arXiv：cs.LG（机器学习，全量分类）· Xiaomian Yang, Sungho Shin·· 15 小时前AI 评分33
重尾噪声下线性系统的学习：单条轨迹的非渐近分析
Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory
AI 导读
研究针对指数稳定系统，在重尾噪声下基于单条观测轨迹给出了向量自回归模型最小二乘估计的非渐近样本复杂度界。在噪声独立同分布、协方差有界且持续激励的假设下，当噪声 p 阶矩有界（p > 2）时，估计误差为 Õ(r^{1/2}T^{-1/2+1/p})，其中 T 为样本数、r 为噪声维度。该分析还统一适用于次指数与次高斯噪声分布，并推广到带外生输入的自回归模型，其误差界的维度因子与模型阶数无关。
正文
Abstract:We establish non-asymptotic sample complexity bounds for the least-squares estimation of vector autoregressive models for exponentially stable systems with heavy-tailed noise based on a single observed trajectory. By assuming i.i.d. noise, bounded noise covariance, and persistent excitation, we show that the estimation error is $\widetilde{\mathcal{O}}(r^{1/2}T^{-1/2+1/p})$ under bounded $p$th moment for $p > 2$, where $T$ is the number of samples, $r$ is the noise dimension, and $\widetilde{\mathcal{O}}(\cdot)$ hides logarithmic terms. We also introduce a unifying approach to sample complexity analysis applicable to broad classes of noise distributions and showcase this by deriving error bounds for sub-exponential and sub-Gaussian noise distributions. Finally, we specialize our analysis to autoregressive models with exogenous inputs and show that the dimension factor of the error bound is independent of the model order.
| Subjects: | Machine Learning (cs.LG); Systems and Control (eess.SY) |
| Cite as: | arXiv:2610.00637 [cs.LG] |
| (or arXiv:2610.00637v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.00637 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Xiaomian Yang [view email]
[v1]
Wed, 30 Sep 2026 19:39:13 UTC (345 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org