想给控制系统加安全证书？这篇用SLS参数化搞定了二次代价，数值实验还比传统方法更稳。
该论文将PAC-Bayesian有限样本保证应用于线性系统的二次轨迹代价控制问题。通过System Level Synthesis参数化显式暴露闭环轨迹映射，使二次代价可证。针对高斯扰动推导了精确单边高斯变换和基于闭环灵敏度的可处理二次上界，并提出了后验局部替代证书。在双积分器数值实验中，该算法作为灵敏度感知的有限样本正则化器，有效降低持有代价和闭环灵敏度。
PAC-Bayesian Certificates for Quadratic Closed-Loop Control
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