Complex Critical Points of Deep Linear Neural Networks
This work provides incremental theoretical insights into the optimization landscape of deep linear networks, primarily of interest to researchers in machine learning theory.
The authors tackled the problem of understanding the complex critical points of deep linear neural networks, improving the bound on the number of such points for single-hidden-layer networks and classifying patterns of zero-coordinate critical points, with computational experiments conducted using HomotopyContinuation.jl.
We extend the work of Mehta, Chen, Tang, and Hauenstein on computing the complex critical points of the loss function of deep linear neutral networks when the activation function is the identity function. For networks with a single hidden layer trained on a single data point we give an improved bound on the number of complex critical points of the loss function. We show that for any number of hidden layers complex critical points with zero coordinates arise in certain patterns which we completely classify for networks with one hidden layer. We report our results of computational experiments with varying network architectures defining small deep linear networks using HomotopyContinuation.jl.