Distributionally Robust PAC-Bayesian Control

TL;DR

This paper introduces a distributionally robust PAC-Bayesian approach for certifying control performance under distribution shifts and unbounded losses, leveraging Wasserstein distance and SLS reparametrization.

Contribution

It combines PAC-Bayes theory with distributionally robust optimization to handle unbounded losses and environment shifts in control systems.

Findings

Provides a sub-Gaussian loss proxy tied to the operator norm of the closed-loop map.

Derives a performance loss bound due to distribution shift.

Offers a computationally tractable framework with safety certificates for real-world deployment.

Abstract

We present a distributionally robust PAC-Bayesian framework for certifying the performance of learning-based finite-horizon controllers. While existing PAC-Bayes control literature typically assumes bounded losses and matching training and deployment distributions, we explicitly address unbounded losses and environmental distribution shifts (the sim-to-real gap). We achieve this by drawing on two modern lines of research, namely the PAC-Bayes generalization theory and distributionally robust optimization via the type-1 Wasserstein distance. By leveraging the System Level Synthesis (SLS) reparametrization, we derive a sub-Gaussian loss proxy and a bound on the performance loss due to distribution shift. Both are tied directly to the operator norm of the closed-loop map. For linear time-invariant systems, this yields a computationally tractable optimization-based framework together with high-probability safety certificates for deployment in real-world environments that differ from those used in training.