# Data-Driven Distributionally Robust Safety Verification Using Barrier Certificates and Conditional Mean Embeddings

arXiv: https://arxiv.org/abs/2403.10497 · HTML: https://mlsanity.com/paper/2403.10497/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 43 references that are in the map.
- First posted on arXiv 15 Mar 2024.

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

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [Safe Learning of Quadrotor Dynamics Using Barrier Certificates](https://mlsanity.com/paper/1710.05472/) (2017)
- [A Rigorous Theory of Conditional Mean Embeddings](https://mlsanity.com/paper/1912.00671/) (2019)
- [Learning Hybrid Control Barrier Functions from Data](https://mlsanity.com/paper/2011.04112/) (2020)
- [Formal Synthesis of Stochastic Systems via Control Barrier Certificates](https://mlsanity.com/paper/1905.04585/) (2019)
- [Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes](https://mlsanity.com/paper/2010.05818/) (2020)
- [Automated Verification and Synthesis of Stochastic Hybrid Systems: A Survey](https://mlsanity.com/paper/2101.07491/) (2021)
- [Formal Controller Synthesis for Continuous-Space MDPs via Model-Free Reinforcement Learning](https://mlsanity.com/paper/2003.00712/) (2020)
- [Data-driven verification and synthesis of stochastic systems via barrier certificates](https://mlsanity.com/paper/2111.10330/) (2021)
- [Formal Synthesis of Controllers for Safety-Critical Autonomous Systems: Developments and Challenges](https://mlsanity.com/paper/2402.13075/) (2024)
- [Formal Policy Synthesis for Continuous-Space Systems via Reinforcement Learning](https://mlsanity.com/paper/2005.01319/) (2020)
- [Data-Driven Abstraction-Based Control Synthesis](https://mlsanity.com/paper/2206.08069/) (2022)
- [Maximum Mean Discrepancy Distributionally Robust Nonlinear Chance-Constrained Optimization with Finite-Sample Guarantee](https://mlsanity.com/paper/2204.11564/) (2022)
- [Robust Control Barrier Functions for Nonlinear Control Systems with Uncertainty: A Duality-based Approach](https://mlsanity.com/paper/2208.05955/) (2022)
- [Distributionally Robust Optimal and Safe Control of Stochastic Systems via Kernel Conditional Mean Embedding](https://mlsanity.com/paper/2304.00644/) (2023)
- [Multi-objective Compositions for Collision-Free Connectivity Maintenance in Teams of Mobile Robots](https://mlsanity.com/paper/1608.06887/) (2016)
- [Bayesian Formal Synthesis of Unknown Systems via Robust Simulation Relations](https://mlsanity.com/paper/2304.07428/) (2023)
- [Unmatched Control Barrier Functions: Certainty Equivalence Adaptive Safety](https://mlsanity.com/paper/2207.13873/) (2022)
- [Formal Verification of Unknown Stochastic Systems via Non-parametric Estimation](https://mlsanity.com/paper/2403.05350/) (2024)
- [Safety-Aware Learning-Based Control of Systems with Uncertainty Dependent Constraints (extended version)](https://mlsanity.com/paper/2210.01374/) (2022)

## Built on by (most cited first, 9 of 9)

- [Safe Reach Set Computation via Neural Barrier Certificates](https://mlsanity.com/paper/2404.18813/) (2024)
- [Data-Driven Abstractions via Binary-Tree Gaussian Processes for Formal Verification](https://mlsanity.com/paper/2407.21029/) (2024)
- [LUCID: Learning-Enabled Uncertainty-Aware Certification of Stochastic Dynamical Systems](https://mlsanity.com/paper/2512.11750/) (2025)
- [Data-Driven Safety Verification using Barrier Certificates and Matrix Zonotopes](https://mlsanity.com/paper/2504.01007/) (2025)
- [Average Reward Reinforcement Learning for Omega-Regular and Mean-Payoff Objectives](https://mlsanity.com/paper/2505.15693/) (2025)
- [BarrierBench: Evaluating Large Language Models for Safety Verification in Dynamical Systems](https://mlsanity.com/paper/2511.09363/) (2025)
- [Kernel-Based Learning of Safety Barriers](https://mlsanity.com/paper/2601.12002/) (2026)
- [Safety Certification is Classification](https://mlsanity.com/paper/2605.06087/) (2026)
- [Kernel-Based Safe Exploration in Deep Reinforcement Learning](https://mlsanity.com/paper/2605.22207/) (2026)

## Abstract (arXiv)

Algorithmic verification of realistic systems to satisfy safety and other temporal requirements has suffered from poor scalability of the employed formal approaches. To design systems with rigorous guarantees, many approaches still rely on exact models of the underlying systems. Since this assumption can rarely be met in practice, models have to be inferred from measurement data or are bypassed completely. Whilst former usually requires the model structure to be known a-priori and immense amounts of data to be available, latter gives rise to a plethora of restrictive mathematical assumptions about the unknown dynamics. In a pursuit of developing scalable formal verification algorithms without shifting the problem to unrealistic assumptions, we employ the concept of barrier certificates, which can guarantee safety of the system, and learn the certificate directly from a compact set of system trajectories. We use conditional mean embeddings to embed data from the system into a reproducing kernel Hilbert space (RKHS) and construct an RKHS ambiguity set that can be inflated to robustify the result w.r.t. a set of plausible transition kernels. We show how to solve the resulting program efficiently using sum-of-squares optimization and a Gaussian process envelope. Our approach lifts the need for restrictive assumptions on the system dynamics and uncertainty, and suggests an improvement in the sample complexity of verifying the safety of a system on a tested case study compared to a state-of-the-art approach.