发表机构
Technion–Israel Institute of Technology（以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对长程相依数据，提出加权经验风险最小化的精确几乎必然学习理论，揭示了学习指数由记忆参数和混沌秩决定，并给出路径常数与簇几何的权重依赖性。
AI 中文摘要
我们针对在长程相依数据上通过正则加权经验风险最小化训练的平滑参数模型，发展了一种精确的几乎必然学习理论。训练观测值来自固定有限窗口的平稳高斯序列，样本权重为正则变化。若总体最小化器处的损失梯度具有维纳混沌秩$m$和非零低频系数，则在长记忆内部区域，有限滞后得分在迭代对数尺度上简化为单个加权埃尔米特混沌。这产生了几乎必然的Bahadur表示、学习参数的精确limsup定律，以及对于$m\ge2$，完整学习轨迹的函数簇集。多项式学习指数由记忆参数和混沌秩决定，且在可容许的幂加权下不变，而尖锐的路径常数和簇几何则依赖于权重。在秩一情形下，对可容许幂指数的全局优化表明每个最优值均为正。时间序列预测和分类示例说明了这些结果。
英文摘要
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.
Comments42 pages, 5 figures
