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We analyze the loss landscape and expressiveness of practical deepconvolutional neural networks (CNNs) with shared weights and max poolinglayers. We show that such CNNs produce linearly independent features at a"wide" layer which has more neurons than the number of training samples. Thiscondition holds e.g. for the VGG network. Furthermore, we provide for such wideCNNs necessary and sufficient conditions for global minima with zero trainingerror. For the case where the wide layer is followed by a fully connected layerwe show that almost every critical point of the empirical loss is a globalminimum with zero training error. Our analysis suggests that both depth andwidth are very important in deep learning. While depth brings morerepresentational power and allows the network to learn high level features,width smoothes the optimization landscape of the loss function in the sensethat a sufficiently wide network has a well-behaved loss surface with almost nobad local minima.

Optimization Landscape and Expressivity of Deep CNNs