Mathematics > Optimization and Control
[Submitted on 24 May 2018 (v1), last revised 29 Oct 2018 (this version, v2)]
Title:On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport
View PDFAbstract: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.
Submission history
From: Lenaic Chizat [view email] [via CCSD proxy][v1] Thu, 24 May 2018 08:28:01 UTC (323 KB)
[v2] Mon, 29 Oct 2018 09:27:29 UTC (1,597 KB)
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