Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian (LQG) control are foundational and extensively researched problems in optimal control. We investigate LQR and LQG problems with semi-adversarial perturbations and time-varying adversarial bandit loss functions. The best-known sublinear regret algorithm of~\cite{gradu2020non} has a $T^{\frac{3}{4}}$ time horizon dependence, and its authors posed an open question about whether a tight rate of $\sqrt{T}$ could be achieved. We answer in the affirmative, giving an algorithm for bandit LQR and LQG which attains optimal regret (up to logarithmic factors) for both known and unknown systems. A central component of our method is a new scheme for bandit convex optimization with memory, which is of independent interest.

翻译：线性二次型调节器（LQR）与线性二次型高斯（LQG）控制是最优控制领域基础且被广泛研究的问题。我们研究了具有半对抗性扰动和时变对抗性赌博机损失函数的LQR与LQG问题。已知次线性遗憾最优算法[1]的时间复杂度依赖为$T^{\frac{3}{4}}$，其作者提出了能否实现$\sqrt{T}$紧致收敛速率的开放问题。我们给出肯定答案，提出了一种针对LQR与LQG的赌博机算法，能够在已知系统和未知系统中均达到最优遗憾（对数因子范围内）。该方法的核心组件是一种全新的带记忆的凸赌博机优化方案，该方案本身具有独立研究价值。