a transient self-correction phase and stationary slowdown.
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).