Regret Bounds for Episodic Risk-Sensitive Linear Quadratic Regulator

TL;DR

This paper develops regret bounds for online adaptive control of risk-sensitive linear quadratic regulators in episodic settings, introducing algorithms with logarithmic and square-root regret guarantees under different assumptions.

Contribution

It provides the first regret bounds for episodic risk-sensitive LQR, using novel analysis of Riccati equations and performance loss in online learning.

Findings

Achieves $ ilde{O}( ext{log } N)$ regret under identifiability

Proposes exploration noise to attain $ ilde{O}( ext{sqrt } N)$ regret without identifiability

First regret bounds established for risk-sensitive LQR in episodic control

Abstract

Risk-sensitive linear quadratic regulator is one of the most fundamental problems in risk-sensitive optimal control. In this paper, we study online adaptive control of risk-sensitive linear quadratic regulator in the finite horizon episodic setting. We propose a simple least-squares greedy algorithm and show that it achieves $\widetilde{\mathcal{O}}(\log N)$ regret under a specific identifiability assumption, where $N$ is the total number of episodes. If the identifiability assumption is not satisfied, we propose incorporating exploration noise into the least-squares-based algorithm, resulting in an algorithm with $\widetilde{\mathcal{O}}(\sqrt{N})$ regret. To our best knowledge, this is the first set of regret bounds for episodic risk-sensitive linear quadratic regulator. Our proof relies on perturbation analysis of less-standard Riccati equations for risk-sensitive linear quadratic control, and a delicate analysis of the loss in the risk-sensitive performance criterion due to applying the suboptimal controller in the online learning process.