Source-Condition Analysis of Kernel Adversarial Estimators

TL;DR

This paper introduces a new analysis of Regularized Adversarial Stabilized estimators using RKHS, providing finite-sample bounds and comparing assumptions with other kernel-based methods in the context of ill-posed inverse problems.

Contribution

It offers the first finite-sample bounds for RAS estimators with RKHS regularization and compares their assumptions with alternative kernel-based estimation approaches.

Findings

Finite-sample bounds for weak error and RMSE established.

Comparison of assumptions with $ ext{L}^2$-penalized RAS and Kernel Maximal Moment estimators.

Enhanced understanding of source conditions for kernel adversarial estimators.

Abstract

In many applications, the target parameter depends on a nuisance function defined by a conditional moment restriction, whose estimation often leads to an ill-posed inverse problem. Classical approaches, such as sieve-based GMM, approximate the restriction using a fixed set of test functions and may fail to capture important aspects of the solution. Adversarial estimators address this limitation by framing estimation as a game between an estimator and an adaptive critic. We study the class of Regularized Adversarial Stabilized (RAS) estimators that employ reproducing kernel Hilbert spaces (RKHSs) for both estimation and testing, with regularization via the RKHS norm. Our first contribution is a novel analysis that establishes finite-sample bounds for both the weak error and the root mean squared error (RMSE) of these estimators under interpretable source conditions, in contrast to existing results. Our second contribution is a detailed comparison of the assumptions underlying this RKHS-norm-regularized approach with those required for (i) RAS estimators using $\mathcal{L}^2$ penalties, and (ii) recently proposed, computationally stable Kernel Maximal Moment estimators.