🤖 AI Summary
This work addresses bandit optimization with β-smooth convex losses under adversarial perturbations subject to a global budget constraint, where perturbations are adaptively generated after the learner selects an action and their cumulative magnitude is bounded by a total budget. The authors generalize existing perturbation models—previously limited to linear losses—to general β-smooth convex losses, proposing an improved standard bandit algorithm. Through a refined analysis of the additional regret introduced by the perturbations, they establish the first explicit expected regret upper bound that depends directly on the perturbation budget. Notably, this bound naturally recovers the known regret guarantees for classical β-smooth bandit convex optimization in the absence of perturbations, thereby unifying the perturbed and unperturbed settings within a single theoretical framework.
📝 Abstract
We study adversarial bandit optimization in which the loss functions may be non-convex and non-smooth. In each round, the learner selects an action and observes only the loss incurred at that action. The loss consists of an underlying convex and $β$-smooth component and an adversarial perturbation that may be chosen after observing the learner's action. The perturbations are subject to a global budget controlling their cumulative magnitude over time.
This framework extends the globally budgeted, post-action perturbation model from underlying linear losses to general convex and $β$-smooth losses. For this broader class, we establish expected regret guarantees that explicitly characterize the effect of the perturbation budget.
To establish these guarantees, we modify a standard bandit optimization algorithm and develop an analysis that controls the additional regret caused by the perturbations. In the absence of perturbations, our results reduce to regret guarantees for the standard bandit convex optimization setting with $β$-smooth losses.