Equivariant Deep Weight Space Alignment

Permutation symmetries of deep networks make simple operations like model averaging and similarity estimation challenging. In many cases, aligning the weights of the networks, i.e., finding optimal permutations between their weights, is necessary. More generally, weight alignment is essential for a wide range of applications, from model merging, through exploring the optimization landscape of deep neural networks, to defining meaningful distance functions between neural networks. Unfortunately, weight alignment is an NP-hard problem. Prior research has mainly focused on solving relaxed versions of the alignment problem, leading to either time-consuming methods or sub-optimal solutions. To accelerate the alignment process and improve its quality, we propose a novel framework aimed at learning to solve the weight alignment problem, which we name Deep-Align. To that end, we first demonstrate that weight alignment adheres to two fundamental symmetries and then, propose a deep architecture that respects these symmetries. Notably, our framework does not require any labeled data. We provide a theoretical analysis of our approach and evaluate Deep-Align on several types of network architectures and learning setups. Our experimental results indicate that a feed-forward pass with Deep-Align produces better or equivalent alignments compared to those produced by current optimization algorithms. Additionally, our alignments can be used as an initialization for other methods to gain even better solutions with a significant speedup in convergence.