# Provably Safe Reinforcement Learning via Action Projection using Reachability Analysis and Polynomial Zonotopes

arXiv: https://arxiv.org/abs/2210.10691 · HTML: https://mlsanity.com/paper/2210.10691/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 61 references that are in the map.
- First posted on arXiv 19 Oct 2022.

## Builds on (most cited first, 22 of 22)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [Deep Reinforcement Learning for Autonomous Driving: A Survey](https://mlsanity.com/paper/2002.00444/) (2020)
- [Safe Reinforcement Learning via Shielding](https://mlsanity.com/paper/1708.08611/) (2017)
- [Sim-to-Real Transfer in Deep Reinforcement Learning for Robotics: a Survey](https://mlsanity.com/paper/2009.13303/) (2020)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [Projection-Based Constrained Policy Optimization](https://mlsanity.com/paper/2010.03152/) (2020)
- [A Closer Look at Invalid Action Masking in Policy Gradient Algorithms](https://mlsanity.com/paper/2006.14171/) (2020)
- [A predictive safety filter for learning-based control of constrained nonlinear dynamical systems](https://mlsanity.com/paper/1812.05506/) (2018)
- [Trial without Error: Towards Safe Reinforcement Learning via Human Intervention](https://mlsanity.com/paper/1707.05173/) (2017)
- [Safe Reinforcement Learning on Autonomous Vehicles](https://mlsanity.com/paper/1910.00399/) (2019)
- [safe-control-gym: a Unified Benchmark Suite for Safe Learning-based Control and Reinforcement Learning in Robotics](https://mlsanity.com/paper/2109.06325/) (2021)
- [Cautious Reinforcement Learning with Logical Constraints](https://mlsanity.com/paper/2002.12156/) (2020)
- [Verifiably Safe Exploration for End-to-End Reinforcement Learning](https://mlsanity.com/paper/2007.01223/) (2020)
- [Neural Simplex Architecture](https://mlsanity.com/paper/1908.00528/) (2019)
- [Reachability-based Trajectory Safeguard (RTS): A Safe and Fast Reinforcement Learning Safety Layer for Continuous Control](https://mlsanity.com/paper/2011.08421/) (2020)
- [Safe Reinforcement Learning Using Black-Box Reachability Analysis](https://mlsanity.com/paper/2204.07417/) (2022)
- [Provably Safe Deep Reinforcement Learning for Robotic Manipulation in Human Environments](https://mlsanity.com/paper/2205.06311/) (2022)
- [Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis](https://mlsanity.com/paper/1901.01780/) (2019)
- [Online Shielding for Stochastic Systems](https://mlsanity.com/paper/2012.09539/) (2020)
- [Fully-Automated Verification of Linear Systems Using Inner- and Outer-Approximations of Reachable Sets](https://mlsanity.com/paper/2209.09321/) (2022)
- [Utilizing Dependencies to Obtain Subsets of Reachable Sets](https://mlsanity.com/paper/1910.08354/) (2019)

## Built on by (most cited first, 26 of 26)

- [Iterative Reachability Estimation for Safe Reinforcement Learning](https://mlsanity.com/paper/2309.13528/) (2023)
- [Empowering Autonomous Driving with Large Language Models: A Safety Perspective](https://mlsanity.com/paper/2312.00812/) (2023)
- [NavRL: Learning Safe Flight in Dynamic Environments](https://mlsanity.com/paper/2409.15634/) (2024)
- [Safety Filtering While Training: Improving the Performance and Sample Efficiency of Reinforcement Learning Agents](https://mlsanity.com/paper/2410.11671/) (2024)
- [A Human-Centered Safe Robot Reinforcement Learning Framework with Interactive Behaviors](https://mlsanity.com/paper/2302.13137/) (2023)
- [Safe Reinforcement Learning on the Constraint Manifold: Theory and Applications](https://mlsanity.com/paper/2404.09080/) (2024)
- [POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled Systems](https://mlsanity.com/paper/2304.01218/) (2023)
- [Verified Safe Reinforcement Learning for Neural Network Dynamic Models](https://mlsanity.com/paper/2405.15994/) (2024)
- [Reinforcement Learning with Adaptive Regularization for Safe Control of Critical Systems](https://mlsanity.com/paper/2404.15199/) (2024)
- [Safe Reinforcement Learning using Data-Driven Predictive Control](https://mlsanity.com/paper/2211.11027/) (2022)
- [DeepBern-Nets: Taming the Complexity of Certifying Neural Networks using Bernstein Polynomial Activations and Precise Bound Propagation](https://mlsanity.com/paper/2305.13508/) (2023)
- [Provably Optimal Reinforcement Learning under Safety Filtering](https://mlsanity.com/paper/2510.18082/) (2025)
- [Provable Traffic Rule Compliance in Safe Reinforcement Learning on the Open Sea](https://mlsanity.com/paper/2402.08502/) (2024)
- [BAP-SRL: Bayesian Adaptive Priority Safe Reinforcement Learning for Vehicle Motion Planning at Mixed Traffic Intersections](https://mlsanity.com/paper/2601.21679/) (2026)
- [Shields to Guarantee Probabilistic Safety in MDPs](https://mlsanity.com/paper/2605.10888/) (2026)
- [X-Safe: Embodiment-Agnostic Safeguarding for Robot Arm Manipulation Policies via Action Masking](https://mlsanity.com/paper/2606.22278/) (2026)
- [Learning Vision-Based Neural Network Controllers with Semi-Probabilistic Safety Guarantees](https://mlsanity.com/paper/2503.00191/) (2025)
- [Leveraging Analytic Gradients in Provably Safe Reinforcement Learning](https://mlsanity.com/paper/2506.01665/) (2025)
- [SEG-Parking: Towards Safe, Efficient, and Generalizable Autonomous Parking via End-to-End Offline Reinforcement Learning](https://mlsanity.com/paper/2509.13956/) (2025)
- [Safety-Aware Multi-Agent Learning for Dynamic Network Bridging](https://mlsanity.com/paper/2404.01551/) (2024)
- [Towards provable probabilistic safety for scalable embodied AI systems](https://mlsanity.com/paper/2506.05171/) (2025)
- [Efficient Safety Verification of Autonomous Vehicles with Neural Network Operator](https://mlsanity.com/paper/2512.04557/) (2025)
- [Conformal Reachability for Safe Control in Unknown Environments](https://mlsanity.com/paper/2602.03799/) (2026)
- [Towards Safe Reinforcement Learning with Reduced Conservativeness: A Case Study on Drone Flight Control](https://mlsanity.com/paper/2608.26852/) (2026)
- [Safe Multi-Robot Coordination via VLM-LLM Reasoning and Reachability Analysis](https://mlsanity.com/paper/2609.27816/) (2026)
- [Reachability-Informed Reinforcement Learning for Multi-Impulse Interplanetary Transfers](https://mlsanity.com/paper/2610.01344/) (2026)

## Abstract (arXiv)

While reinforcement learning produces very promising results for many applications, its main disadvantage is the lack of safety guarantees, which prevents its use in safety-critical systems. In this work, we address this issue by a safety shield for nonlinear continuous systems that solve reach-avoid tasks. Our safety shield prevents applying potentially unsafe actions from a reinforcement learning agent by projecting the proposed action to the closest safe action. This approach is called action projection and is implemented via mixed-integer optimization. The safety constraints for action projection are obtained by applying parameterized reachability analysis using polynomial zonotopes, which enables to accurately capture the nonlinear effects of the actions on the system. In contrast to other state-of-the-art approaches for action projection, our safety shield can efficiently handle input constraints and dynamic obstacles, eases incorporation of the spatial robot dimensions into the safety constraints, guarantees robust safety despite process noise and measurement errors, and is well suited for high-dimensional systems, as we demonstrate on several challenging benchmark systems.