  The energy landscape of high-dimensional non-convex optimization problems is
crucial to understanding the effectiveness of modern deep neural network
architectures. Recent works have experimentally shown that two different
solutions found after two runs of a stochastic training are often connected by
very simple continuous paths (e.g., linear) modulo a permutation of the
weights. In this paper, we provide a framework theoretically explaining this
empirical observation. Based on convergence rates in Wasserstein distance of
empirical measures, we show that, with high probability, two wide enough
two-layer neural networks trained with stochastic gradient descent are linearly
connected. Additionally, we express upper and lower bounds on the width of each
layer of two deep neural networks with independent neuron weights to be
linearly connected. Finally, we empirically demonstrate the validity of our
approach by showing how the dimension of the support of the weight distribution
of neurons, which dictates Wasserstein convergence rates is correlated with
linear mode connectivity.
