对抗环境下的自适应离散化：Lipschitz赌博机、动态定价与拍卖调优
Adaptive Discretization against an Adversary: Lipschitz bandits, Dynamic Pricing, and Auction Tuning
AI总结：
针对对抗收益下Lipschitz赌博机的自适应离散化研究空白，提出Adversarial Zooming算法，推导了兼具最坏情况最优性和实例依赖性的悔界，并将其拓展应用于动态定价、拍卖调优等非平滑对抗场景。
摘要：
Lipschitz赌博机是多臂赌博机的一个重要研究方向，针对[0,1]区间这类具有结构的大型动作空间展开研究，这类问题中相似动作的收益具有相似性。该领域的核心思路是对动作空间进行自适应离散化，逐步“聚焦”到更具潜力的区域，目标是在保证最坏情况下近最优性能的同时，充分利用“更优”的问题实例。目前该问题的随机版本已有充分研究，但带对抗收益的通用版本仍缺乏相关结论。\n 我们提出了首个适用于对抗版本的自适应离散化算法（Adversarial Zooming，对抗缩放算法），并推导了依赖实例的悔界。具体而言，我们的结果既涵盖了对抗版本的最坏情况最优悔界，也包含了随机版本的依赖实例的悔界。\n 我们将该算法应用于多个基础场景，包括动态定价和拍卖保留价调优，所有场景均基于对抗收益模型。尽管这些领域通常不满足Lipschitz条件，但我们的分析仅需要更弱的Lipschitz变体条件，无需额外的平滑性假设即可得到有意义的悔界。值得注意的是，我们将结果扩展到了具有非平滑收益结构的多产品动态定价场景，该场景甚至不满足单侧Lipschitz条件。
英文摘要：
Lipschitz bandits is a prominent version of multi-armed bandits that studies large, structured action spaces such as the $[0,1]$ interval, where similar actions are guaranteed to have similar rewards. A central theme here is the adaptive discretization of the action space, which gradually ``zooms in'' on the more promising regions thereof. The goal is to take advantage of ``nicer'' problem instances, while retaining near-optimal worst-case performance. While the stochastic version of the problem is well-understood, the general version with adversarial rewards is not. We provide the first algorithm (\emph{Adversarial Zooming}) for adaptive discretization in the adversarial version, and derive instance-dependent regret bounds. In particular, we recover the worst-case optimal regret bound for the adversarial version, and the instance-dependent regret bound for the stochastic version. We apply our algorithm to several fundamental applications -- including dynamic pricing and auction reserve tuning -- all under adversarial reward models. While these domains often violate Lipschitzness, our analysis only requires a weaker version thereof, allowing for meaningful regret bounds without additional smoothness assumptions. Notably, we extend our results to multi-product dynamic pricing with non-smooth reward structures, a setting which does not even satisfy one-sided Lipschitzness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
