[Paper Review] Causal Inference of General Treatment Effects using Neural Networks with A Diverging Number of Confounders

Xiaohong Chen, Ying Liu+2 more|arXiv (Cornell University)|Sep 15, 2020

Advanced Causal Inference Techniques39 references4 citations

TL;DR

This paper proposes a neural network-based method for efficient causal inference of general treatment effects—such as average, quantile, and asymmetric least squares treatment effects—when the number of confounders grows with sample size. By introducing a mixed smoothness class and leveraging ReLU neural networks to estimate the propensity score, the method achieves root-n consistency and asymptotic normality, overcoming the curse of dimensionality without requiring sparsity.

ABSTRACT

Semiparametric efficient estimation of various multi-valued causal effects, including quantile treatment effects, is important in economic, biomedical, and other social sciences. Under the unconfoundedness condition, adjustment for confounders requires estimating the nuisance functions relating outcome or treatment to confounders nonparametrically. This paper considers a generalized optimization framework for efficient estimation of general treatment effects using artificial neural networks (ANNs) to approximate the unknown nuisance function of growing-dimensional confounders. We establish a new approximation error bound for the ANNs to the nuisance function belonging to a mixed smoothness class without a known sparsity structure. We show that the ANNs can alleviate the "curse of dimensionality" under this circumstance. We establish the root-$n$ consistency and asymptotic normality of the proposed general treatment effects estimators, and apply a weighted bootstrap procedure for conducting inference. The proposed methods are illustrated via simulation studies and a real data application.

Motivation & Objective

• To address the challenge of estimating general treatment effects when the number of confounders diverges with sample size.
• To overcome the curse of dimensionality in nonparametric estimation of nuisance functions under unconfoundedness.
• To develop a root-n consistent and asymptotically normal estimator for multi-valued treatment effects using neural networks.
• To enable valid inference via a weighted bootstrap procedure without requiring complex asymptotic variance estimation.
• To establish theoretical guarantees for neural network approximation under a mixed smoothness class without known sparsity.

Proposed method

• Proposes a generalized optimization framework that directly minimizes a loss function involving an artificial neural network (ANN)-approximated propensity score.
• Uses ReLU-based feedforward neural networks to model the nuisance function (propensity score) in high-dimensional confounder settings.
• Introduces a mixed smoothness class as a subset of the Barron class, enabling tighter bounds on the Fourier transform moment of the nuisance function.
• Derives a new approximation error bound for ANNs under this mixed smoothness class, showing robustness to increasing dimensionality.
• Establishes root-n consistency and asymptotic normality of the treatment effect estimator under mild regularity conditions.
• Employs a weighted bootstrap procedure for inference, avoiding the need to compute complex asymptotic variance formulas, especially for quantile and asymmetric treatment effects.

Experimental results

Research questions

• RQ1 Can neural networks effectively mitigate the curse of dimensionality in nonparametric estimation of nuisance functions when the number of confounders grows with sample size?
• RQ2 What function space assumptions are sufficient to ensure fast approximation rates for neural networks in high-dimensional causal inference?
• RQ3 Does the proposed ANN-based estimator achieve root-n consistency and asymptotic normality for general treatment effects under diverging confounders?
• RQ4 How does the performance of the proposed method compare to existing methods (e.g., GLM, GAM, RF, GBM, DNN) in finite samples for quantile and average treatment effects?
• RQ5 Can a weighted bootstrap procedure provide more accurate confidence sets than asymptotic variance-based inference for complex treatment effects like quantile treatment effects?

Key findings

• The proposed ANN-based estimator achieves root-n consistency and asymptotic normality for general treatment effects, even when the number of confounders grows with sample size.
• The approximation error bound for ANNs in the mixed smoothness class depends on the dimension in a controlled way, enabling effective high-dimensional function estimation.
• In simulation studies, the ANN-based method outperforms GLM, GAM, RF, GBM, and DNN in terms of bias, empirical standard deviation, and coverage rates for quantile treatment effects.
• For n=5000, the ANN method achieved a coverage rate of 0.935 for Q1, 0.940 for Q2, and 0.935 for Q3 under Model 2, with empirical standard deviations around 0.16–0.17.
• The weighted bootstrap confidence sets based on the ANN estimator showed higher accuracy than asymptotic variance-based intervals, with coverage rates consistently above 0.92 in all scenarios.
• The method demonstrated robustness across different models and sample sizes, maintaining low bias and good coverage even under high-dimensional confounding with p=10.