On the Equivalence between Neyman Orthogonality and Pathwise Differentiability

TL;DR

This paper establishes a formal equivalence between Neyman orthogonality and pathwise differentiability within the semiparametric framework, clarifying their relationship and structural differences through theoretical analysis and examples.

Contribution

It identifies an implicit regularity condition that links Neyman orthogonality and pathwise differentiability, unifying two key concepts in semiparametric inference.

Findings

Proves the formal equivalence under a local product structure assumption.

Shows the two concepts impose different structural requirements.

Provides detailed examples illustrating the theoretical results.

Abstract

It has been frequently observed that Neyman orthogonality, the central device underlying double/debiased machine learning (Chernozhukov et al., 2018), and pathwise differentiability, a cornerstone concept from semiparametric theory, often lead to the same debiased estimators in practice. Despite the widespread adoption of both ideas, the precise nature of this equivalence has remained elusive, with the two concepts having been developed in largely separate traditions. In this work, we revisit the semiparametric framework of van der Laan and Robins (2003) and identify an implicit regularity assumption on the relationship between target and nuisance parameters -- a local product structure -- that allows us to establish a formal equivalence between Neyman orthogonality and pathwise differentiability. We also show that the two directions of this equivalence impose fundamentally different structural requirements. Finally, we illustrate the theory through three detailed examples of estimating the average treatment effect and expected density in a nonparametric model, as well as the slope in a partially linear model. This helps clarify the relationship between these two foundational frameworks and provides a useful reference for practitioners working at their intersection.