# A PAC-Bayesian Framework for Optimal Control with Stability Guarantees

**Authors:** Mahrokh Ghoddousi Boroujeni, Clara Lucía Galimberti, Andreas Krause, Giancarlo Ferrari-Trecate  
**arXiv:** [2403.17790](https://arxiv.org/abs/2403.17790)  
**Date:** 2024-03-26  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2403.17790v1/

## Key points

- Stochastic Nonlinear Optimal Control (SNOC) focuses on minimizing a cost function to address uncertainties in nonlinear systems
- Traditional approach involves minimizing an empirical cost from a finite dataset, but faces challenges with out-of-sample uncertainties and small training datasets
- SNOC policies can overfit, leading to discrepancies between empirical and true costs during control deployment
- A novel approach using PAC-Bayes theory provides generalization bounds for SNOC, allowing for optimal controller design with prior knowledge integration
- This approach ensures closed-loop stability by incorporating recent parametrizations of stabilizing controllers for nonlinear systems
- Demonstrated effectiveness through the design of neural network controllers in cooperative robotics tasks
- Offers a principled way to improve control policies, combat overfitting, and enhance performance and reliability of SNOC systems

## Summary

Stochastic Nonlinear Optimal Control (SNOC) is a critical area of research that focuses on minimizing a cost function to account for random uncertainties impacting the dynamics of nonlinear systems. This problem has traditionally been approached by minimizing an empirical cost derived from a finite dataset of sampled disturbances. However, this method poses challenges in quantifying control performance against out-of-sample uncertainties, especially when dealing with small training datasets. SNOC policies are susceptible to overfitting in such scenarios, leading to significant discrepancies between the empirical cost and the true cost incurred during control deployment. To address these issues and ensure reliability in real-world applications, a novel approach leveraging PAC-Bayes theory has been introduced. This framework provides rigorous generalization bounds for SNOC, allowing for the design of optimal controllers that incorporate prior knowledge into the synthesis process. By integrating recent parametrizations of stabilizing controllers for nonlinear systems, this approach inherently guarantees closed-loop stability. The effectiveness of this proposed method in incorporating prior knowledge and mitigating overfitting has been demonstrated through the design of neural network controllers for tasks in cooperative robotics. The refined approach offers a principled way to improve control policies while combatting overfitting, ultimately enhancing the overall performance and reliability of SNOC systems in practical applications.