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The strategy of pre-training a large model on a diverse dataset, thenfine-tuning for a particular application has yielded impressive results incomputer vision, natural language processing, and robotic control. Thisstrategy has vast potential in adaptive control, where it is necessary torapidly adapt to changing conditions with limited data. Toward concretelyunderstanding the benefit of pre-training for adaptive control, we study theadaptive linear quadratic control problem in the setting where the learner hasprior knowledge of a collection of basis matrices for the dynamics. This basisis misspecified in the sense that it cannot perfectly represent the dynamics ofthe underlying data generating process. We propose an algorithm that uses thisprior knowledge, and prove upper bounds on the expected regret after $T$interactions with the system. In the regime where $T$ is small, the upperbounds are dominated by a term scales with either $\texttt{poly}(\log T)$ or$\sqrt{T}$, depending on the prior knowledge available to the learner. When $T$is large, the regret is dominated by a term that grows with $\delta T$, where$\delta$ quantifies the level of misspecification. This linear term arises dueto the inability to perfectly estimate the underlying dynamics using themisspecified basis, and is therefore unavoidable unless the basis matrices arealso adapted online. However, it only dominates for large $T$, after thesublinear terms arising due to the error in estimating the weights for thebasis matrices become negligible. We provide simulations that validate ouranalysis. Our simulations also show that offline data from a collection ofrelated systems can be used as part of a pre-training stage to estimate amisspecified dynamics basis, which is in turn used by our adaptive controller.

Nonasymptotic Regret Analysis of Adaptive Linear Quadratic Control with Model Misspecification