  The loss landscapes of deep neural networks are not well understood due to
their high nonconvexity. Empirically, the local minima of these loss functions
can be connected by a learned curve in model space, along which the loss
remains nearly constant; a feature known as mode connectivity. Yet, current
curve finding algorithms do not consider the influence of symmetry in the loss
surface created by model weight permutations. We propose a more general
framework to investigate the effect of symmetry on landscape connectivity by
accounting for the weight permutations of the networks being connected. To
approximate the optimal permutation, we introduce an inexpensive heuristic
referred to as neuron alignment. Neuron alignment promotes similarity between
the distribution of intermediate activations of models along the curve. We
provide theoretical analysis establishing the benefit of alignment to mode
connectivity based on this simple heuristic. We empirically verify that the
permutation given by alignment is locally optimal via a proximal alternating
minimization scheme. Empirically, optimizing the weight permutation is critical
for efficiently learning a simple, planar, low-loss curve between networks that
successfully generalizes. Our alignment method can significantly alleviate the
recently identified robust loss barrier on the path connecting two adversarial
robust models and find more robust and accurate models on the path.

Machine Learning;Optimization and Control;Machine Learning