  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.
