# Safe Learning of Uncertain Environments

arXiv: https://arxiv.org/abs/2103.01413 · HTML: https://mlsanity.com/paper/2103.01413/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 33 references that are in the map.
- First posted on arXiv 2 Mar 2021.

## Builds on (most cited first, 14 of 14)

- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [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)
- [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)
- [Convergent Policy Optimization for Safe Reinforcement Learning](https://mlsanity.com/paper/1910.12156/) (2019)
- [Safety-Constrained Reinforcement Learning for MDPs](https://mlsanity.com/paper/1510.05880/) (2015)
- [MAMPS: Safe Multi-Agent Reinforcement Learning via Model Predictive Shielding](https://mlsanity.com/paper/1910.12639/) (2019)
- [Safe Multi-Agent Interaction through Robust Control Barrier Functions with Learned Uncertainties](https://mlsanity.com/paper/2004.05273/) (2020)
- [Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes](https://mlsanity.com/paper/2010.05818/) (2020)
- [Input-to-State Safety With Control Barrier Functions](https://mlsanity.com/paper/1803.03035/) (2018)
- [Bayesian Safe Learning and Control with Sum-of-Squares Analysis and Polynomial Kernels](https://mlsanity.com/paper/2004.00662/) (2020)
- [Collision Avoidance Based on Robust Lexicographic Task Assignment](https://mlsanity.com/paper/2002.11247/) (2020)

## Abstract (arXiv)

In many learning based control methodologies, learning the unknown dynamic model precedes the control phase, while the aim is to control the system such that it remains in some safe region of the state space. In this work, our aim is to guarantee safety while learning and control proceed simultaneously. Specifically, we consider the problem of safe learning in nonlinear control-affine systems subject to unknown additive uncertainty. We first model the uncertainty as a Gaussian noise and use state measurements to learn its mean and covariance. We provide rigorous time-varying bounds on the mean and covariance of the uncertainty and employ them to modify the control input via an optimization program with potentially time-varying safety constraints. We show that with an arbitrarily large probability we can guarantee that the state will remain in the safe set, while learning and control are carried out simultaneously, provided that a feasible solution exists for the optimization problem. We provide a secondary formulation of this optimization that is computationally more efficient. This is based on tightening the safety constraints to counter the uncertainty about the learned mean and covariance. The magnitude of the tightening can be decreased as our confidence in the learned mean and covariance increases (i.e., as we gather more measurements about the environment). Extensions of the method are provided for non-Gaussian process noise with unknown mean and covariance as well as Gaussian uncertainties with state-dependent mean and covariance to accommodate more general environments.