核化谱滤波器学习算法的差异原理分析
Analyzing the discrepancy principle for kernelized spectral filter learning algorithms
摘要：
本文研究了非参数回归问题中最优迭代次数未知且使用迭代学习算法的提前停止规则的构造。更确切地说，我们研究了差异原则，以及基于平滑残差的修改，包括梯度下降的核化谱滤波器学习算法。我们的主要理论界是经验估计误差（固定设计）和预测误差（随机设计）的预言不等式。从这些有限样本界限可以看出，经典的差异原则在统计上适应于在硬学习场景中发生的慢速率，而平滑的差异原则在更快速率的范围内是自适应的（分别是）。更高的平滑度参数）。我们的方法依赖于偏差不等式的停止规则在固定的设计设置，结合改变范数参数来处理随机设计设置。
英文摘要：
We investigate the construction of early stopping rules in the nonparametric regression problem where iterative learning algorithms are used and the optimal iteration number is unknown. More precisely, we study the discrepancy principle, as well as modifications based on smoothed residuals, for kernelized spectral filter learning algorithms including gradient descent. Our main theoretical bounds are oracle inequalities established for the empirical estimation error (fixed design), and for the prediction error (random design). From these finite-sample bounds it follows that the classical discrepancy principle is statistically adaptive for slow rates occurring in the hard learning scenario, while the smoothed discrepancy principles are adaptive over ranges of faster rates (resp. higher smoothness parameters). Our approach relies on deviation inequalities for the stopping rules in the fixed design setting, combined with change-of-norm arguments to deal with the random design setting.
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