Anytime Model Selection in Linear Bandits

TL;DR

This paper introduces ALEXP, a new model selection algorithm for linear bandits that significantly improves regret dependence on the number of models by emulating full-information feedback, with anytime guarantees.

Contribution

The paper develops ALEXP, an online learning algorithm for model selection in linear bandits with exponentially improved regret dependence on the number of models, using a novel analysis of the Lasso.

Findings

ALEXP achieves $ ext{log} M$ regret dependence, improving over previous polynomial bounds.

ALEXP provides anytime regret guarantees without prior knowledge of the horizon.

The approach establishes a new connection between online learning and high-dimensional statistics.

Abstract

Model selection in the context of bandit optimization is a challenging problem, as it requires balancing exploration and exploitation not only for action selection, but also for model selection. One natural approach is to rely on online learning algorithms that treat different models as experts. Existing methods, however, scale poorly ($\text{poly}M$) with the number of models $M$ in terms of their regret. Our key insight is that, for model selection in linear bandits, we can emulate full-information feedback to the online learner with a favorable bias-variance trade-off. This allows us to develop ALEXP, which has an exponentially improved ($\log M$) dependence on $M$ for its regret. ALEXP has anytime guarantees on its regret, and neither requires knowledge of the horizon $n$, nor relies on an initial purely exploratory stage. Our approach utilizes a novel time-uniform analysis of the Lasso, establishing a new connection between online learning and high-dimensional statistics.