# Distributionally Robust PAC-Bayesian Control

arXiv: https://arxiv.org/abs/2604.10588 · HTML: https://mlsanity.com/paper/2604.10588/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 23 references that are in the map.
- First posted on arXiv 12 Apr 2026.

## Builds on (most cited first, 13 of 13)

- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Distributionally Robust Optimization: A Review](https://mlsanity.com/paper/1908.05659/) (2019)
- [JuMP: A Modeling Language for Mathematical Optimization](https://mlsanity.com/paper/1508.01982/) (2015)
- [Don't Unroll Adjoint: Differentiating SSA-Form Programs](https://mlsanity.com/paper/1810.07951/) (2018)
- [A System Level Approach to Controller Synthesis](https://mlsanity.com/paper/1610.04815/) (2016)
- [PAC-Bayes-Chernoff bounds for unbounded losses](https://mlsanity.com/paper/2401.01148/) (2024)
- [A PAC-Bayesian Framework for Optimal Control with Stability Guarantees](https://mlsanity.com/paper/2403.17790/) (2024)
- [Risk-averse model predictive control](https://mlsanity.com/paper/1704.00342/) (2017)
- [PAC-Bayesian Optimal Control with Stability and Generalization Guarantees](https://mlsanity.com/paper/2512.02858/) (2025)

## Built on by (most cited first, 1 of 1)

- [PAC-Bayesian Certificates for Quadratic Closed-Loop Control](https://mlsanity.com/paper/2606.28281/) (2026)

## Abstract (arXiv)

We present a distributionally robust PAC-Bayesian framework for certifying the performance of learning-based finite-horizon controllers. While existing PAC-Bayes control literature typically assumes bounded losses and matching training and deployment distributions, we explicitly address unbounded losses and environmental distribution shifts (the sim-to-real gap). We achieve this by drawing on two modern lines of research, namely the PAC-Bayes generalization theory and distributionally robust optimization via the type-1 Wasserstein distance. By leveraging the System Level Synthesis (SLS) reparametrization, we derive a sub-Gaussian loss proxy and a bound on the performance loss due to distribution shift. Both are tied directly to the operator norm of the closed-loop map. For linear time-invariant systems, this yields a computationally tractable optimization-based framework together with high-probability safety certificates for deployment in real-world environments that differ from those used in training.