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Deep learning has achieved notable success in various fields, including imageand speech recognition. One of the factors in the successful performance ofdeep learning is its high feature extraction ability. In this study, we focuson the adaptivity of deep learning; consequently, we treat the variableexponent Besov space, which has a different smoothness depending on the inputlocation $x$. In other words, the difficulty of the estimation is not uniformwithin the domain. We analyze the general approximation error of the variableexponent Besov space and the approximation and estimation errors of deeplearning. We note that the improvement based on adaptivity is remarkable whenthe region upon which the target function has less smoothness is small and thedimension is large. Moreover, the superiority to linear estimators is shownwith respect to the convergence rate of the estimation error.

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