# Sample-efficient Safe Learning for Online Nonlinear Control with Control Barrier Functions

arXiv: https://arxiv.org/abs/2207.14419 · HTML: https://mlsanity.com/paper/2207.14419/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 42 references that are in the map.
- First posted on arXiv 29 Jul 2022.

## Builds on (most cited first, 28 of 28)

- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Benchmarking Deep Reinforcement Learning for Continuous Control](https://mlsanity.com/paper/1604.06778/) (2016)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [An Information-Theoretic Analysis of Thompson Sampling](https://mlsanity.com/paper/1403.5341/) (2014)
- [A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems](https://mlsanity.com/paper/1705.01292/) (2017)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Information Theoretic Regret Bounds for Online Nonlinear Control](https://mlsanity.com/paper/2006.12466/) (2020)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [Safe Exploration in Finite Markov Decision Processes with Gaussian Processes](https://mlsanity.com/paper/1606.04753/) (2016)
- [Learning for Safety-Critical Control with Control Barrier Functions](https://mlsanity.com/paper/1912.10099/) (2019)
- [Reinforcement Learning for Safety-Critical Control under Model Uncertainty, using Control Lyapunov Functions and Control Barrier Functions](https://mlsanity.com/paper/2004.07584/) (2020)
- [Safety-Critical Model Predictive Control with Discrete-Time Control Barrier Function](https://mlsanity.com/paper/2007.11718/) (2020)
- [Safe Learning of Quadrotor Dynamics Using Barrier Certificates](https://mlsanity.com/paper/1710.05472/) (2017)
- [An Online Learning Approach to Model Predictive Control](https://mlsanity.com/paper/1902.08967/) (2019)
- [Adaptive Safety with Control Barrier Functions](https://mlsanity.com/paper/1910.00555/) (2019)
- [The Robotarium: A remotely accessible swarm robotics research testbed](https://mlsanity.com/paper/1609.04730/) (2016)
- [Safe Multi-Agent Interaction through Robust Control Barrier Functions with Learned Uncertainties](https://mlsanity.com/paper/2004.05273/) (2020)
- [Barrier-Certified Adaptive Reinforcement Learning with Applications to Brushbot Navigation](https://mlsanity.com/paper/1801.09627/) (2018)
- [Multi-Robot Collision Avoidance under Uncertainty with Probabilistic Safety Barrier Certificates](https://mlsanity.com/paper/1912.09957/) (2019)
- [Probabilistic Safety-Assured Adaptive Merging Control for Autonomous Vehicles](https://mlsanity.com/paper/2104.14159/) (2021)
- [Probabilistic Safety Constraints for Learned High Relative Degree System Dynamics](https://mlsanity.com/paper/1912.10116/) (2019)

## Built on by (most cited first, 3 of 3)

- [GRID: A Platform for General Robot Intelligence Development](https://mlsanity.com/paper/2310.00887/) (2023)
- [Computationally and Sample Efficient Safe Reinforcement Learning Using Adaptive Conformal Prediction](https://mlsanity.com/paper/2503.17678/) (2025)
- [Certifiable Safe Model-Based Reinforcement Learning with Control-Affine Dynamics Approximation](https://mlsanity.com/paper/2607.16501/) (2026)

## Abstract (arXiv)

Reinforcement Learning (RL) and continuous nonlinear control have been successfully deployed in multiple domains of complicated sequential decision-making tasks. However, given the exploration nature of the learning process and the presence of model uncertainty, it is challenging to apply them to safety-critical control tasks due to the lack of safety guarantee. On the other hand, while combining control-theoretical approaches with learning algorithms has shown promise in safe RL applications, the sample efficiency of safe data collection process for control is not well addressed. In this paper, we propose a provably sample efficient episodic safe learning framework for online control tasks that leverages safe exploration and exploitation in an unknown, nonlinear dynamical system. In particular, the framework 1) extends control barrier functions (CBFs) in a stochastic setting to achieve provable high-probability safety under uncertainty during model learning and 2) integrates an optimism-based exploration strategy to efficiently guide the safe exploration process with learned dynamics for near optimal control performance. We provide formal analysis on the episodic regret bound against the optimal controller and probabilistic safety with theoretical guarantees. Simulation results are provided to demonstrate the effectiveness and efficiency of the proposed algorithm.