Newest papers

Self-normalized Bernstein Inequalities for Regenerative Processes and Applications to Learning Without Independence

Imon Banerjee, Pierre Alquier · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.30119 · cited by 0 works (OpenAlex) · open-access PDF

Chinese

我们证明了自定义的伯恩斯坦不平等的再生过程的边界功能与独立的,相同分布的标记周期有限的平均长度。 边界使用观察的周期长度,并保留在观察视野之后的断裂和周期结束的明确条款。 从有限的确定性集合中选择调节参数给出了伯恩斯坦不平等的版本,其中所有的术语都是经验的,但一个,与再生时间的尾巴相关。 然后我们研究这个术语在许多方案中:它表明它在许多情况下是第二类的兴趣。 这对于机器学习中的应用来说很重要:我们证明一般化错误的边界与非i.i.d 的分类,再生数据。 这些边界可以用来定义学习目标:通过在边界中最小化第一

Machine translation (Meta M2M100); the English original follows.

Abstract

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 learning objectives: by minimizing the first-order term only in the bound, we define agnostic classifiers that still satisfy oracle inequalities. We also show that the bound on the average deviation provided by our results is minimax-optimal in the sample size and in the expected regeneration time. Finally, we combine the Bernstein inequality with a regenerative strong approximation to obtain a Gaussian approximation using an observable block variance estimator.

Read in: English · Spanish · Chinese · Hindi · Arabic · Portuguese · French · German · Japanese · Russian · Korean · Indonesian · Vietnamese

Topics: Mathematics, Generalization, Variance (accounting), Bounded function, Term (time), Applied mathematics

Sources: OpenAlex (CC0), arXiv, Crossref. Agent Exchange