Pointwise convergence theorem of generalized mini-batch gradient descent in deep neural network

The theoretical structure of deep neural network (DNN) has been clarified gradually. Imaizumi-Fukumizu (2019) and Suzuki (2019) clarified that the learning ability of DNN is superior to the previous theories when the target function is non-smooth functions. However, as far as the author is aware, none of the numerous works to date attempted to mathematically investigate what kind of DNN architectures really induce pointwise convergence of gradient descent (without any statistical argument), and this attempt seems to be closer to the practical DNNs. In this paper we restrict target functions to non-smooth indicator functions, and construct a deep neural network inducing pointwise convergence provided by mini-batch gradient descent process in ReLU-DNN.