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Deep neural networks are highly expressive machine learning models with theability to interpolate arbitrary datasets. Deep nets are typically optimizedvia first-order methods and the optimization process crucially depends on thecharacteristics of the network as well as the dataset. This work sheds light onthe relation between the network size and the properties of the dataset with anemphasis on deep residual networks (ResNets). Our contribution is that if thenetwork Jacobian is full rank, gradient descent for the quadratic loss andsmooth activation converges to the global minima even if the network width $m$of the ResNet scales linearly with the sample size $n$, and independently fromthe network depth. To the best of our knowledge, this is the first work whichprovides a theoretical guarantee for the convergence of neural networks in the$m=\Omega(n)$ regime.

Stronger Convergence Results for Deep Residual Networks: Network Width Scales Linearly with Training Data Size