Exact Solutions of a Deep Linear Network

This work finds the exact solutions to a deep linear network with weight decay and stochastic neurons, a fundamental model for understanding the landscape of neural networks. Our result implies that weight decay strongly interacts with the model architecture and can create bad minima in a network with more than $1$ hidden layer, qualitatively different for a network with only $1$ hidden layer. As an application, we also analyze stochastic nets and show that their prediction variance vanishes to zero as the stochasticity, the width, or the depth tends to infinity.