Estimation error analysis of deep learning on the regression problem on the variable exponent Besov space

TL;DR

This paper analyzes the approximation and estimation errors of deep learning in variable exponent Besov spaces, highlighting its adaptivity and superiority over linear estimators, especially in high-dimensional, non-uniform smoothness scenarios.

Contribution

It provides a theoretical analysis of deep learning's adaptivity in variable exponent Besov spaces, demonstrating improved convergence rates over linear estimators.

Findings

Deep learning shows remarkable adaptivity in variable smoothness regions.

Superiority over linear estimators in convergence rate.

Effectiveness increases with smaller low-smoothness regions and higher dimensions.

Abstract

Deep learning has achieved notable success in various fields, including image and speech recognition. One of the factors in the successful performance of deep learning is its high feature extraction ability. In this study, we focus on the adaptivity of deep learning; consequently, we treat the variable exponent Besov space, which has a different smoothness depending on the input location $x$. In other words, the difficulty of the estimation is not uniform within the domain. We analyze the general approximation error of the variable exponent Besov space and the approximation and estimation errors of deep learning. We note that the improvement based on adaptivity is remarkable when the region upon which the target function has less smoothness is small and the dimension is large. Moreover, the superiority to linear estimators is shown with respect to the convergence rate of the estimation error.