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Deep learning has shown high performances in various types of tasks fromvisual recognition to natural language processing, which indicates superiorflexibility and adaptivity of deep learning. To understand this phenomenontheoretically, we develop a new approximation and estimation error analysis ofdeep learning with the ReLU activation for functions in a Besov space and itsvariant with mixed smoothness. The Besov space is a considerably generalfunction space including the Holder space and Sobolev space, and especially cancapture spatial inhomogeneity of smoothness. Through the analysis in the Besovspace, it is shown that deep learning can achieve the minimax optimal rate andoutperform any non-adaptive (linear) estimator such as kernel ridge regression,which shows that deep learning has higher adaptivity to the spatialinhomogeneity of the target function than other estimators such as linear ones.In addition to this, it is shown that deep learning can avoid the curse ofdimensionality if the target function is in a mixed smooth Besov space. We alsoshow that the dependency of the convergence rate on the dimensionality is tightdue to its minimax optimality. These results support high adaptivity of deeplearning and its superior ability as a feature extractor.

Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality