Weighted Empirical Risk Minimization for Machine Learning under Long-Range Dependence: Exact Pathwise Rates and Learning-Error Geometry — Paper

Abstract

We develop an exact almost-sure learning theory for smooth parametric models trained by regularly weighted empirical risk minimization on long-range dependent data. The training observations are generated from a fixed finite window of a stationary Gaussian sequence, and the sample weights are regularly varying. If the loss gradient at the population minimizer has Wiener-chaos rank $m$ and a nonzero low-frequency coefficient, then, in the long-memory interior regime, the finite-lag score reduces on the iterated-logarithm scale to a single weighted Hermite chaos. This yields an almost-sure Bahadur representation, an exact limsup law for the learned parameter, and, for $m\ge2$, the functional cluster set of the complete learning trajectory. The polynomial learning exponent is determined by the memory parameter and the chaos rank and is invariant under the admissible power weighting, whereas the sharp pathwise constant and cluster geometry depend on the weights. In the rank-one case, global optimization over the admissible power exponents shows that every optimizer is positive. Time-series prediction and classification examples illustrate the results.

Authors 1

Elina Moldavskaya

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Official page: arxiv.org/abs/2609.10767
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Arxiv announce type: cross
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
arXiv id: 2609.10767
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Categories: stat.ML, cs.LG
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
PDF: arxiv.org/pdf/2609.10767
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Primary category: stat.ML
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Published: 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh

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2 h ago

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Authors: Elina Moldavskaya

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Value	Valid from → to	Status	Source	Confidence	Extractor
stat.ML, cs.LG	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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11 Sept 2026
New paperPaperWeighted Empirical Risk Minimization for Machine Learning under Long-Range Dependence: Exact Pathwise Rates and Learning-Error Geometry
New paper: Weighted Empirical Risk Minimization for Machine Learning under Long-Range Dependence: Exact Pathwise Rates and Learning-Error Geometry
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arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.LG	feed	T1· Official	26 min ago	1

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