Weighted Empirical Risk Minimization for Machine Learning under Long-Range Dependence: Exact Pathwise Rates and Learning-Error Geometry
Updated 1 h ago · first seen 11 Sept 2026
paper_01M294FQ3FYEW34SXX19TH98EE
- Published
- 11 Sept 2026
- T1 · 1 h ago
- arXiv
- 2609.10767
- T1 · 1 h ago
- Category
- stat.ML
- T1 · 1 h ago
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
Specification
- Official page
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
- Arxiv announce type
- cross
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
- arXiv id
- 2609.10767
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
- Categories
- stat.ML, cs.LG
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
- Primary category
- stat.ML
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
- Published
- 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
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1 h ago
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- Elina Moldavskaya
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Official pageofficial_url1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| https://arxiv.org/abs/2609.10767 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Abstractabstract1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 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. | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Arxiv announce typearxiv_announce_type1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| cross | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
arXiv idarxiv_id1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 2609.10767 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Categoriescategories1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| stat.ML, cs.LG | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
PDFpdf_url1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| https://arxiv.org/pdf/2609.10767 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Primary categoryprimary_category1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| stat.ML | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Publishedpublished_at1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 11 Sept 2026 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Claims are temporal and append-only: a new observation closes the previous claim (valid_to) instead of overwriting it. Conflicting claims from different sources are kept side by side and flagged — never averaged. Methodology →
New paper: Weighted Empirical Risk Minimization for Machine Learning under Long-Range Dependence: Exact Pathwise Rates and Learning-Error Geometry
arxiv
| Source | Document | Type | Tier | Last observed | Snapshots |
|---|---|---|---|---|---|
| arXiv (Atom API + RSS) | rss.arxiv.org/rss/cs.LG | feed | T1· Official | 1 h ago | 1 |
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