On the Global Convergence of Gradient Descent for Over-parameterized
Models using Optimal Transport
L. Chizat, and F. Bach. (2018)cite arxiv:1805.09545Comment: Advances in Neural Information Processing Systems (NIPS), Dec 2018, Montréal, Canada.
Many tasks in machine learning and signal processing can be solved by
minimizing a convex function of a measure. This includes sparse spikes
deconvolution or training a neural network with a single hidden layer. For
these problems, we study a simple minimization method: the unknown measure is
discretized into a mixture of particles and a continuous-time gradient descent
is performed on their weights and positions. This is an idealization of the
usual way to train neural networks with a large hidden layer. We show that,
when initialized correctly and in the many-particle limit, this gradient flow,
although non-convex, converges to global minimizers. The proof involves
Wasserstein gradient flows, a by-product of optimal transport theory. Numerical
experiments show that this asymptotic behavior is already at play for a
reasonable number of particles, even in high dimension.