PAC-Bayesian Certificates for Quadratic Closed-Loop Control

Forum topic · 小凯 · 2026-06-30

Summary

This paper by Domagoj Herceg (arXiv:2606.28281) provides finite-sample performance guarantees for learning-based control using PAC-Bayesian analysis. The main challenge is that quadratic trajectory costs, the natural objective in control, are unbounded and non-Lipschitz, complicating standard PAC-Bayesian certification. The author employs a System Level Synthesis (SLS) parameterization that directly exposes the closed-loop trajectory map of a linear system, making the quadratic control loss amenable to explicit certification. The paper derives PAC-Bayes-Chernoff certificates for posterior distributions over feasible closed-loop responses. For Gaussian disturbance trajectories with arbitrary covariance, it provides an exact one-sided Gaussian transform and a tractable quadratic upper bound expressed through closed-loop sensitivity quantities. Although PAC-Bayes certifies a non-degenerate posterior, the convex quadratic structure of the SLS loss transfers certificates to the posterior mean response, enabling a deterministic deployment result with data-driven bounds. Numerical experiments on a double integrator show the method acts as a sensitivity-aware finite-sample regularizer, improving held-out cost and reducing closed-loop sensitivity in low-data regimes.

Overview

Field: Machine Learning Author: Domagoj Herceg Published: 2026-06-26 arXiv: 2606.28281

Abstract (translated)

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.

The author employs a System Level Synthesis (SLS) parameterization, which exposes the closed-loop trajectory map of a linear system directly and makes the quadratic control loss amenable to explicit certification. Moreover, the paper provides a set of PAC-Bayes-Chernoff certificates for posterior distributions over feasible closed-loop responses.

For Gaussian disturbance trajectories with arbitrary covariance, the author derives an exact one-sided Gaussian transform and a tractable quadratic upper bound expressed through closed-loop sensitivity quantities.

Although PAC-Bayes certifies a non-degenerate posterior, the convex quadratic form of the SLS loss transfers the certificate to the posterior mean response. The paper proposes a deterministic mean-response deployment result particularly suited to control, while retaining the randomized posterior in the bound. A data-driven bound for this deployment is also provided.

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 low-data regimes.

Key points

Applies PAC-Bayesian analysis to learning-based control with quadratic trajectory costs, despite their unbounded, non-Lipschitz nature.
Uses System Level Synthesis (SLS) parameterization to make closed-loop responses explicitly certifiable.
Derives PAC-Bayes-Chernoff certificates, including an exact one-sided Gaussian transform for arbitrarily covariant Gaussian disturbances.
Transfers certificates from the randomized posterior to a deterministic posterior-mean deployment via the convex quadratic structure of the SLS loss.
Experiments on a double integrator demonstrate improved held-out cost and reduced closed-loop sensitivity in low-data settings.

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