We study deep neural networks and their use in semiparametric inference. We establish novel nonasymptotic high probability bounds for deep feedforward neural nets. These deliver rates of convergence that 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 nonasymptotic high probability bounds, and the subsequent semiparametric inference, treat the current standard architecture: fully connected feedforward neural networks (multilayer perceptrons), with the now‐common rectified linear unit activation function, unbounded weights, and a depth explicitly diverging with the sample size. We discuss other architectures as well, including fixed‐width, very deep networks. We establish the 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, and demonstrate the effectiveness of deep learning with an empirical application to direct mail marketing.


Max H. Farrell, Tengyuan Liang, and Sanjog Misra. 2021. “Deep Neural Networks for Estimation and Inference.” Econometrica 89 (1): 181–213.

   title={Deep Neural Networks for Estimation and Inference},
   publisher={The Econometric Society},
   author={Farrell, Max H. and Liang, Tengyuan and Misra, Sanjog},
   pages={181–213} }