Optimal Rates for Bandit Nonstochastic Control
This solves a foundational problem in optimal control for researchers and practitioners, offering optimal regret guarantees in adversarial bandit settings.
The paper addresses the open question of achieving optimal regret rates in bandit Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian (LQG) control with adversarial perturbations, providing an algorithm that attains a regret of √T (up to logarithmic factors), improving upon the previous best of T^(3/4).
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.