On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport.
ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 31 (NIPS 2018)(2018)
摘要
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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关键词
gradient descent,machine learning,neural networks,global convergence,neural network,optimal transport,asymptotic behavior,signal processing,convex function,gradient flow,the proof
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