# Bayesian Safe Learning and Control with Sum-of-Squares Analysis and Polynomial Kernels

arXiv: https://arxiv.org/abs/2004.00662 · HTML: https://mlsanity.com/paper/2004.00662/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2020.

## Built on by (most cited first, 7 of 7)

- [Symbolic Abstractions From Data: A PAC Learning Approach](https://mlsanity.com/paper/2104.13901/) (2021)
- [Learning-enhanced robust controller synthesis with rigorous statistical and control-theoretic guarantees](https://mlsanity.com/paper/2105.03397/) (2021)
- [Machine Learning for Smart and Energy-Efficient Buildings](https://mlsanity.com/paper/2211.14889/) (2022)
- [Bayesian inference for data-efficient, explainable, and safe robotic motion planning: A review](https://mlsanity.com/paper/2307.08024/) (2023)
- [Combining Bayesian Inference and Reinforcement Learning for Agent Decision Making: A Review](https://mlsanity.com/paper/2505.07911/) (2025)
- [Safe Learning of Uncertain Environments](https://mlsanity.com/paper/2103.01413/) (2021)
- [Learning Safety Filters for Unknown Discrete-Time Linear Systems](https://mlsanity.com/paper/2111.00631/) (2021)

## Abstract (arXiv)

We propose an iterative method to safely learn the unmodeled dynamics of a nonlinear system using Bayesian Gaussian process (GP) models with polynomial kernel functions. The method maintains safety by ensuring that the system state stays within the region of attraction (ROA) of a stabilizing control policy while collecting data. A quadratic programming based exploration control policy is computed to keep the exploration trajectory inside an inner-approximation of the ROA and to maximize the information gained from the trajectory. A prior GP model, which incorporates prior information about the unknown dynamics, is used to construct an initial stabilizing policy. As the GP model is updated with data, it is used to synthesize a new policy and a larger ROA, which increases the range of safe exploration. The use of polynomial kernels allows us to compute ROA inner-approximations and stabilizing control laws for the model using sum-of-squares programming. We also provide a probabilistic guarantee of safety which ensures that the policy computed using the learned model stabilizes the true dynamics with high confidence.