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 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 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.

对于自适应控制中的预先训练策略，我们研究了一个自适应线性二次控制问题，在这个设置中学习者具有动力学的一组基矩阵的先验知识。我们提出了一种使用这个先验知识的算法，并对通过与系统进行T次交互后的预期遗憾给出了上界。

自适应线性二次控制的非渐进遗憾分析及模型误差