Sparsified-Learning for High-Dimensional Heavy-Tailed Locally Stationary Time Series, Concentration and Oracle Inequalities
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| Format: | Preprint |
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2025
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| author | Wang, Yingjie Alaya, Mokhtar Z. Bouzebda, Salim Liu, Xinsheng |
| author_facet | Wang, Yingjie Alaya, Mokhtar Z. Bouzebda, Salim Liu, Xinsheng |
| contents | Sparse learning is ubiquitous in many machine learning tasks. It aims to regularize the goodness-of-fit objective by adding a penalty term to encode structural constraints on the model parameters. In this paper, we develop a flexible sparse learning framework tailored to high-dimensional heavy-tailed locally stationary time series (LSTS). The data-generating mechanism incorporates a regression function that changes smoothly over time and is observed under noise belonging to the class of sub-Weibull and regularly varying distributions. We introduce a sparsity-inducing penalized estimation procedure that combines additive modeling with kernel smoothing and define an additive kernel-smoothing hypothesis class. In the presence of locally stationary dynamics, we assume exponentially decaying $β$-mixing coefficients to derive concentration inequalities for kernel-weighted sums of locally stationary processes with heavy-tailed noise. We further establish nonasymptotic prediction-error bounds, yielding both slow and fast convergence rates under different sparsity structures, including Lasso and total variation penalization with the least-squares loss. To support our theoretical results, we conduct numerical experiments on simulated LSTS with sub-Weibull and Pareto noise, highlighting how tail behavior affects prediction error across different covariate-dimensions as the sample size increases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06477 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparsified-Learning for High-Dimensional Heavy-Tailed Locally Stationary Time Series, Concentration and Oracle Inequalities Wang, Yingjie Alaya, Mokhtar Z. Bouzebda, Salim Liu, Xinsheng Machine Learning Statistics Theory Sparse learning is ubiquitous in many machine learning tasks. It aims to regularize the goodness-of-fit objective by adding a penalty term to encode structural constraints on the model parameters. In this paper, we develop a flexible sparse learning framework tailored to high-dimensional heavy-tailed locally stationary time series (LSTS). The data-generating mechanism incorporates a regression function that changes smoothly over time and is observed under noise belonging to the class of sub-Weibull and regularly varying distributions. We introduce a sparsity-inducing penalized estimation procedure that combines additive modeling with kernel smoothing and define an additive kernel-smoothing hypothesis class. In the presence of locally stationary dynamics, we assume exponentially decaying $β$-mixing coefficients to derive concentration inequalities for kernel-weighted sums of locally stationary processes with heavy-tailed noise. We further establish nonasymptotic prediction-error bounds, yielding both slow and fast convergence rates under different sparsity structures, including Lasso and total variation penalization with the least-squares loss. To support our theoretical results, we conduct numerical experiments on simulated LSTS with sub-Weibull and Pareto noise, highlighting how tail behavior affects prediction error across different covariate-dimensions as the sample size increases. |
| title | Sparsified-Learning for High-Dimensional Heavy-Tailed Locally Stationary Time Series, Concentration and Oracle Inequalities |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2504.06477 |