We provide adaptive inference methods, based on $\ell _1$ regularization, for regular (semiparametric) and nonregular (nonparametric) linear functionals of the conditional expectation function. Examples of regular functionals include average treatment effects, policy effects, and derivatives. Examples of nonregular functionals include average treatment effects, policy effects, and derivatives conditional on a covariate subvector fixed at a point. We construct a Neyman orthogonal equation for the target parameter that is approximately invariant to small perturbations of the nuisance parameters. To achieve this property, we include the Riesz representer for the functional as an additional nuisance parameter. Our analysis yields weak ‘double sparsity robustness’: either the approximation to the regression or the approximation to the representer can be ‘completely dense’ as long as the other is sufficiently ‘sparse’. Our main results are nonasymptotic and imply asymptotic uniform validity over large classes of models, translating into honest confidence bands for both global and local parameters.
Deep Neural Networks for Estimation and Inference
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.
 Award ID(s):
 2042473
 Publication Date:
 NSFPAR ID:
 10328957
 Journal Name:
 Econometrica
 Volume:
 89
 Issue:
 1
 Page Range or eLocationID:
 181 to 213
 ISSN:
 00129682
 Sponsoring Org:
 National Science Foundation
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We consider the problem of finding a twolayer neural network with sigmoid, rectified linear unit (ReLU), or binary step activation functions that “fits” a training data set as accurately as possible as quantified by the training error; and study the following question: does a low training error guarantee that the norm of the output layer (outer norm) itself is small? We answer affirmatively this question for the case of nonnegative output weights. Using a simple covering number argument, we establish that under quite mild distributional assumptions on the input/label pairs; any such network achieving a small training error on polynomially many data necessarily has a wellcontrolled outer norm. Notably, our results (a) have a polynomial (in d) sample complexity, (b) are independent of the number of hidden units (which can potentially be very high), (c) are oblivious to the training algorithm; and (d) require quite mild assumptions on the data (in particular the input vector X ∈ Rd need not have independent coordinates). We then leverage our bounds to establish generalization guarantees for such networks through fatshattering dimension, a scalesensitive measure of the complexity class that the network architectures we investigate belong to. Notably, our generalization bounds also have goodmore »

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