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In this paper, we explore some basic questions on the complexity of trainingneural networks with ReLU activation function. We show that it is NP-hard totrain a two-hidden layer feedforward ReLU neural network. If dimension of theinput data and the network topology is fixed, then we show that there exists apolynomial time algorithm for the same training problem. We also show that ifsufficient over-parameterization is provided in the first hidden layer of ReLUneural network, then there is a polynomial time algorithm which finds weightssuch that output of the over-parameterized ReLU neural network matches with theoutput of the given data.

Complexity of Training ReLU Neural Network