# Safe Adaptive Learning-based Control for Constrained Linear Quadratic Regulators with Regret Guarantees

arXiv: https://arxiv.org/abs/2111.00411 · HTML: https://mlsanity.com/paper/2111.00411/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 56 references that are in the map.
- First posted on arXiv 31 Oct 2021.

## Builds on (most cited first, 23 of 23)

- [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)
- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems](https://mlsanity.com/paper/1705.01292/) (2017)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Risk-sensitive Reinforcement Learning](https://mlsanity.com/paper/1311.2097/) (2013)
- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Online Control with Adversarial Disturbances](https://mlsanity.com/paper/1902.08721/) (2019)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Improper Learning for Non-Stochastic Control](https://mlsanity.com/paper/2001.09254/) (2020)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Logarithmic Regret for Online Control](https://mlsanity.com/paper/1909.05062/) (2019)
- [Linear model predictive safety certification for learning-based control](https://mlsanity.com/paper/1803.08552/) (2018)
- [Safely Learning to Control the Constrained Linear Quadratic Regulator](https://mlsanity.com/paper/1809.10121/) (2018)
- [Online Optimal Control with Affine Constraints](https://mlsanity.com/paper/2010.04891/) (2020)
- [LMI Properties and Applications in Systems, Stability, and Control Theory](https://mlsanity.com/paper/1903.08599/) (2019)
- [Geometric Exploration for Online Control](https://mlsanity.com/paper/2010.13178/) (2020)
- [Linear robust adaptive model predictive control: Computational complexity and conservatism -- extended version](https://mlsanity.com/paper/1909.01813/) (2019)
- [Robust Adaptive Model Predictive Control: Performance and Parameter Estimation](https://mlsanity.com/paper/1911.00865/) (2019)
- [Adaptive MPC under Time Varying Uncertainty: Robust and Stochastic](https://mlsanity.com/paper/1909.13473/) (2019)
- [Adaptive Model Predictive Control for A Class of Constrained Linear Systems with Parametric Uncertainties](https://mlsanity.com/paper/1901.03930/) (2019)

## Built on by (most cited first, 11 of 11)

- [Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis](https://mlsanity.com/paper/2309.14648/) (2023)
- [Non-asymptotic System Identification for Linear Systems with Nonlinear Policies](https://mlsanity.com/paper/2306.10369/) (2023)
- [Learning-Based Adaptive Control for Stochastic Linear Systems with Input Constraints](https://mlsanity.com/paper/2209.07040/) (2022)
- [Stability Bounds for Learning-Based Adaptive Control of Discrete-Time Multi-Dimensional Stochastic Linear Systems with Input Constraints](https://mlsanity.com/paper/2304.00569/) (2023)
- [Exact Recovery for System Identification with More Corrupt Data than Clean Data](https://mlsanity.com/paper/2305.10506/) (2023)
- [Learning Decentralized Linear Quadratic Regulators with √T Regret](https://mlsanity.com/paper/2210.08886/) (2022)
- [Anytime-Competitive Reinforcement Learning with Policy Prior](https://mlsanity.com/paper/2311.01568/) (2023)
- [Rate-Optimal Regret for the Safe Learning-based Control of the Constrained Linear Quadratic Regulator](https://mlsanity.com/paper/2604.22158/) (2026)
- [Learning Safety Filters for Unknown Discrete-Time Linear Systems](https://mlsanity.com/paper/2111.00631/) (2021)
- [Rate-matching the regret lower-bound in the linear quadratic regulator with unknown dynamics](https://mlsanity.com/paper/2202.05799/) (2022)
- [When Can Safe Controllers Adapt? Information before Commitment](https://mlsanity.com/paper/2607.16895/) (2026)

## Abstract (arXiv)

We study the adaptive control of an unknown linear system with a quadratic cost function subject to safety constraints on both the states and actions. The challenges of this problem arise from the tension among safety, exploration, performance, and computation. To address these challenges, we propose a polynomial-time algorithm that guarantees feasibility and constraint satisfaction with high probability under proper conditions. Our algorithm is implemented on a single trajectory and does not require system restarts. Further, we analyze the regret of our learning algorithm compared to the optimal safe linear controller with known model information. The proposed algorithm can achieve a tilde O(T2/3) regret, where T is the number of stages and tilde O(·) absorbs some logarithmic terms of T.