In this work, we propose to train a deep neural network by distributed optimization over a graph. Two nonlinear functions are considered: the rectified linear unit (ReLU) and a linear unit with both lower and upper cutoffs (DCutLU). The problem reformulation over a graph is realized by explicitly representing ReLU or DCutLU using a set of slack variables. We then apply the alternating direction method of multipliers (ADMM) to update the weights of the network layerwise by solving subproblems of the reformulated problem. Empirical results suggest that the ADMM-based method is less sensitive to overfitting than the stochastic gradient descent (SGD) and Adam methods.