Nearest Neighbor Matching as Least Squares Density Ratio Estimation and Riesz Regression

TL;DR

This paper establishes a theoretical connection between Nearest Neighbor matching, density-ratio estimation, and Riesz regression, providing a unified framework for debiased machine learning methods.

Contribution

It proves that NN matching can be viewed as Riesz regression via density-ratio estimation, unifying these approaches under a common theoretical framework.

Findings

NN matching is equivalent to LSIF density-ratio estimation.

Riesz regression can be derived from LSIF framework.

NN matching can be interpreted as Riesz regression for debiased ML.

Abstract

This study proves that Nearest Neighbor (NN) matching can be interpreted as an instance of Riesz regression for automatic debiased machine learning. Lin et al. (2023) shows that NN matching is an instance of density-ratio estimation with their new density-ratio estimator. Chernozhukov et al. (2024) develops Riesz regression for automatic debiased machine learning, which directly estimates the Riesz representer (or equivalently, the bias-correction term) by minimizing the mean squared error. In this study, we first prove that the density-ratio estimation method proposed in Lin et al. (2023) is essentially equivalent to Least-Squares Importance Fitting (LSIF) proposed in Kanamori et al. (2009) for direct density-ratio estimation. Furthermore, we derive Riesz regression using the LSIF framework. Based on these results, we derive NN matching from Riesz regression. This study is based on our work Kato (2025a) and Kato (2025b).