[论文] Self-normalized Bernstein Inequalities for Regenerative Processes and ...

论文概要

研究领域: ML 作者: Imon Banerjee, Pierre Alquier 发布时间: 2026-10-21 arXiv: 2610.30119

中文摘要

我们证明了一个关于再生过程（具有独立同分布标记循环且循环长度均值有限）有界函数的自归一化 Bernstein 不等式。该界限使用观测到的循环长度，并保留了截断和超出观测视野的循环的显式项。从有限确定性集合中选择调参，可以得到一个除一个与再生时间尾部相关的项外全为经验项的 Bernstein 不等式版本。我们在多种情形下研究了这一项：结果表明在许多感兴趣的情形中它是二阶项。这对机器学习应用很重要：我们证明了使用非独立同分布的再生数据进行分类的泛化误差界限。这些界限可用于定义学习目标：仅通过最小化界限中的一阶项，我们定义了仍满足预言机不等式的不可知分类器。我们还证明了我们的结果给出的平均偏差界限在样本量和期望再生时间方面是极小化最优的。最后，我们将 Bernstein 不等式与再生强逼近相结合，使用可观测的块方差估计量获得了高斯逼近。

原文摘要

We prove a self-normalized Bernstein inequality for bounded functions of a regenerative process with independent, identically distributed marked cycles of finite mean length. The bound uses observed cycle lengths and retains explicit terms for truncation and the cycle ending after the observation horizon. Selecting a tuning parameter from a finite deterministic collection gives a version of Bernstein inequality where all terms are empirical but one, related to the tails of the regeneration times. We then study this term in many regimes: it turns out that it is of second-order in many cases of interest. This is important for applications in machine learning: we prove generalization error bounds for classification with non-i.i.d., regenerative data. These bounds can be used to define learnin...

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