动态变化环境中的在线学习
Online Learning in Dynamically Changing Environments
- Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对动态变化环境中的在线学习问题，引入成本为K的动态变化过程，并证明了有限VC维假设类在绝对损失和可混合损失下的紧遗憾界，为分布盲机制下的非平稳样本遗憾分析迈出第一步。
AI中文摘要：
我们研究了当样本来自一般未知非平稳过程时的在线学习和在线遗憾最小化问题。我们引入了成本为$K$的动态变化过程的概念，其中过程的条件边际分布可以任意变化，但在$T$轮中不同条件边际分布的数量以$K$为界。对于此类过程，我们证明了在绝对损失（即期望误分类损失）下，任何有限VC维类$\mathcal{H}$的期望最坏情况遗憾的紧界（至多相差$\sqrt{\log T}$因子）为$O(\sqrt{KT\cdot\mathsf{VC}(\mathcal{H})\log T})$。然后，我们通过建立紧界（至多相差$\log^3 T$因子）的遗憾界$O(K\cdot\mathsf{VC}(\mathcal{H})\log^3 T)$，将此界改进到一般的可混合损失。我们将这些结果扩展到具有未知参考测度的一般平滑对手过程，通过证明在一般有界凸损失下，一维阈值函数的次线性遗憾界。我们的结果可以被视为在分布盲（通用）机制下对非平稳样本进行遗憾分析的第一步。这也带来了一个新的视角，将假设类复杂性的研究转向生成数据的过程的复杂性研究。
英文摘要：
We study the problem of online learning and online regret minimization when samples are drawn from a general unknown non-stationary process. We introduce the concept of a dynamic changing process with cost $K$, where the conditional marginals of the process can vary arbitrarily, but that the number of different conditional marginals is bounded by $K$ over $T$ rounds. For such processes we prove a tight (upto $\sqrt{\log T}$ factor) bound $O(\sqrt{KT\cdot\mathsf{VC}(\mathcal{H})\log T})$ for the expected worst case regret of any finite VC-dimensional class $\mathcal{H}$ under absolute loss (i.e., the expected miss-classification loss). We then improve this bound for general mixable losses, by establishing a tight (up to $\log^3 T$ factor) regret bound $O(K\cdot\mathsf{VC}(\mathcal{H})\log^3 T)$. We extend these results to general smooth adversary processes with unknown reference measure by showing a sub-linear regret bound for $1$-dimensional threshold functions under a general bounded convex loss. Our results can be viewed as a first step towards regret analysis with non-stationary samples in the distribution blind (universal) regime. This also brings a new viewpoint that shifts the study of complexity of the hypothesis classes to the study of the complexity of processes generating data.
