  We present a computational method for empirically characterizing the training
loss level-sets of deep neural networks. Our method numerically constructs a
path in parameter space that is constrained to a set with a fixed near-zero
training loss. By measuring regularization functions and test loss at different
points within this path, we examine how different points in the parameter space
with the same fixed training loss compare in terms of generalization ability.
We also compare this method for finding regularized points with the more
typical method, that uses objective functions which are weighted sums of
training loss and regularization terms. We apply dimensionality reduction to
the traversed paths in order to visualize the loss level sets in a
well-regularized region of parameter space. Our results provide new information
about the loss landscape of deep neural networks, as well as a new strategy for
reducing test loss.
