提出最优的非参数工具变量回归方法，突破传统收敛瓶颈。
Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal
- 用核方法构建两阶段最小二乘，解决非参数工具变量问题。
- 在标准假设下实现强L2范数下的最优学习速率。
- 适合研究因果推断与高维数据建模的学者参考。
我们研究了基于核方法的工具变量（KIV）算法，这是一种用于非参数工具变量回归的核型两阶段最小二乘法。我们提供了覆盖可识别与不可识别情形的收敛性分析：当结构函数不可识别时，证明KIV估计量收敛至再生核希尔伯特空间中最小范数的工具变量解。关键在于，我们建立了强L2范数下的收敛性，而非仅在伪范数下。通过一个连接条件量化统计难度，该条件比较了内生解释变量的协方差结构与工具变量诱导的结构，给出可解释的病态程度度量。在标准特征值衰减与源假设下，我们推导出KIV在强L2范数下的学习速率，并证明其在固定光滑度类上达到极小极大最优。最后，将第一阶段的Tikhonov正则化替换为一般谱正则化，避免饱和现象，提升对更平滑第一阶段目标的速率。匹配的下界表明，工具变量回归相对于普通核岭回归不可避免地存在速率下降。
原文摘要 · Abstract (English)
We study the kernel instrumental variable (KIV) algorithm, a kernel-based two-stage least-squares method for nonparametric instrumental variable regression. We provide a convergence analysis covering both identified and non-identified regimes: when the structural function is not identified, we show that the KIV estimator converges to the minimum-norm IV solution in the reproducing kernel Hilbert space associated with the kernel. Crucially, we establish convergence in the strong $L_2$ norm, rather than only in a pseudo-norm. We quantify statistical difficulty through a link condition that compares the covariance structure of the endogenous regressor with that induced by the instrument, yielding an interpretable measure of ill-posedness. Under standard eigenvalue-decay and source assumptions, we derive strong $L_2$ learning rates for KIV and prove that they are minimax-optimal over fixed smoothness classes. Finally, we replace the stage-1 Tikhonov step by general spectral regularization, thereby avoiding saturation and improving rates for smoother first-stage targets. The matching lower bound shows that instrumental regression induces an unavoidable slowdown relative to ordinary kernel ridge regression.
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