Thompson Sampling for Linear-Quadratic Control Problems

TL;DR

This paper analyzes the regret of Thompson sampling in linear quadratic control problems, revealing a trade-off that leads to a regret of order T^{2/3}, worse than existing methods.

Contribution

It provides the first analysis of Thompson sampling's regret in LQ control, highlighting a fundamental trade-off affecting its performance.

Findings

Thompson sampling achieves regret of O(T^{2/3}) in LQ control.

Optimism-in-face-of-uncertainty outperforms Thompson sampling with O(√T) regret.

Trade-off between sampling frequency and control policy switches impacts regret.

Abstract

We consider the exploration-exploitation tradeoff in linear quadratic (LQ) control problems, where the state dynamics is linear and the cost function is quadratic in states and controls. We analyze the regret of Thompson sampling (TS) (a.k.a. posterior-sampling for reinforcement learning) in the frequentist setting, i.e., when the parameters characterizing the LQ dynamics are fixed. Despite the empirical and theoretical success in a wide range of problems from multi-armed bandit to linear bandit, we show that when studying the frequentist regret TS in control problems, we need to trade-off the frequency of sampling optimistic parameters and the frequency of switches in the control policy. This results in an overall regret of $O(T^{2/3})$, which is significantly worse than the regret $O(\sqrt{T})$ achieved by the optimism-in-face-of-uncertainty algorithm in LQ control problems.