Distributionally Robust PAC-Bayesian Control

Domagoj Herceg; Duarte Antunes

cs.LGApr 12, 2026

Distributionally Robust PAC-Bayesian Control

Authors: Domagoj Herceg, Duarte Antunes

Organizations: The authors are with the Department of Mechanical Engineering, Eindhoven university of Technology, Eindhoven, The Netherlands.

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.

Figures & tables

Fig. 1: Decomposition of the robust PAC-Bayes certificate cost for various training sample sizes. Gibbs risk is defined as empirical cost averaged over the optimized posterior

Q

. As the PAC-Bayes complexity depends on the controller performance, the resulting cost is self-balancing without the need to resort to tricks such as saturation.

Fig. 2: Comparison of vanilla PAC-Bayes (

\rho=0

) vs robust PAC-Bayes (

\rho=0.08

). The vanilla PAC-Bayes bound (black dashed) does not properly upper-bound the test risk (gray solid) due to unmodeled distribution shift, while the robust bound (orange dashed) provides a valid bound on the test risk (blue solid). We can also observe that DRO PAC-Bayes results in a controller that has lower empirical cost in addition to correct theoretical guarantee. Both empirical test costs are validated against the same deployment distribution shift, suggesting the twofold benefit of our DRO PAC-Bayes approach. We empirically observe the same behavior over different

\rho

values. We report mean values for test cost.

Distribution Shift Robustness

Distribution shift robustness studies how to keep machine-learning predictions reliable when deployment data differ from training data.

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Jun 26, 2026J/K move · Enter open · S save

PAC-Bayesian Certificates for Quadratic Closed-Loop Control

Domagoj Herceg

Department of Mechanical Engineering Eindhoven University of Technology The Netherlands

PAC-Bayesian Generalization Bounds Data-Driven Control Optimal Control

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PAC-Bayesian bounds provide finite-sample guarantees for data-dependent randomized predictors, but applying them to learning-based control is difficult because the natural objective is a quadratic trajectory cost. Such losses are unbounded, non-Lipschitz , and lead to response-dependent Chernoff terms. We employ System Level Synthesis parameterization, which exposes the closed-loop trajectory map of a linear system directly and makes the quadratic control loss amenable to explicit certification. Moreover, we provide a set of PAC-Bayes-Chernoff certificates for posterior distributions over feasible closed-loop responses. For Gaussian disturbance trajectories with arbitrary covariance, we derive an exact one-sided Gaussian transform and a tractable quadratic upper bound expressed through closed-loop sensitivity quantities. We also derive a posterior-localized surrogate for settings where pointwise closed-loop response certificates are unavailable or have support related admissibility issues. Although PAC-Bayes certifies a non-degenerate posterior, the convex quadratic form of the SLS loss transfers the certificate to the posterior mean response. We present a deterministic mean response deployment result that is particularly suitable for control while retaining the stochastic posterior in the bound. Additionally, we provide a data-driven bound for this deployment, transitioning away from an oracle bound. Minimizing this bound naturally results in a learning algorithm for control selection from data. Numerical experiments on a double integrator show that the algorithm acts as a sensitivity-aware finite-sample regularizer, improving held-out cost and reducing closed-loop sensitivity in the low-data regime

Abstract

Distribution Shift Robustness

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