Simultaneous Linear Connectivity of Neural Networks Modulo Permutation

Core Concepts

Neural networks exhibit permutation symmetry, where reordering neurons in each layer does not change the underlying function they compute. This contributes to the non-convexity of the networks' loss landscapes. Recent work has argued that permutation symmetries are the only sources of non-convexity, meaning there are essentially no loss barriers between trained networks if they are permuted appropriately. This work refines these arguments into three distinct claims of increasing strength, and provides empirical evidence for the strongest claim.

Abstract

The paper investigates different notions of linear connectivity of neural networks modulo permutation. It makes the following key observations: Existing evidence only supports "weak linear connectivity" - that for each pair of networks, there exist permutations that linearly connect them. The stronger claim of "strong linear connectivity" - that for each network, there exists one permutation that simultaneously connects it with other networks - is both intuitively and practically more desirable, as it would imply a convex loss landscape after accounting for permutation. The paper introduces an intermediate claim of "simultaneous weak linear connectivity" - that for certain sequences of networks, there exists one permutation that simultaneously aligns matching pairs of networks from these sequences. The paper provides empirical evidence for simultaneous weak linear connectivity: It shows that a single permutation can align SGD training trajectories, meaning the networks exhibit low loss barriers at each step of optimization. It also demonstrates that the same permutation can align sequences of iteratively pruned networks. Furthermore, the paper provides the first evidence towards strong linear connectivity, by showing that barriers decrease with increasing network width when interpolating among three networks. The paper also discusses limitations of weight matching and activation matching algorithms used for aligning networks, and how they relate to network stability and feature emergence during training.

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Simultaneous linear connectivity of neural networks modulo permutation

by Ekansh Sharm... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.06498.pdf

Deeper Inquiries

What are the implications of strong linear connectivity modulo permutation for model compression, transfer learning, and ensemble methods

Strong linear connectivity modulo permutation has significant implications for various areas in machine learning. Model Compression: Strong linear connectivity implies that a single permutation can align multiple independently trained networks, enabling the compression of multiple models into a single representation. This can lead to more efficient model storage and deployment, especially in resource-constrained environments. Transfer Learning: With strong linear connectivity, a single permutation can simultaneously align multiple networks, making it easier to transfer knowledge between models. This can enhance transfer learning capabilities by enabling the seamless transfer of representations learned across different models. Ensemble Methods: Strong linear connectivity allows for the creation of ensembles by aligning multiple networks. By combining the predictions of linearly connected models, ensemble methods can leverage diverse representations learned by each model, leading to improved performance and robustness. Overall, strong linear connectivity modulo permutation enhances model efficiency, transferability, and ensemble diversity in various machine learning applications.

How do the permutations that align networks relate to the underlying structure and representations learned by the networks

The permutations that align networks provide insights into the underlying structure and representations learned by the networks. Symmetry in Representations: Permutations that align networks reveal the symmetry present in the learned representations. The fact that networks can be aligned through permutations indicates that the networks capture similar underlying patterns and features, albeit in different orders or arrangements. Consistency in Learning: The ability to find permutations that align networks suggests that the networks learn consistent representations despite variations in initialization or training procedures. This consistency in learned representations highlights the robustness and generalizability of the models. Interpretability: Understanding the permutations that align networks can provide insights into the interpretability of the learned representations. By analyzing how different permutations affect the alignment and performance of networks, researchers can gain a deeper understanding of the features encoded by the models. In essence, the permutations that align networks offer a window into the structure and consistency of the representations learned by neural networks.

Can the insights on simultaneous weak linear connectivity be extended to other types of neural network architectures beyond feedforward networks, such as recurrent or graph neural networks

The insights on simultaneous weak linear connectivity can be extended to various types of neural network architectures beyond feedforward networks. Recurrent Neural Networks (RNNs): Similar principles of simultaneous weak linear connectivity can be applied to RNNs. By aligning sequences of RNNs trained on different initializations or datasets, researchers can explore the continuity of representations across time steps and sequences. Graph Neural Networks (GNNs): In GNNs, simultaneous weak linear connectivity can help align networks trained on different graph structures or node features. This alignment can facilitate the transfer of knowledge between graphs and enhance the interpretability of GNN representations. Transformer Networks: For transformer architectures, simultaneous weak linear connectivity can aid in understanding the evolution of attention patterns and representations across layers. Aligning transformer networks can reveal the consistency of learned representations and improve model interpretability. By extending the concept of simultaneous weak linear connectivity to diverse neural network architectures, researchers can gain deeper insights into the continuity and consistency of learned representations across different models and settings.