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.

翻译：我们研究了核工具变量（KIV）算法，这是一种基于核的非参数工具变量回归的两阶段最小二乘方法。我们提供了涵盖可识别与不可识别两种情形的收敛性分析：当结构函数不可识别时，我们证明KIV估计量收敛于再生核希尔伯特空间中与核相关的极小范数IV解。关键之处在于，我们建立了强$L_2$范数下的收敛性，而不仅限于伪范数。我们通过一种连接条件量化统计难度，该条件比较了内生回归变量的协方差结构与工具变量诱导的协方差结构，从而得到一种可解释的病态性度量。在标准特征值衰减和源假设下，我们推导了KIV的强$L_2\)学习速率，并证明这些速率在固定光滑类上达到极小化最优。最后，我们用通用谱正则化替代第一阶段的Tikhonov步骤，从而避免饱和效应并改善对更光滑第一阶段目标的学习速率。匹配的下界表明，与普通核岭回归相比，工具变量回归会不可避免地导致速率减缓。