PAC-Bayesian Certificates for Quadratic Closed-Loop Control

Forum topic · 小凯 · 2026-06-30

Summary

This paper (arXiv:2606.28281) by Domagoj Herceg develops PAC-Bayesian finite-sample certificates for learning-based control of linear systems. PAC-Bayes bounds typically certify data-dependent randomized predictors, but applying them to control is challenging because the quadratic trajectory cost objective is unbounded and non-Lipschitz, producing response-dependent Chernoff terms. The author uses a System Level Synthesis (SLS) parameterization, which directly exposes the closed-loop trajectory map of a linear system and makes the quadratic control loss amenable to explicit certification. The work provides a family of PAC-Bayes-Chernoff certificates for posterior distributions over feasible closed-loop responses, and for Gaussian disturbance trajectories with arbitrary covariance derives an exact one-sided Gaussian transform plus a tractable quadratic upper bound expressed through closed-loop sensitivity quantities. Although PAC-Bayes certifies randomized posteriors, the convex quadratic structure of the SLS loss allows certificates to transfer to the posterior mean response, yielding a deterministic deployment result with data-driven bounds. Numerical experiments on a double integrator show the algorithm 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

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, an exact one-sided Gaussian transform is derived, along with a tractable quadratic upper bound expressed through closed-loop sensitivity quantities.

Although PAC-Bayes certifies non-degenerate posteriors, the convex quadratic form of the SLS loss transfers the certificates to the posterior mean response. The paper presents a deterministic mean-response deployment result particularly suited to control, while retaining the stochastic posterior in the bound, and provides data-driven bounds for this deployment.

Numerical experiments on a double integrator demonstrate that the algorithm acts as a sensitivity-aware finite-sample regularizer, improving held-out cost and reducing closed-loop sensitivity in low-data regimes.

Highlights

Applies PAC-Bayesian analysis to quadratic trajectory costs in learning-based control, overcoming unboundedness and non-Lipschitzness.
Uses System Level Synthesis parameterization to expose the closed-loop trajectory map and enable explicit certification.
Derives exact one-sided Gaussian transforms and tractable quadratic bounds for arbitrary-covariance Gaussian disturbances.
Transfers certificates to a deterministic posterior-mean deployment with data-driven bounds.
Experiments on a double integrator show improved held-out cost and reduced closed-loop sensitivity in low-data settings.