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Riesz representers for the rest of us

Nicholas T. Williams, Oliver J. Hines, Kara E. Rudolph

TL;DR

The paper addresses challenges in semiparametric causal estimation when incorporating flexible machine learning by reframing estimands through the Riesz representation theorem. It shows how estimands can be written as inner products with Riesz representers, linking them to IPW-type weights and enabling a simple recursive algorithm to derive the EIF. It then formalizes Riesz regression as a method to learn representers directly from data using a Riesz loss, and discusses practical considerations and potential benefits in high-dimensional mediation and longitudinal settings. Through worked examples, the authors provide both theoretical grounding and practical guidance for applying Riesz-based methods in epidemiology, highlighting pathways to automatic debiasing and improved estimation when nuisance parameters are complex.

Abstract

The application of semiparametric efficient estimators, particularly those that leverage machine learning, is rapidly expanding within epidemiology and causal inference. This literature is increasingly invoking the Riesz representation theorem and Riesz regression. This paper aims to introduce the Riesz representation theorem to an epidemiologic audience, explaining what it is and why it's useful, using step-by-step worked examples.

Riesz representers for the rest of us

TL;DR

The paper addresses challenges in semiparametric causal estimation when incorporating flexible machine learning by reframing estimands through the Riesz representation theorem. It shows how estimands can be written as inner products with Riesz representers, linking them to IPW-type weights and enabling a simple recursive algorithm to derive the EIF. It then formalizes Riesz regression as a method to learn representers directly from data using a Riesz loss, and discusses practical considerations and potential benefits in high-dimensional mediation and longitudinal settings. Through worked examples, the authors provide both theoretical grounding and practical guidance for applying Riesz-based methods in epidemiology, highlighting pathways to automatic debiasing and improved estimation when nuisance parameters are complex.

Abstract

The application of semiparametric efficient estimators, particularly those that leverage machine learning, is rapidly expanding within epidemiology and causal inference. This literature is increasingly invoking the Riesz representation theorem and Riesz regression. This paper aims to introduce the Riesz representation theorem to an epidemiologic audience, explaining what it is and why it's useful, using step-by-step worked examples.

Paper Structure

This paper contains 8 sections, 1 theorem, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. What is the Riesz representation theorem?
  4. Estimands as bounded linear functionals
  5. An alternate expression using Riesz representers
  6. Using Riesz representers to derive the EIF
  7. Riesz regression
  8. Discussion

Key Result

Theorem 1

Every bounded linear functional $\theta = \mathsf{m}_1(\mathsf{Q}_1)$ (where $\mathsf{Q}_1$ is the final iterated expectation) can be expressed as an inner product of the initial outcome regression $\mathsf{Q}_K$ (or the outcome of that regression e.g., $Y$) and weights $\alpha(X)$.

Theorems & Definitions (7)

  • Example 1: Mean outcome among the treated
  • Example 2: Average treatment effect
  • Example 3: Mean outcome under control for the treated
  • Example 4: Natural direct effect
  • Theorem 1: Riesz representation theorem---Informal
  • Definition 1: Linear mapping
  • Definition 2: Bounded linear functional