# Almost Surely √T Regret for Adaptive LQR

arXiv: https://arxiv.org/abs/2301.05537 · HTML: https://mlsanity.com/paper/2301.05537/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 41 references that are in the map.
- First posted on arXiv 13 Jan 2023.

## Builds on (most cited first, 10 of 10)

- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Learning Linear-Quadratic Regulators Efficiently with only √T Regret](https://mlsanity.com/paper/1902.06223/) (2019)
- [On Adaptive Linear-Quadratic Regulators](https://mlsanity.com/paper/1806.10749/) (2018)
- [Input Perturbations for Adaptive Control and Learning](https://mlsanity.com/paper/1811.04258/) (2018)
- [On the Sample Complexity of Stabilizing LTI Systems on a Single Trajectory](https://mlsanity.com/paper/2202.07187/) (2022)
- [Exact Asymptotics for Linear Quadratic Adaptive Control](https://mlsanity.com/paper/2011.01364/) (2020)
- [An Online Approach to Physical Watermark Design](https://mlsanity.com/paper/1911.01868/) (2019)
- [Safe and Efficient Switching Mechanism Design for Uncertified Linear Controller](https://mlsanity.com/paper/2210.14595/) (2022)

## Abstract (arXiv)

The Linear-Quadratic Regulation (LQR) problem with unknown system parameters has been widely studied, but it has remained unclear whether tilde O(√T) regret, which is the best known dependence on time, can be achieved almost surely. In this paper, we propose an adaptive LQR controller with almost surely tilde O(√T) regret upper bound. The controller features a circuit-breaking mechanism, which circumvents potential safety breach and guarantees the convergence of the system parameter estimate, but is shown to be triggered only finitely often and hence has negligible effect on the asymptotic performance of the controller. The proposed controller is also validated via simulation on Tennessee Eastman Process~(TEP), a commonly used industrial process example.