# Probabilistic Reach-Avoid for Bayesian Neural Networks

arXiv: https://arxiv.org/abs/2310.01951 · HTML: https://mlsanity.com/paper/2310.01951/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 66 references that are in the map.
- First posted on arXiv 3 Oct 2023.

## Builds on (most cited first, 30 of 35)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model](https://mlsanity.com/paper/1911.08265/) (2019)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Safe Reinforcement Learning via Shielding](https://mlsanity.com/paper/1708.08611/) (2017)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [Safe Controller Optimization for Quadrotors with Gaussian Processes](https://mlsanity.com/paper/1509.01066/) (2015)
- [Adv-BNN: Improved Adversarial Defense through Robust Bayesian Neural Network](https://mlsanity.com/paper/1810.01279/) (2018)
- [Uncertainty Quantification with Statistical Guarantees in End-to-End Autonomous Driving Control](https://mlsanity.com/paper/1909.09884/) (2019)
- [Learning and Policy Search in Stochastic Dynamical Systems with Bayesian Neural Networks](https://mlsanity.com/paper/1605.07127/) (2016)
- [Robustness of Bayesian Neural Networks to Gradient-Based Attacks](https://mlsanity.com/paper/2002.04359/) (2020)
- [Probabilistic Safety for Bayesian Neural Networks](https://mlsanity.com/paper/2004.10281/) (2020)
- [Formal Verification of Neural Network Controlled Autonomous Systems](https://mlsanity.com/paper/1810.13072/) (2018)
- [Certified Reinforcement Learning with Logic Guidance](https://mlsanity.com/paper/1902.00778/) (2019)
- [Statistical Guarantees for the Robustness of Bayesian Neural Networks](https://mlsanity.com/paper/1903.01980/) (2019)
- [Neural Simplex Architecture](https://mlsanity.com/paper/1908.00528/) (2019)
- [Reinforcement Learning with Probabilistic Guarantees for Autonomous Driving](https://mlsanity.com/paper/1904.07189/) (2019)
- [Bayesian Inference with Certifiable Adversarial Robustness](https://mlsanity.com/paper/2102.05289/) (2021)
- [Robust Explanation Constraints for Neural Networks](https://mlsanity.com/paper/2212.08507/) (2022)
- [Robustness Guarantees for Bayesian Inference with Gaussian Processes](https://mlsanity.com/paper/1809.06452/) (2018)
- [Safe Control with Neural Network Dynamic Models](https://mlsanity.com/paper/2110.01110/) (2021)
- [Make Sure You're Unsure: A Framework for Verifying Probabilistic Specifications](https://mlsanity.com/paper/2102.09479/) (2021)
- [Verifiably Safe Off-Model Reinforcement Learning](https://mlsanity.com/paper/1902.05632/) (2019)
- [Probabilistic Guarantees for Safe Deep Reinforcement Learning](https://mlsanity.com/paper/2005.07073/) (2020)
- [Infinite Time Horizon Safety of Bayesian Neural Networks](https://mlsanity.com/paper/2111.03165/) (2021)
- [Efficiency through Uncertainty: Scalable Formal Synthesis for Stochastic Hybrid Systems](https://mlsanity.com/paper/1901.01576/) (2019)
- [Adversarial Robustness Guarantees for Classification with Gaussian Processes](https://mlsanity.com/paper/1905.11876/) (2019)

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

- [Stochastic Weight Sharing for Bayesian Neural Networks](https://mlsanity.com/paper/2505.17856/) (2025)
- [Risk-Averse Certification of Bayesian Neural Networks](https://mlsanity.com/paper/2411.19729/) (2024)
- [Probabilistic Verification of Neural Networks via Efficient Probabilistic Hull Generation](https://mlsanity.com/paper/2604.21556/) (2026)
- [Hybrid Robustness Verification for Spatio-Temporal Neural Networks](https://mlsanity.com/paper/2606.09746/) (2026)
- [Certification-Enhanced Generalization Bounds](https://mlsanity.com/paper/2610.06238/) (2026)

## Abstract (arXiv)

Model-based reinforcement learning seeks to simultaneously learn the dynamics of an unknown stochastic environment and synthesise an optimal policy for acting in it. Ensuring the safety and robustness of sequential decisions made through a policy in such an environment is a key challenge for policies intended for safety-critical scenarios. In this work, we investigate two complementary problems: first, computing reach-avoid probabilities for iterative predictions made with dynamical models, with dynamics described by Bayesian neural network (BNN); second, synthesising control policies that are optimal with respect to a given reach-avoid specification (reaching a "target" state, while avoiding a set of "unsafe" states) and a learned BNN model. Our solution leverages interval propagation and backward recursion techniques to compute lower bounds for the probability that a policy's sequence of actions leads to satisfying the reach-avoid specification. Such computed lower bounds provide safety certification for the given policy and BNN model. We then introduce control synthesis algorithms to derive policies maximizing said lower bounds on the safety probability. We demonstrate the effectiveness of our method on a series of control benchmarks characterized by learned BNN dynamics models. On our most challenging benchmark, compared to purely data-driven policies the optimal synthesis algorithm is able to provide more than a four-fold increase in the number of certifiable states and more than a three-fold increase in the average guaranteed reach-avoid probability.