Online Least Squares Estimation with Self-Normalized Processes: An Application to Bandit Problems

TL;DR

This paper introduces a new analysis technique for online least squares estimation using self-normalized processes, leading to tighter confidence bounds and improved regret guarantees in bandit problems.

Contribution

It provides a simplified proof of tail bounds for vector martingales and develops improved confidence sets for online decision algorithms, enhancing regret bounds in bandit problems.

Findings

Tighter confidence sets for least squares estimates.

Improved regret bounds for UCB algorithms.

Bounds applicable to small sample sizes and various bandit settings.

Abstract

The analysis of online least squares estimation is at the heart of many stochastic sequential decision making problems. We employ tools from the self-normalized processes to provide a simple and self-contained proof of a tail bound of a vector-valued martingale. We use the bound to construct a new tighter confidence sets for the least squares estimate. We apply the confidence sets to several online decision problems, such as the multi-armed and the linearly parametrized bandit problems. The confidence sets are potentially applicable to other problems such as sleeping bandits, generalized linear bandits, and other linear control problems. We improve the regret bound of the Upper Confidence Bound (UCB) algorithm of Auer et al. (2002) and show that its regret is with high-probability a problem dependent constant. In the case of linear bandits (Dani et al., 2008), we improve the problem dependent bound in the dimension and number of time steps. Furthermore, as opposed to the previous result, we prove that our bound holds for small sample sizes, and at the same time the worst case bound is improved by a logarithmic factor and the constant is improved.