Regret Bounds for Episodic Risk-Sensitive Linear Quadratic Regulator
This work addresses a fundamental problem in risk-sensitive optimal control, providing theoretical guarantees for online learning in control systems, though it is incremental as it extends regret analysis to a risk-sensitive variant.
The paper tackles the online adaptive control of risk-sensitive linear quadratic regulator in episodic settings, proposing algorithms that achieve $\widetilde{\mathcal{O}}(\log N)$ regret under an identifiability assumption and $\widetilde{\mathcal{O}}(\sqrt{N})$ regret otherwise, marking the first regret bounds for this problem.
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.