Piecewise Strong Convexity of Neural Networks
This work addresses an open problem from COLT 2015, providing theoretical insights into optimization landscapes for neural networks, which is incremental but important for researchers in machine learning theory.
The authors tackled the problem of understanding the loss surface of ReLU neural networks with weight decay regularization, proving that the regularized loss is piecewise strongly convex on a set containing global minimizers, which leads to isolated local minima and no non-zero critical points below a training error threshold in linear networks.
We study the loss surface of a feed-forward neural network with ReLU non-linearities, regularized with weight decay. We show that the regularized loss function is piecewise strongly convex on an important open set which contains, under some conditions, all of its global minimizers. This is used to prove that local minima of the regularized loss function in this set are isolated, and that every differentiable critical point in this set is a local minimum, partially addressing an open problem given at the Conference on Learning Theory (COLT) 2015; our result is also applied to linear neural networks to show that with weight decay regularization, there are no non-zero critical points in a norm ball obtaining training error below a given threshold. We also include an experimental section where we validate our theoretical work and show that the regularized loss function is almost always piecewise strongly convex when restricted to stochastic gradient descent trajectories for three standard image classification problems.