Deep Neural Networks for Estimation and Inference

Abstract: We study deep neural networks and their use in semiparametric inference. We establish novel rates of convergence for deep feedforward neural nets. Our new rates are sufficiently fast (in some cases minimax optimal) to allow us to establish valid second-step inference after first-step estimation with deep learning, a result also new to the literature. Our estimation rates and semiparametric inference results handle the current standard architecture: fully connected feedforward neural networks (multi-layer perceptrons), with the now-common rectified linear unit activation function and a depth explicitly diverging with the sample size. We discuss other architectures as well, including fixed-width, very deep networks. We establish nonasymptotic bounds for these deep nets for a general class of nonparametric regression-type loss functions, which includes as special cases least squares, logistic regression, and other generalized linear models. We then apply our theory to develop semiparametric inference, focusing on causal parameters for concreteness, such as treatment effects, expected welfare, and decomposition effects. Inference in many other semiparametric contexts can be readily obtained. We demonstrate the effectiveness of deep learning with a Monte Carlo analysis and an empirical application to direct mail marketing.

• The paper establishes theoretical foundations for using deep neural networks (DNNs), specifically ReLU MLPs, in semiparametric estimation and inference, proving they can yield valid statistical inference.
• It derives nonasymptotic bounds and rapid convergence rates for these DNNs under general nonparametric loss functions, supporting their use in complex econometric and statistical analyses.
• An empirical application on direct mail marketing data demonstrates the practical utility of DNNs for estimating causal parameters like average treatment effects and optimizing marketing policies.

Overview of "Deep Neural Networks for Estimation and Inference"

This paper investigates the application of deep neural networks (DNNs) in the context of semiparametric inference, providing theoretical advancements that establish convergence rates and valid inference results critical for econometric and statistical investigations. The researchers focus specifically on fully connected feedforward neural networks, commonly known as multi-layer perceptrons (MLPs), equipped with the rectified linear unit (ReLU) activation function. These setups have gained significant traction in machine learning applications due to their computational efficiency and empirical success. The authors contribute by deriving nonasymptotic bounds under general nonparametric loss functions and demonstrating the efficiency and feasibility of using DNNs in causal inference.

Theoretical Contributions

The primary theoretical contributions of this paper are twofold: first, it establishes nonasymptotic bounds for MLP‐ReLU architectures, showcasing their sufficiency for valid second-step inference, and second, it lays out the implications of these results for estimating causal parameters such as treatment effects. The paper leverages a localization analysis with scale-insensitive complexity measures to establish these bounds, avoiding more restrictive conditions typically found in neural network or sieve analyses. This approach allows for bounding the empirical processes via Rademacher complexity and symmetrization techniques, culminating in convergence rates notably faster than the conventional $n^{-1/4}$ in certain settings, thereby supporting semiparametric inference tasks.

In examining ReLU-based DNNs, the paper verifies that the architecture adheres to the essential statistical properties required for high-quality nonparametric estimation. For instance, it manages to draw precise connections between network architecture, approximation capabilities, and statistical convergence rates, indicating that, in many cases, these networks can fully adapt to function complexities encountered in econometric data.

Empirical Evaluation

An empirical application on a large-scale direct mail marketing data set is presented to illustrate the practical relevance of the theoretical findings. The data comprises nearly 300,000 consumers, allowing the examination of treatment effects of a catalog mailing on consumer spending. The results from implementing various DNN architectures demonstrate the method's utility, providing unbiased and efficient estimates of the average treatment effect and expected profit from different mailing policies.

Implications for Causal Inference

The application of deep learning in causal inference problems emerges as one of the key discussions in the paper, showcased through detailed analysis and empirical validation. By focusing on average treatment effects, the authors illustrate how DNNs can be utilized to model complex, high-dimensional relationships between covariates and outcomes, leading to improved treatment effect estimators. The application is robust to violations of linear model assumptions, therefore accommodating non-linear and interaction effects effortlessly.

Additionally, the authors highlight potential advancements in targeting strategies within marketing contexts, showing that learned models can effectively optimize decision-making strategies by predicting the causal impacts of different policies or interventions. This aligns well with contemporary needs in economics and related fields where observational data is abundant but poses significant inferential challenges due to its non-experimental nature.

Future Prospects

The paper paves the way for future exploration into adaptive DNN architectures tailored for specific econometric tasks, which may further refine the balance between computational efficiency and statistical precision. Moreover, the theoretical framework developed herein can potentially be extended to accommodate other activation functions and network structures, broadening the applicability of DNNs in handling diverse categories of semiparametric estimation problems that statisticians and economists frequently encounter.

In conclusion, the paper demonstrates how deep learning frameworks, particularly those incorporating deep neural networks, can transition from black box predictive tools to part of the rigorous econometrician’s toolkit, offering a bridge between high-dimensional learning capabilities and valid economic inference.