A Unified Framework for Debiased Machine Learning: Riesz Representer Fitting under Bregman Divergence

Masahiro Kato. 2026. Preprint (arXiv:2601.07752). 日本語版

Debiased machine learning estimation of a causal effect or a structural parameter turns on the Riesz representer, the correction term that cancels the bias, and estimating that representer is the central step. We propose Generalized Riesz Regression, which fits a model for the Riesz representer by minimizing a Bregman divergence, and this brings estimation methods that grew up separately into one framework.

The problem

Estimating the Riesz representer has been approached from several directions with different ancestries. Riesz regression minimizes a squared distance, the tailored loss is built on the KL divergence, balancing methods match the moments of the covariates directly, and other approaches borrow an estimator from density ratio estimation. As long as the relations among them are not worked out, there is no way to say which method to use when, or where the differences in their properties come from.

The proposed method

We fix a model for the representer together with a link function, and formulate estimation as minimizing an empirical Bregman divergence. Choose squared distance as the divergence and Riesz regression falls out as a special case. Choose the KL divergence and the tailored loss falls out the same way. With an appropriate pairing of divergence and link function, the dual of the estimation problem coincides with covariate balancing, which we call automatic covariate balancing.

Main results and conditions

Under that same pairing, we show that the sample mean of the outcome weighted by the estimated representer satisfies Neyman orthogonality without any estimate of the regression function, which we call automatic Neyman orthogonalization. The property reduces the estimation error of the orthogonal score, and it also makes clear where debiased machine learning and targeted maximum likelihood estimation (TMLE) part ways. These properties hold under the condition that the divergence and the model specification (the link function) are compatible.

Relation to earlier work

With debiased machine learning, Chernozhukov and coauthors established the framework for causal inference via machine learning and orthogonal scores (2018, Double/debiased machine learning for treatment and structural parameters), and the same group proposed Riesz regression, which estimates the correction term directly as a regression (2024, Automatic Debiased Machine Learning via Riesz Regression). The leading way to balance covariates directly is the covariate balancing propensity score of Imai and Ratkovic (2014, Covariate Balancing Propensity Score), and targeted maximum likelihood estimation (van der Laan and Rubin, 2006, Targeted Maximum Likelihood Learning) is a separate route that folds the correction into the estimation itself. Our contribution is to account for all of these along one axis, the choice of Bregman divergence, and to spell out the conditions under which balancing and orthogonalization hold automatically. A study in the same line recasts Riesz regression as direct density ratio estimation (accepted at Finance Research Letters, Riesz Regression As Direct Density Ratio Estimation).

Where it applies

This covers the estimation of causal effects such as the average treatment effect, and of structural parameters more broadly, whenever machine learning is used. The framework is implemented as the Python package genriesz, and the related line of work is collected on the Generalized Riesz Regression page.

Paper and materials

BibTeX

@misc{unified-framework-debiased-ml-bregman,
  author       = {Masahiro Kato},
  title        = {A Unified Framework for Debiased Machine Learning: Riesz Representer Fitting under Bregman Divergence},
  year         = {2026},
  eprint       = {2601.07752},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2601.07752},
}