On the minimax optimality and superiority of deep neural network learning over sparse parameter spaces

Deep learning has been applied to various tasks in the field of machine learning and has shown superiority to other common procedures such as kernel methods. To provide a better theoretical understanding of the reasons for its success, we discuss the performance of deep learning and other methods on a nonparametric regression problem with a Gaussian noise. Whereas existing theoretical studies of deep learning have been based mainly on mathematical theories of well-known function classes such as H\"{o}lder and Besov classes, we focus on function classes with discontinuity and sparsity, which are those naturally assumed in practice. To highlight the effectiveness of deep learning, we compare deep learning with a class of linear estimators representative of a class of shallow estimators. It is shown that the minimax risk of a linear estimator on the convex hull of a target function class does not differ from that of the original target function class. This results in the suboptimality of linear methods over a simple but non-convex function class, on which deep learning can attain nearly the minimax-optimal rate. In addition to this extreme case, we consider function classes with sparse wavelet coefficients. On these function classes, deep learning also attains the minimax rate up to log factors of the sample size, and linear methods are still suboptimal if the assumed sparsity is strong. We also point out that the parameter sharing of deep neural networks can remarkably reduce the complexity of the model in our setting.Comment: 33 page

Harmless Overparametrization in Two-layer Neural Networks

Overparametrized neural networks, where the number of active parameters is larger than the sample size, prove remarkably effective in modern deep learning practice. From the classical perspective, however, much fewer parameters are sufficient for optimal estimation and prediction, whereas overparametrization can be harmful even in the presence of explicit regularization. To reconcile this conflict, we present a generalization theory for overparametrized ReLU networks by incorporating an explicit regularizer based on the scaled variation norm. Interestingly, this regularizer is equivalent to the ridge from the angle of gradient-based optimization, but is similar to the group lasso in terms of controlling model complexity. By exploiting this ridge-lasso duality, we show that overparametrization is generally harmless to two-layer ReLU networks. In particular, the overparametrized estimators are minimax optimal up to a logarithmic factor. By contrast, we show that overparametrized random feature models suffer from the curse of dimensionality and thus are suboptimal

Adaptive deep learning for nonlinear time series models

In this paper, we develop a general theory for adaptive nonparametric estimation of the mean function of a non-stationary and nonlinear time series model using deep neural networks (DNNs). We first consider two types of DNN estimators, non-penalized and sparse-penalized DNN estimators, and establish their generalization error bounds for general non-stationary time series. We then derive minimax lower bounds for estimating mean functions belonging to a wide class of nonlinear autoregressive (AR) models that include nonlinear generalized additive AR, single index, and threshold AR models. Building upon the results, we show that the sparse-penalized DNN estimator is adaptive and attains the minimax optimal rates up to a poly-logarithmic factor for many nonlinear AR models. Through numerical simulations, we demonstrate the usefulness of the DNN methods for estimating nonlinear AR models with intrinsic low-dimensional structures and discontinuous or rough mean functions, which is consistent with our theory.Comment: 49 pages, 1 figur