Anytime Model Selection in Linear Bandits

Parnian Kassraie, Nicolas Emmenegger, Andreas Krause, Aldo Pacchiano

Keywords: Model Selection, Linear Bandits, Anytime Martingale Bounds

TL;DR: For model selection in Linear Bandits, we present exponentially improved guarantees that are uniformly valid over an unbounded time horizon.

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 ($\mathrm{poly}M$) with the number of models $M$ in terms of their regret. We develop \alexp, an anytime algorithm, which has an exponentially improved ($\log M$) dependence on $M$ for its regret. We neither require knowledge of the horizon $n$, nor rely on an initial purely exploratory stage. Our approach utilizes a novel time-uniform analysis of the Lasso, by defining a self-normalized martingale sequence based on the empirical process error, establishing a new connection between interactive learning and high-dimensional statistics.

Submission Number: 70