In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. Recently, many flexible machine learning methods have been developed for instrumental variable estimation. However, these methods have at least one of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) only obtaining estimation error rates in terms of pseudometrics (\emph{e.g.,} projected norm) rather than valid metrics (\emph{e.g.,} $L_2$ norm); or (3) imposing the so-called closedness condition that requires a certain conditional expectation operator to be sufficiently smooth. In this paper, we present the first method and analysis that can avoid all three limitations, while still permitting general function approximation. Specifically, we propose a new penalized minimax estimator that can converge to a fixed IV solution even when there are multiple solutions, and we derive a strong $L_2$ error rate for our estimator under lax conditions. Notably, this guarantee only needs a widely-used source condition and realizability assumptions, but not the so-called closedness condition. We argue that the source condition and the closedness condition are inherently conflicting, so relaxing the latter significantly improves upon the existing literature that requires both conditions. Our estimator can achieve this improvement because it builds on a novel formulation of the IV estimation problem as a constrained optimization problem.

翻译：本文研究工具变量（IV）回归的非参数估计问题。近年来，许多灵活的机器学习方法被开发用于工具变量估计。然而，这些方法至少存在以下限制之一：（1）要求IV回归具有唯一识别性；（2）仅能获得基于伪度量（例如投影范数）的估计误差率，而非有效度量（例如$L_2$范数）下的误差率；或（3）施加所谓的封闭性条件，要求特定条件期望算子具有充分光滑性。本文提出首个能够同时避免上述三种限制的方法与分析框架，同时仍允许通用函数逼近。具体而言，我们设计了一种新的惩罚极小化极大估计量，即使存在多个解时仍能收敛至固定IV解，并在宽松条件下推导出该估计量的强$L_2$误差率。值得注意的是，该保证仅需广泛使用的源条件与可实现性假设，而无需封闭性条件。我们论证了源条件与封闭性条件本质上是相互矛盾的，因此放松后者显著改进了现有文献中需要同时满足这两个条件的方法。我们的估计量能够实现这一改进，源于其基于将IV估计问题重新表述为约束优化问题的新颖框架。