# Optimization Landscape and Expressivity of Deep CNNs

arXiv: https://arxiv.org/abs/1710.10928 · HTML: https://mlsanity.com/paper/1710.10928/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2017.

## Built on by (most cited first, 2 of 2)

- [Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization](https://mlsanity.com/paper/2102.12430/) (2021)
- [GridDehazeNet+: An Enhanced Multi-Scale Network with Intra-Task Knowledge Transfer for Single Image Dehazing](https://mlsanity.com/paper/2103.13998/) (2021)

## Abstract (arXiv)

We analyze the loss landscape and expressiveness of practical deep convolutional neural networks (CNNs) with shared weights and max pooling layers. We show that such CNNs produce linearly independent features at a "wide" layer which has more neurons than the number of training samples. This condition holds e.g. for the VGG network. Furthermore, we provide for such wide CNNs necessary and sufficient conditions for global minima with zero training error. For the case where the wide layer is followed by a fully connected layer we show that almost every critical point of the empirical loss is a global minimum with zero training error. Our analysis suggests that both depth and width are very important in deep learning. While depth brings more representational power and allows the network to learn high level features, width smoothes the optimization landscape of the loss function in the sense that a sufficiently wide network has a well-behaved loss surface with almost no bad local minima.