# Nonasymptotic Regret Analysis of Adaptive Linear Quadratic Control with Model Misspecification arXiv: https://arxiv.org/abs/2401.00073 · HTML: https://mlsanity.com/paper/2401.00073/ ## Key facts - Cited by 6 papers in the map (data through 14 Oct 2026). - Builds on 26 of its 39 references that are in the map. - First posted on arXiv 29 Dec 2023. ## Builds on (most cited first, 26 of 26) - [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018) - [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020) - [RoboNet: Large-Scale Multi-Robot Learning](https://mlsanity.com/paper/1910.11215/) (2019) - [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020) - [On the Theory of Transfer Learning: The Importance of Task Diversity](https://mlsanity.com/paper/2006.11650/) (2020) - [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020) - [Information Theoretic Regret Bounds for Online Nonlinear Control](https://mlsanity.com/paper/2006.12466/) (2020) - [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 Linear-Quadratic Regulators Efficiently with only √T Regret](https://mlsanity.com/paper/1902.06223/) (2019) - [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022) - [The Nonstochastic Control Problem](https://mlsanity.com/paper/1911.12178/) (2019) - [Logarithmic Regret Bound in Partially Observable Linear Dynamical Systems](https://mlsanity.com/paper/2003.11227/) (2020) - [Logarithmic Regret for Learning Linear Quadratic Regulators Efficiently](https://mlsanity.com/paper/2002.08095/) (2020) - [Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator](https://mlsanity.com/paper/2003.07937/) (2020) - [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023) - [Multi-Task Imitation Learning for Linear Dynamical Systems](https://mlsanity.com/paper/2212.00186/) (2022) - [Model-free Learning with Heterogeneous Dynamical Systems: A Federated LQR Approach](https://mlsanity.com/paper/2308.11743/) (2023) - [Reinforcement Learning with Fast Stabilization in Linear Dynamical Systems](https://mlsanity.com/paper/2007.12291/) (2020) - [Joint Learning of Linear Time-Invariant Dynamical Systems](https://mlsanity.com/paper/2112.10955/) (2021) - [Minimal Expected Regret in Linear Quadratic Control](https://mlsanity.com/paper/2109.14429/) (2021) - [Sample-Efficient Linear Representation Learning from Non-IID Non-Isotropic Data](https://mlsanity.com/paper/2308.04428/) (2023) - [Regret Lower Bounds for Learning Linear Quadratic Gaussian Systems](https://mlsanity.com/paper/2201.01680/) (2022) - [Learning to Control Linear Systems can be Hard](https://mlsanity.com/paper/2205.14035/) (2022) - [Minimax Adaptive Control for a Finite Set of Linear Systems](https://mlsanity.com/paper/2011.10814/) (2020) - [An Online Learning Analysis of Minimax Adaptive Control](https://mlsanity.com/paper/2307.07268/) (2023) ## Built on by (most cited first, 6 of 6) - [Coreset-Based Task Selection for Sample-Efficient Meta-Reinforcement Learning](https://mlsanity.com/paper/2502.02332/) (2025) - [Regret Analysis of Multi-task Representation Learning for Linear-Quadratic Adaptive Control](https://mlsanity.com/paper/2407.05781/) (2024) - [Learning to Stabilize Unknown LTI Systems on a Single Trajectory under Stochastic Noise](https://mlsanity.com/paper/2406.00234/) (2024) - [Adversarially Robust Multitask Adaptive Control](https://mlsanity.com/paper/2511.05444/) (2025) - [Optimistic Online LQR via Intrinsic Rewards](https://mlsanity.com/paper/2603.28938/) (2026) - [Logarithmic Regret for Nonlinear Control](https://mlsanity.com/paper/2501.10261/) (2025) ## Abstract (arXiv) The strategy of pre-training a large model on a diverse dataset, then fine-tuning for a particular application has yielded impressive results in computer vision, natural language processing, and robotic control. This strategy has vast potential in adaptive control, where it is necessary to rapidly adapt to changing conditions with limited data. Toward concretely understanding the benefit of pre-training for adaptive control, we study the adaptive linear quadratic control problem in the setting where the learner has prior knowledge of a collection of basis matrices for the dynamics. This basis is misspecified in the sense that it cannot perfectly represent the dynamics of the underlying data generating process. We propose an algorithm that uses this prior knowledge, and prove upper bounds on the expected regret after T interactions with the system. In the regime where T is small, the upper bounds are dominated by a term that scales with either textttpoly(log T) or √T, depending on the prior knowledge available to the learner. When T is large, the regret is dominated by a term that grows with δ T, where δ quantifies the level of misspecification. This linear term arises due to the inability to perfectly estimate the underlying dynamics using the misspecified basis, and is therefore unavoidable unless the basis matrices are also adapted online. However, it only dominates for large T, after the sublinear terms arising due to the error in estimating the weights for the basis matrices become negligible. We provide simulations that validate our analysis. Our simulations also show that offline data from a collection of related systems can be used as part of a pre-training stage to estimate a misspecified dynamics basis, which is in turn used by our adaptive controller.