Optimal Rates for Bandit Nonstochastic Control

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 timevarying adversarial bandit loss functions. The best-known sublinear regret algorithm of Gradu et al. [2020] has a T 3 4 time horizon dependence, and the authors posed an open question about whether a tight rate of √ 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. 1 The LQR/LQG dynamics can be generalized to time-varying linear dynamical systems. Here we restrict ourselves to linear time-invariant systems for simplicity. 37th Conference on Neural Information Processing Systems (NeurIPS 2023).