Papers › Convergence of Shallow ReLU Networks on Weakly Interacting Data
Convergence of Shallow ReLU Networks on Weakly Interacting Data
Léo Dana, Francis Bach, Loucas Pillaud-Vivien
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-{\L}ojasiewicz viewpoint along the gradient-flow trajectory, which provides an exponential rate of convergence of 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 1/n and 1/(√(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 1/(log(n)).
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