Estimation error analysis of deep learning on the regression problem on the variable exponent Besov space
This work addresses theoretical error analysis for deep learning in non-uniform smoothness settings, which is incremental as it builds on existing approximation theory.
The paper tackles the problem of analyzing estimation errors for deep learning in regression on variable exponent Besov spaces, where smoothness varies with input location, and shows that deep learning's adaptivity leads to superior convergence rates compared to linear estimators, especially when regions of low smoothness are small and dimensions are high.
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.