Proving Linear Mode Connectivity of Neural Networks via Optimal Transport

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.

A General Framework For Proving The Equivariant Strong Lottery Ticket Hypothesis

The Strong Lottery Ticket Hypothesis (SLTH) stipulates the existence of a subnetwork within a sufficiently overparameterized (dense) neural network that -- when initialized randomly and without any training -- achieves the accuracy of a fully trained target network. Recent work by \citet{da2022proving} demonstrates that the SLTH can also be extended to translation equivariant networks -- i.e. CNNs -- with the same level of overparametrization as needed for SLTs in dense networks. However, modern neural networks are capable of incorporating more than just translation symmetry, and developing general equivariant architectures such as rotation and permutation has been a powerful design principle. In this paper, we generalize the SLTH to functions that preserve the action of the group $G$ -- i.e. $G$-equivariant network -- and prove, with high probability, that one can prune a randomly initialized overparametrized $G$-equivariant network to a $G$-equivariant subnetwork that approximates another fully trained $G$-equivariant network of fixed width and depth. We further prove that our prescribed overparametrization scheme is also optimal as a function of the error tolerance. We develop our theory for a large range of groups, including important ones such as subgroups of the Euclidean group $\text{E}(n)$ and subgroups of the symmetric group $G \leq \mathcal{S}_n$ -- allowing us to find SLTs for MLPs, CNNs, $\text{E}(2)$-steerable CNNs, and permutation equivariant networks as specific instantiations of our unified framework which completely extends prior work. Empirically, we verify our theory by pruning overparametrized $\text{E}(2)$-steerable CNNs and message passing GNNs to match the performance of trained target networks within a given error tolerance.