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.

翻译：稀疏学习在众多机器学习任务中无处不在。其目标是通过在拟合优度目标函数中添加惩罚项，对模型参数施加结构性约束。本文针对高维重尾局部平稳时间序列（LSTS）构建了一个灵活的稀疏学习框架。数据生成机制包含一个随时间平滑变化的回归函数，观测噪声属于亚韦布尔分布和正则变化分布类别。我们提出了一种结合加性建模与核平滑的稀疏诱导惩罚估计方法，并定义了加性核平滑假设类。在存在局部平稳动态特性的情况下，我们假设混合系数呈指数衰减，从而推导出具有重尾噪声的局部平稳过程的核加权和的集中不等式。进一步建立了非渐近预测误差界，在包括Lasso惩罚和全变差惩罚（结合最小二乘损失）在内的不同稀疏结构下，分别得到了慢速和快速收敛速率。为验证理论结果，我们对具有亚韦布尔噪声和帕累托噪声的模拟LSTS进行了数值实验，揭示了随着样本量增加，尾部行为如何影响不同协变量维度下的预测误差。