Near-Optimal Regret for the Safe Learning-based Control of the Constrained Linear Quadratic Regulator

• URL: http://arxiv.org/abs/2604.22158v1
• Date: Fri, 24 Apr 2026 02:14:29 GMT
• Title: Near-Optimal Regret for the Safe Learning-based Control of the Constrained Linear Quadratic Regulator
• Abstract summary: We study the problem of adaptive control of the linear quadratic regulator (LQR) with constraints that must be satisfied at every time step.<n>We contribute to this problem by showing $tildeO(sqrtT)$ regret and satisfaction of chance constraints.<n>Our proposed algorithm for this problem uses an SDP to select an optimistic policy, and then "scales back" this policy until it is verifiably-safe.
• Score: 2.4961059113841952
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of adaptive control of the stochastic linear quadratic regulator (LQR) with constraints that must be satisfied at every time step. Prior work on the multidimensional problem has shown $\tilde{O}(T^{2/3})$ regret and satisfaction of robust constraints, leaving open the question of whether $\tilde{O}(\sqrt{T})$ regret can be attained in the constrained LQR setting. We contribute to this problem by showing $\tilde{O}(\sqrt{T})$ regret and satisfaction of chance constraints. This type of constraints allow us to handle unbounded noise and also enable analytical techniques not directly applicable to robust constraints. Our proposed algorithm for this problem uses an SDP to select an optimistic policy, and then "scales back" this policy until it is verifiably-safe. Our theoretical analysis establishes regret and constraint guarantees via a key lemma that bounds the system covariance in terms of the chosen policy. This covariance-based analysis is in contrast with the cost-to-go based analysis that is typically used in adaptive LQR.

Related papers

• Optimistic Safety for Online Convex Optimization with Unknown Linear Constraints [31.526232903811533]
  We introduce an algorithm that we term Optimistically Safe OCO (OSOCO) and show it enjoys $tildeO(sqrtT)$ regret and no constraint violation. In the case of static linear constraints, this improves on the previous best known $tildeO(T2/3)$ regret under the same assumptions. In the case of time-varying constraints, our work supplements existing results that show $O(sqrtT)$ regret and $O(sqrtT)$ cumulative violation
  arXiv  Detail & Related papers  (2024-03-09T04:01:39Z)
• Almost Surely $\sqrt{T}$ Regret for Adaptive LQR [3.8499701725610285]
  We propose an adaptive LQR controller with almost surely $tilde mathcalO(sqrtT)$ regret upper bound.<n>The controller features a circuit-breaking mechanism, which circumvents potential safety breach and guarantees the convergence of the parameter estimate.
  arXiv  Detail & Related papers  (2023-01-13T13:35:43Z)
• A Unifying Framework for Online Optimization with Long-Term Constraints [62.35194099438855]
  We study online learning problems in which a decision maker has to take a sequence of decisions subject to $m$ long-term constraints. The goal is to maximize their total reward, while at the same time achieving small cumulative violation across the $T$ rounds. We present the first best-of-both-world type algorithm for this general class problems, with no-regret guarantees both in the case in which rewards and constraints are selected according to an unknown model, and in the case in which they are selected at each round by an adversary.
  arXiv  Detail & Related papers  (2022-09-15T16:59:19Z)
• Thompson Sampling Achieves $\tilde O(\sqrt{T})$ Regret in Linear Quadratic Control [85.22735611954694]
  We study the problem of adaptive control of stabilizable linear-quadratic regulators (LQRs) using Thompson Sampling (TS) We propose an efficient TS algorithm for the adaptive control of LQRs, TSAC, that attains $tilde O(sqrtT)$ regret, even for multidimensional systems.
  arXiv  Detail & Related papers  (2022-06-17T02:47:53Z)
• Dynamic Regret Minimization for Control of Non-stationary Linear Dynamical Systems [18.783925692307054]
  We present an algorithm that achieves the optimal dynamic regret of $tildemathcalO(sqrtST)$ where $S$ is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems.
  arXiv  Detail & Related papers  (2021-11-06T01:30:51Z)
• Safe Adaptive Learning-based Control for Constrained Linear Quadratic Regulators with Regret Guarantees [11.627320138064684]
  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. Our algorithm is implemented on a single trajectory and does not require system restarts.
  arXiv  Detail & Related papers  (2021-10-31T05:52:42Z)
• Finite-time System Identification and Adaptive Control in Autoregressive Exogenous Systems [79.67879934935661]
  We study the problem of system identification and adaptive control of unknown ARX systems. We provide finite-time learning guarantees for the ARX systems under both open-loop and closed-loop data collection.
  arXiv  Detail & Related papers  (2021-08-26T18:00:00Z)
• Efficient Optimistic Exploration in Linear-Quadratic Regulators via Lagrangian Relaxation [107.06364966905821]
  We study the exploration-exploitation dilemma in the linear quadratic regulator (LQR) setting. Inspired by the extended value iteration algorithm used in optimistic algorithms for finite MDPs, we propose to relax the optimistic optimization of ofulq. We show that an $epsilon$-optimistic controller can be computed efficiently by solving at most $Obig(log (1/epsilon)big)$ Riccati equations.
  arXiv  Detail & Related papers  (2020-07-13T16:30:47Z)
• Stochastic Bandits with Linear Constraints [69.757694218456]
  We study a constrained contextual linear bandit setting, where the goal of the agent is to produce a sequence of policies. We propose an upper-confidence bound algorithm for this problem, called optimistic pessimistic linear bandit (OPLB)
  arXiv  Detail & Related papers  (2020-06-17T22:32:19Z)
• Naive Exploration is Optimal for Online LQR [49.681825576239355]
  We show that the optimal regret scales as $widetildeTheta(sqrtd_mathbfu2 d_mathbfx T)$, where $T$ is the number of time steps, $d_mathbfu$ is the dimension of the input space, and $d_mathbfx$ is the dimension of the system state. Our lower bounds rule out the possibility of a $mathrmpoly(logT)$-regret algorithm, which had been
  arXiv  Detail & Related papers  (2020-01-27T03:44:54Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.