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Adversarial Estimation of Riesz Representers

Victor Chernozhukov, Whitney Newey, Rahul Singh, Vasilis Syrgkanis

TL;DR

This work introduces an adversarial, direct estimator for the Riesz representer of mean-square continuous linear functionals, enabling inference with general, non-Donsker function spaces such as neural networks and random forests. A nonasymptotic $L_2$-rate is developed in terms of a critical-radius quantity, unifying fast rates across neural networks, RFs, and RKHSs, and accommodating mis-specification. The paper then shows how these rates verify the conditions needed for targeted and debiased semiparametric inference, including stability- and complexity-based guarantees that avoid sample splitting in some settings. Computational analysis and simulations demonstrate nominal coverage in nonlinear and high-dimensional scenarios, and an empirical application extends semiparametric estimation to assess heterogeneous effects of matching grants across political environments. Together, the results provide a flexible, theoretically-grounded framework for accurate semiparametric inference with modern machine learning function spaces.

Abstract

Many causal parameters are linear functionals of an underlying regression. The Riesz representer is a key component in the asymptotic variance of a semiparametrically estimated linear functional. We propose an adversarial framework to estimate the Riesz representer using general function spaces. We prove a nonasymptotic mean square rate in terms of an abstract quantity called the critical radius, then specialize it for neural networks, random forests, and reproducing kernel Hilbert spaces as leading cases. Our estimators are highly compatible with targeted and debiased machine learning with sample splitting; our guarantees directly verify general conditions for inference that allow mis-specification. We also use our guarantees to prove inference without sample splitting, based on stability or complexity. Our estimators achieve nominal coverage in highly nonlinear simulations where some previous methods break down. They shed new light on the heterogeneous effects of matching grants.

Adversarial Estimation of Riesz Representers

TL;DR

This work introduces an adversarial, direct estimator for the Riesz representer of mean-square continuous linear functionals, enabling inference with general, non-Donsker function spaces such as neural networks and random forests. A nonasymptotic -rate is developed in terms of a critical-radius quantity, unifying fast rates across neural networks, RFs, and RKHSs, and accommodating mis-specification. The paper then shows how these rates verify the conditions needed for targeted and debiased semiparametric inference, including stability- and complexity-based guarantees that avoid sample splitting in some settings. Computational analysis and simulations demonstrate nominal coverage in nonlinear and high-dimensional scenarios, and an empirical application extends semiparametric estimation to assess heterogeneous effects of matching grants across political environments. Together, the results provide a flexible, theoretically-grounded framework for accurate semiparametric inference with modern machine learning function spaces.

Abstract

Many causal parameters are linear functionals of an underlying regression. The Riesz representer is a key component in the asymptotic variance of a semiparametrically estimated linear functional. We propose an adversarial framework to estimate the Riesz representer using general function spaces. We prove a nonasymptotic mean square rate in terms of an abstract quantity called the critical radius, then specialize it for neural networks, random forests, and reproducing kernel Hilbert spaces as leading cases. Our estimators are highly compatible with targeted and debiased machine learning with sample splitting; our guarantees directly verify general conditions for inference that allow mis-specification. We also use our guarantees to prove inference without sample splitting, based on stability or complexity. Our estimators achieve nominal coverage in highly nonlinear simulations where some previous methods break down. They shed new light on the heterogeneous effects of matching grants.

Paper Structure

This paper contains 41 sections, 40 theorems, 116 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction and related work
  2. Adversarial estimation over general function spaces
  3. Nonasymptotic mean square rate via critical radius
  4. Semiparametric inference
  5. Analysis of computational error
  6. Simulated and real data analysis
  7. Discussion
  8. Proof of main results
  9. Adversarial estimation and fast rate
  10. Semiparametric inference
  11. Application: Asset pricing
  12. Fundamental asset pricing equation
  13. Matching symbols
  14. Detailed comparisons
  15. Sparse linear case
  16. ...and 26 more sections

Key Result

Corollary 1

Consider the population limit of our criterion where $n\to \infty$ and $\lambda,\mu \to 0$: $\max_{f\in {\mathcal{F}}} \mathbb{E}\left[m(Z; f) - a(X)\cdot f(X)\right] - \|f\|_{2}^2.$ This limit equals $\frac{1}{4} \|a-a_0\|_2^2$.

Figures (2)

  • Figure 1: Heterogeneous effects on dollars donated {$n=25859$, $dim(W)=15$}.
  • Figure 2: Heterogeneous effects on whether donated {$n=25859$, $dim(W)=15$}.

Theorems & Definitions (78)

  • Corollary 1: Population limit
  • Theorem 1: Mean square rate
  • Corollary 2: Weaker metric rate
  • Corollary 3: Mean square rate without norm regularization
  • Corollary 4: Union of hypothesis spaces
  • Corollary 5: Neural network Riesz representer rate
  • Corollary 6: Random forest Riesz representer rate
  • Corollary 7: RKHS Riesz representer rate
  • Corollary 8: Normality via sample splitting
  • Theorem 2: Normality via estimator stability
  • ...and 68 more