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Deep learning has gained huge empirical successes in large-scaleclassification problems. In contrast, there is a lack of statisticalunderstanding about deep learning methods, particularly in the minimaxoptimality perspective. For instance, in the classical smooth decision boundarysetting, existing deep neural network (DNN) approaches are rate-suboptimal, andit remains elusive how to construct minimax optimal DNN classifiers. Moreover,it is interesting to explore whether DNN classifiers can circumvent the curseof dimensionality in handling high-dimensional data. The contributions of thispaper are two-fold. First, based on a localized margin framework, we discoverthe source of suboptimality of existing DNN approaches. Motivated by this, wepropose a new deep learning classifier using a divide-and-conquer technique:DNN classifiers are constructed on each local region and then aggregated to aglobal one. We further propose a localized version of the classical Tsybakov'snoise condition, under which statistical optimality of our new classifier isestablished. Second, we show that DNN classifiers can adapt to low-dimensionaldata structures and circumvent the curse of dimensionality in the sense thatthe minimax rate only depends on the effective dimension, potentially muchsmaller than the actual data dimension. Numerical experiments are conducted onsimulated data to corroborate our theoretical results.

Minimax Optimal Deep Neural Network Classifiers Under Smooth Decision Boundary