Adversarial Bandit Optimization with Globally Bounded Perturbations to Linear Losses

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.

Adversarial bandit optimization for approximately linear functions

We consider a bandit optimization problem for nonconvex and non-smooth functions, where in each trial the loss function is the sum of a linear function and a small but arbitrary perturbation chosen after observing the player's choice. We give both expected and high probability regret bounds for the problem. Our result also implies an improved high-probability regret bound for the bandit linear optimization, a special case with no perturbation. We also give a lower bound on the expected regret.