On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport

Lenaic Chizat (1) , Francis Bach (1)

1 SIERRA - Statistical Machine Learning and Parsimony

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.

Mots clés: Optimal transport; Wasserstein gradient flows; Convex optimization; Non-convex optimization

Domaines: Machine Learning [stat.ML]; Optimisation et contrôle [math.OC]; Réseau de neurones [cs.NE]

Lenaic Chizat, Francis Bach. On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport. Advances in Neural Information Processing Systems (NIPS), Dec 2018, Montréal, Canada. ⟨hal-01798792v1⟩