On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport
This provides a theoretical foundation for training over-parameterized models like neural networks, addressing a fundamental problem in machine learning optimization, though it is incremental in extending existing convergence results.
The paper tackles the global convergence of gradient descent for over-parameterized models, such as neural networks with a single hidden layer, by discretizing measures into particles and performing continuous-time gradient descent. It shows that, with correct initialization and in the many-particle limit, this non-convex gradient flow converges to global minimizers, as supported by numerical experiments in high dimensions.
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.