Optimism-Based Adaptive Regulation of Linear-Quadratic Systems

TL;DR

This paper provides finite-time regret bounds for optimism-based adaptive control policies in linear-quadratic systems, addressing the exploration-exploitation trade-off with novel probabilistic techniques.

Contribution

It establishes the first non-asymptotic, high-probability regret bounds for optimism-based adaptive regulation of linear-quadratic systems under mild assumptions.

Findings

Optimal up to logarithmic factors in finite time

Handles heavy-tailed noise distributions

Requires only stabilizability of the system

Abstract

The main challenge for adaptive regulation of linear-quadratic systems is the trade-off between identification and control. An adaptive policy needs to address both the estimation of unknown dynamics parameters (exploration), as well as the regulation of the underlying system (exploitation). To this end, optimism-based methods which bias the identification in favor of optimistic approximations of the true parameter are employed in the literature. A number of asymptotic results have been established, but their finite time counterparts are few, with important restrictions. This study establishes results for the worst-case regret of optimism-based adaptive policies. The presented high probability upper bounds are optimal up to logarithmic factors. The non-asymptotic analysis of this work requires very mild assumptions; (i) stabilizability of the system's dynamics, and (ii) limiting the degree of heaviness of the noise distribution. To establish such bounds, certain novel techniques are developed to comprehensively address the probabilistic behavior of dependent random matrices with heavy-tailed distributions.