PAC-Bayesian Certificates for Quadratic Closed-Loop Control

Forum topic · 小凯 · 2026-06-30

Summary

This paper introduces PAC-Bayesian certificates for learning-based control with quadratic trajectory costs. PAC-Bayesian bounds provide finite-sample guarantees for data-dependent randomized predictors, but quadratic control losses are unbounded and non-Lipschitz, complicating certification. The authors use System Level Synthesis (SLS) parameterization, which directly exposes the closed-loop trajectory map of linear systems and makes the quadratic control loss amenable to explicit certification. They provide PAC-Bayes-Chernoff certificates for posterior distributions over feasible closed-loop responses. For Gaussian disturbance trajectories with arbitrary covariance, they derive an exact one-sided Gaussian transform and a tractable quadratic upper bound expressed through closed-loop sensitivity quantities. Although PAC-Bayes certifies randomized posteriors, the convex quadratic form of the SLS loss transfers certificates to the posterior mean response, enabling deterministic deployment while retaining randomness in the bound, with an additional data-driven bound for this deployment. 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.

PAC-Bayesian Certificates for Quadratic Closed-Loop Control

Paper: arXiv:2606.28281 Author: Domagoj Herceg Field: Machine Learning

Overview

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 authors employ 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.

Key Contributions

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 and 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 certificates to the posterior mean response, yielding a deterministic mean-response deployment result particularly suited to control while retaining the randomized posterior in the bound.
A data-driven bound for this deterministic deployment.

Experiments

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

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PAC-Bayesian Certificates for Quadratic Closed-Loop Control

Summary

PAC-Bayesian Certificates for Quadratic Closed-Loop Control

Overview

Key Contributions

Experiments

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