Tight Rates for Bandit Control Beyond Quadratics

Unlike classical control theory, such as Linear Quadratic Control (LQC), real-world control problems are highly complex. These problems often involve adversarial perturbations, bandit feedback models, and non-quadratic, adversarially chosen cost functions. A fundamental yet unresolved question is whether optimal regret can be achieved for these general control problems. The standard approach to addressing this problem involves a reduction to bandit convex optimization with memory. In the bandit setting, constructing a gradient estimator with low variance is challenging due to the memory structure and non-quadratic loss functions.In this paper, we provide an affirmative answer to this question. Our main contribution is an algorithm that achieves an $\tilde{O}(\sqrt{T})$ optimal regret for bandit non-stochastic control with strongly-convex and smooth cost functions in the presence of adversarial perturbations, improving the previously known $\tilde{O}(T^{2/3})$ regret bound from \citep{cassel2020bandit}. Our algorithm overcomes the memory issue by reducing the problem to Bandit Convex Optimization (BCO) without memory and addresses general strongly-convex costs using recent advancements in BCO from \citep{suggala2024second}. Along the way, we develop an improved algorithm for BCO with memory, which may be of independent interest.