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
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : 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.
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Communication dans un congrès
Advances in Neural Information Processing Systems (NIPS), Dec 2018, Montréal, Canada
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Contributeur : Lénaïc Chizat <>
Soumis le : samedi 27 octobre 2018 - 08:19:47
Dernière modification le : vendredi 9 novembre 2018 - 08:51:39

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  • HAL Id : hal-01798792, version 2
  • ARXIV : 1805.09545

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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-01798792v2〉

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