在再生核希尔伯特空间中建立广义非参数回归的收敛理论，揭示其抗维数诅咒机制。
Generalized nonparametric regression in reproducing kernel Hilbert spaces: Consistency and rates of convergence
- 基于正则化M-估计构建泛化非参数回归方法，兼容凸与非凸损失函数。
- 给出精确收敛速率，偏差由源条件决定，方差与模型误设无关。
- 适用于高维函数空间，可解释为何避免维数诅咒，适合理论研究者。
我们在再生核希尔伯特空间中发展了正则化M-估计的完整理论。在对损失函数施加弱条件的前提下，建立了估计量的存在性与可测性，涵盖广泛的凸与非凸损失，包括有界鲁棒损失。进一步证明了精确的收敛速率，其偏差-方差分解由一个新颖的复杂度度量控制。我们发现方差独立于模型误设，而偏差依赖于学习理论中的源条件参数。针对张量积Sobolev空间，获得了新收敛速率，关联到具有主导混合光滑性的函数空间，显著扩展了现有结果，并解释了该估计器如何规避维数诅咒。本方法融合泛函分析与经验过程理论，实现目标函数的渐近线性化，无需闭式解或全局Lipschitz假设，可能具有独立研究价值。估计器用C++实现，理论得到数值实验支持。
原文摘要 · Abstract (English)
We develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces. Under mild conditions on the loss we establish existence and measurability of the estimator, covering a wide range of convex and non-convex losses, including bounded robust losses. We further prove sharp rates of convergence with an explicit bias-variance decomposition governed by a novel complexity measure. We show that the variance is independent of misspecification, while the bias depends on a source condition parameter known in the learning literature. For tensor product Sobolev spaces we obtain new rates that connect to spaces of functions with dominating mixed smoothness, substantially extending existing results and explaining why these estimators circumvent the curse of dimensionality. Our methodology, combining elements from both functional analysis and empirical process theory, allows for an asymptotic linearisation of the objective function that avoids both closed-form solutions and global Lipschitz assumptions, and may be of independent interest. The estimators are implemented in C++ and theory is supported by numerical experiments.
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