Riesz Regression As Direct Density Ratio Estimation

TL;DR

This paper reveals that Riesz regression is mathematically equivalent to density ratio estimation, enabling the transfer of theoretical results and techniques from DRE to Riesz regression in causal inference tasks.

Contribution

It establishes the equivalence between Riesz regression and density ratio estimation, facilitating the application of existing DRE methods and analyses to Riesz regression.

Findings

Riesz representer can be expressed as a signed density ratio.

Riesz regression objective matches least-squares importance fitting.

Enables transfer of DRE convergence and regularization results.

Abstract

This study clarifies the relationship between Riesz regression [Chernozhukov et al., 2021] and density ratio estimation (DRE) in causal inference problems, such as average treatment effect estimation. We first show that the Riesz representer can be written as a signed density ratio and then demonstrate that the Riesz regression objective coincides with the least-squares importance fitting criterion [Kanamori et al., 2009]. Although Riesz regression applies to a broad class of representer estimation problems, this equivalence with DRE allows us to transfer existing DRE results, including convergence rate analyses, generalizations based on Bregman divergence minimization, and regularization techniques for flexible models such as neural networks.