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Recent works have shown that gradient descent can find a global minimum forover-parameterized neural networks where the widths of all the hidden layersscale polynomially with $N$ ($N$ being the number of training samples). In thispaper, we prove that, for deep networks, a single layer of width $N$ followingthe input layer suffices to ensure a similar guarantee. In particular, all theremaining layers are allowed to have constant widths, and form a pyramidaltopology. We show an application of our result to the widely used LeCun'sinitialization and obtain an over-parameterization requirement for the singlewide layer of order $N^2.$

Global Convergence of Deep Networks with One Wide Layer Followed by Pyramidal Topology