🤖 AI Summary
This work addresses adversarial bandit optimization under a global perturbation budget, where in each round the loss consists of a linear function plus an action-dependent perturbation term, with the total perturbation constrained globally. Within this non-convex and non-smooth setting, the paper establishes—for the first time—both expected and high-probability regret upper bounds under a global perturbation budget, improving upon the classical high-probability regret bound in the unperturbed case. Additionally, it provides a matching lower bound on the expected regret. The analysis combines techniques from adversarial bandits, perturbation modeling, and refined probabilistic arguments, offering rigorous theoretical guarantees for online decision-making in perturbed environments.
📝 Abstract
We study a class of adversarial bandit optimization problems in which the loss functions may be non-convex and non-smooth. In each round, the learner observes a loss that consists of an underlying linear component together with an additional perturbation applied after the learner selects an action. The perturbations are measured relative to the linear losses and are constrained by a global budget that bounds their cumulative magnitude over time. Under this model, we establish both expected and high-probability regret guarantees. As a special case of our analysis, we recover an improved high-probability regret bound for classical bandit linear optimization, which corresponds to the setting without perturbations. We further complement our upper bounds by proving a lower bound on the expected regret.