Lipschitz bandit is a variant of stochastic bandits that deals with a continuous arm set defined on a metric space, where the reward function is subject to a Lipschitz constraint. In this paper, we introduce a new problem of Lipschitz bandits in the presence of adversarial corruptions where an adaptive adversary corrupts the stochastic rewards up to a total budget $C$. The budget is measured by the sum of corruption levels across the time horizon $T$. We consider both weak and strong adversaries, where the weak adversary is unaware of the current action before the attack, while the strong one can observe it. Our work presents the first line of robust Lipschitz bandit algorithms that can achieve sub-linear regret under both types of adversary, even when the total budget of corruption $C$ is unrevealed to the agent. We provide a lower bound under each type of adversary, and show that our algorithm is optimal under the strong case. Finally, we conduct experiments to illustrate the effectiveness of our algorithms against two classic kinds of attacks.

本文提出了第一行对抗下的强、弱 Lipschitz bandit 算法，能在无总损坏预算的情况下，在两种类型的对手下实现次线性后悔，但强对手情况下表现最优，并通过对两种经典攻击的试验证明了该算法的有效性。

鲁棒利普希茨贝叶斯在对抗扰动下的决策