A PAC-Bayes Approach for Controlling Unknown Linear Discrete-time Systems

2026-05-11

AI Summary

This work addresses the challenge of designing data-driven controllers with theoretical guarantees for stochastic linear discrete-time systems whose parameters follow unknown distributions—a setting where existing methods fall short. The paper introduces, for the first time, a PAC-Bayes framework to learn randomized controllers, deriving data-dependent generalization bounds that apply even when the quadratic cost function is unbounded. Efficient algorithms are developed for both finite and infinite controller hypothesis spaces. The resulting controllers achieve performance comparable to the classical LQG optimum in the ideal LQG setting, while providing rigorous probabilistic performance guarantees. Notably, this approach overcomes the restrictive assumption of bounded cost functions inherent in prior methods.

Abstract

This paper presents a PAC-Bayes framework for learning controllers for unknown stochastic linear discrete-time systems, where the system parameters are drawn from a fixed but unknown distribution. We derive a data-dependent high probability bound on the performance of any learned (stochastic) controller, and propose novel efficient learning algorithms with theoretical guarantees, which can be implemented for both finite and infinite controller spaces. Compared to prior work, our bound holds for unbounded quadratic cost. In the special case where LQG is optimal, our numerical results suggest that the learned controllers achieve comparable performance to LQG.

Problem

• PAC-Bayes
• linear discrete-time systems
• unknown dynamics
• stochastic control
• performance guarantee

Innovation

• PAC-Bayes
• linear discrete-time systems
• stochastic control
• unbounded quadratic cost
• data-dependent bounds