The first algorithm for the Linear Quadratic (LQ) control problem with an unknown system model, featuring a regret of $\mathcal{O}(\sqrt{T})$, was introduced by Abbasi-Yadkori and Szepesv\'ari (2011). Recognizing the computational complexity of this algorithm, subsequent efforts (see Cohen et al. (2019), Mania et al. (2019), Faradonbeh et al. (2020a), and Kargin et al.(2022)) have been dedicated to proposing algorithms that are computationally tractable while preserving this order of regret. Although successful, the existing works in the literature lack a fully adaptive exploration-exploitation trade-off adjustment and require a user-defined value, which can lead to overall regret bound growth with some factors. In this work, noticing this gap, we propose the first fully adaptive algorithm that controls the number of policy updates (i.e., tunes the exploration-exploitation trade-off) and optimizes the upper-bound of regret adaptively. Our proposed algorithm builds on the SDP-based approach of Cohen et al. (2019) and relaxes its need for a horizon-dependant warm-up phase by appropriately tuning the regularization parameter and adding an adaptive input perturbation. We further show that through careful exploration-exploitation trade-off adjustment there is no need to commit to the widely-used notion of strong sequential stability, which is restrictive and can introduce complexities in initialization.

提出了一种解决具有未知系统模型的线性二次（LQ）控制问题的算法，其遗憾为O(√T)，并在此基础上提出了首个完全自适应的算法，同时控制策略更新次数和自适应地优化遗憾上限，避免了计算复杂性问题。

线性二次系统控制的全自适应保证后悔算法