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

## TL;DR

This paper demonstrates that deep neural networks are minimax optimal for certain sparse and discontinuous function classes, outperforming linear estimators and highlighting the benefits of parameter sharing.

## Contribution

It provides a theoretical analysis showing deep learning's superiority over linear methods on sparse and discontinuous function classes, with insights into model complexity reduction.

## Key findings

- Deep learning attains nearly minimax-optimal rates on sparse function classes.
- Linear estimators are suboptimal on non-convex and sparse function classes.
- Parameter sharing in neural networks reduces model complexity significantly.

## Abstract

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.