Convergence of the Gradient Flow for Shallow ReLU Networks on Weakly Interacting Data

Léo Dana, Loucas Pillaud-Vivien, Francis Bach

Keywords: Shallow Neural Networks, Gradient Flow, High-dimension, Convergence rate

TL;DR: We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on n data points, when the input lies in very high dimension.

Abstract: We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on $n$ data points. Our main contribution leverages the high dimensionality of the ambient space, which implies low correlation of the input samples, to demonstrate that a network with width of order $\log(n)$ neurons suffices for global convergence with high probability. Our analysis uses a Polyak–Łojasiewicz viewpoint along the gradient-flow trajectory, which provides an exponential rate of convergence of $\frac{1}{n}$. When the data are exactly orthogonal, we give further refined characterizations of the convergence speed, proving its asymptotic behavior lies between the orders $\frac{1}{n}$ and $\frac{1}{\sqrt{n}}$, and exhibiting a phase-transition phenomenon in the convergence rate, during which it evolves from the lower bound to the upper, and in a relative time of order $\frac{1}{\log(n)}$.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 16591