Structured Linear Contextual Bandits: A Sharp and Geometric Smoothed Analysis

TL;DR

This paper introduces a smoothed setting for structured linear contextual bandits with Gaussian noise perturbations, proposing simple greedy algorithms and providing unified regret bounds that leverage geometric properties of the parameter structure.

Contribution

It presents a unified analysis of greedy algorithms for structured linear bandits under Gaussian smoothing, with sharper regret bounds and geometric insights.

Findings

Greedy algorithms perform well in the smoothed setting.

Regret bounds depend on Gaussian widths related to structure.

Sharper bounds achieved for unstructured parameters.

Abstract

Bandit learning algorithms typically involve the balance of exploration and exploitation. However, in many practical applications, worst-case scenarios needing systematic exploration are seldom encountered. In this work, we consider a smoothed setting for structured linear contextual bandits where the adversarial contexts are perturbed by Gaussian noise and the unknown parameter $\theta^*$ has structure, e.g., sparsity, group sparsity, low rank, etc. We propose simple greedy algorithms for both the single- and multi-parameter (i.e., different parameter for each context) settings and provide a unified regret analysis for $\theta^*$ with any assumed structure. The regret bounds are expressed in terms of geometric quantities such as Gaussian widths associated with the structure of $\theta^*$. We also obtain sharper regret bounds compared to earlier work for the unstructured $\theta^*$ setting as a consequence of our improved analysis. We show there is implicit exploration in the smoothed setting where a simple greedy algorithm works.