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

Numerical Exploration of Training Loss Level-Sets in Deep Neural Networks