PAC-Bayesian Certificates for Quadratic Closed-Loop Control
2026-06-26 • Machine Learning
Machine Learning
AI summaryⓘ
The authors explore how PAC-Bayesian methods, which give guarantees for random decision-making based on data, can be applied to control systems where the goal is to minimize quadratic costs over time. They use a technique called System Level Synthesis to directly work with closed-loop system responses, making it easier to certify these quadratic costs. They develop new mathematical bounds and certificates that handle uncertainties from disturbances and link the randomized predictions to a deterministic control strategy. Their approach also leads to a practical learning algorithm that improves control performance, especially when data is limited, as shown in simulations with a double integrator system.
PAC-Bayesian boundsSystem Level Synthesisclosed-loop controlquadratic costChernoff boundsGaussian disturbancesposterior distributionfinite-sample guaranteescontrol sensitivitylearning-based control
Authors
Domagoj Herceg
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. 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