The Goldilocks zone: Towards better understanding of neural network loss landscapes

TL;DR

This paper investigates the loss landscapes of neural networks, identifying a 'Goldilocks zone' characterized by high convexity and positive curvature, which correlates with better initialization and faster training.

Contribution

The study introduces the concept of the Goldilocks zone in neural network loss landscapes and links it to initialization quality and training efficiency.

Findings

The Goldilocks zone exhibits an excess of positive eigenvalues of the Hessian.

High convexity measures in this zone correlate with better network initialization.

Initializing networks within this zone leads to faster training on MNIST.

Abstract

We explore the loss landscape of fully-connected and convolutional neural networks using random, low-dimensional hyperplanes and hyperspheres. Evaluating the Hessian, $H$, of the loss function on these hypersurfaces, we observe 1) an unusual excess of the number of positive eigenvalues of $H$, and 2) a large value of $\mathrm{Tr}(H) / ||H||$ at a well defined range of configuration space radii, corresponding to a thick, hollow, spherical shell we refer to as the \textit{Goldilocks zone}. We observe this effect for fully-connected neural networks over a range of network widths and depths on MNIST and CIFAR-10 datasets with the $\mathrm{ReLU}$ and $\tanh$ non-linearities, and a similar effect for convolutional networks. Using our observations, we demonstrate a close connection between the Goldilocks zone, measures of local convexity/prevalence of positive curvature, and the suitability of a network initialization. We show that the high and stable accuracy reached when optimizing on random, low-dimensional hypersurfaces is directly related to the overlap between the hypersurface and the Goldilocks zone, and as a corollary demonstrate that the notion of intrinsic dimension is initialization-dependent. We note that common initialization techniques initialize neural networks in this particular region of unusually high convexity/prevalence of positive curvature, and offer a geometric intuition for their success. Furthermore, we demonstrate that initializing a neural network at a number of points and selecting for high measures of local convexity such as $\mathrm{Tr}(H) / ||H||$, number of positive eigenvalues of $H$, or low initial loss, leads to statistically significantly faster training on MNIST. Based on our observations, we hypothesize that the Goldilocks zone contains an unusually high density of suitable initialization configurations.