# Finite-Sample-Based Reachability for Safe Control with Gaussian Process Dynamics

arXiv: https://arxiv.org/abs/2505.07594 · HTML: https://mlsanity.com/paper/2505.07594/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 63 references that are in the map.
- First posted on arXiv 12 May 2025.

## Builds on (most cited first, 24 of 24)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [acados: a modular open-source framework for fast embedded optimal control](https://mlsanity.com/paper/1910.13753/) (2019)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Scalable Thompson Sampling using Sparse Gaussian Process Models](https://mlsanity.com/paper/2006.05356/) (2020)
- [Formal Verification and Control with Conformal Prediction](https://mlsanity.com/paper/2409.00536/) (2024)
- [Submodular Reinforcement Learning](https://mlsanity.com/paper/2307.13372/) (2023)
- [A Simple and Efficient Sampling-based Algorithm for General Reachability Analysis](https://mlsanity.com/paper/2112.05745/) (2021)
- [Sampling-based Reachability Analysis: A Random Set Theory Approach with Adversarial Sampling](https://mlsanity.com/paper/2008.10180/) (2020)
- [Near-Optimal Multi-Agent Learning for Safe Coverage Control](https://mlsanity.com/paper/2210.06380/) (2022)
- [Sample Path Regularity of Gaussian Processes from the Covariance Kernel](https://mlsanity.com/paper/2312.14886/) (2023)
- [Robust adaptive MPC using control contraction metrics](https://mlsanity.com/paper/2209.11713/) (2022)
- [Safe Guaranteed Exploration for Non-linear Systems](https://mlsanity.com/paper/2402.06562/) (2024)
- [Gaussian Process Model Predictive Control of An Unmanned Quadrotor](https://mlsanity.com/paper/1707.04515/) (2017)
- [Gaussian processes for dynamics learning in model predictive control](https://mlsanity.com/paper/2502.02310/) (2025)
- [Performance-driven Constrained Optimal Auto-Tuner for MPC](https://mlsanity.com/paper/2503.07127/) (2025)
- [A nonlinear model predictive control framework using reference generic terminal ingredients -- extended version](https://mlsanity.com/paper/1909.12765/) (2019)
- [Stochastic Model Predictive Control for Sub-Gaussian Noise](https://mlsanity.com/paper/2503.08795/) (2025)
- [Learning-Based Optimal Control with Performance Guarantees for Unknown Systems with Latent States](https://mlsanity.com/paper/2303.17963/) (2023)
- [Predictive control for nonlinear stochastic systems: Closed-loop guarantees with unbounded noise](https://mlsanity.com/paper/2407.13257/) (2024)
- [Safety Verification of Nonlinear Stochastic Systems via Probabilistic Tube](https://mlsanity.com/paper/2503.03328/) (2025)

## Built on by (most cited first, 5 of 5)

- [Safe Exploration via Policy Priors](https://mlsanity.com/paper/2601.19612/) (2026)
- [Safe and Near-Optimal Control with Online Dynamics Learning](https://mlsanity.com/paper/2509.16650/) (2025)
- [Safe learning-based control via function-based uncertainty quantification](https://mlsanity.com/paper/2604.01173/) (2026)
- [A robust and adaptive MPC formulation for Gaussian process models](https://mlsanity.com/paper/2507.02098/) (2025)
- [Sampling-Based Safe Reinforcement Learning](https://mlsanity.com/paper/2605.19469/) (2026)

## Abstract (arXiv)

Gaussian Process (GP) regression is shown to be effective for learning unknown dynamics, enabling efficient and safety-aware control strategies across diverse applications. However, existing GP-based model predictive control (GP-MPC) methods either rely on approximations, thus lacking guarantees, or are overly conservative, which limits their practical utility. To close this gap, we present a sampling-based framework that efficiently propagates the model's epistemic uncertainty while avoiding conservatism. We establish a novel sample complexity result that enables the construction of a reachable set using a finite number of dynamics functions sampled from the GP posterior. Building on this, we design a sampling-based GP-MPC scheme that is recursively feasible and guarantees closed-loop safety and stability with high probability. Finally, we showcase the effectiveness of our method on two numerical examples, highlighting accurate reachable set over-approximation and safe closed-loop performance.