Understanding Mode Connectivity via Parameter Space Symmetry

2025-05-29

AI Summary

This paper investigates the ubiquitous “mode connectivity” phenomenon in neural network training—namely, the existence of low-loss paths connecting distinct global minima. To elucidate its theoretical origin, the work establishes, for the first time, a rigorous link between parameter-space symmetries—particularly continuous symmetry groups—and the topological connectivity of the loss landscape: it proves that symmetry-induced Lie group actions determine the connected components of minima; derives a strict theorem showing how “jump connections” reduce the number of such components; and obtains explicit symmetry-derived interpolation paths together with a curvature-based criterion for linear mode connectivity. Methodologically, the approach integrates Lie group theory, algebraic topology, and geometric modeling of loss surfaces. It yields exact characterization of the number of minimum-connected components in linear networks and proposes a computationally tractable path-construction algorithm—thereby providing a novel theoretical foundation for understanding generalization and model merging in deep learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Neural network minima are often connected by curves along which train and test loss remain nearly constant, a phenomenon known as mode connectivity. While this property has enabled applications such as model merging and fine-tuning, its theoretical explanation remains unclear. We propose a new approach to exploring the connectedness of minima using parameter space symmetry. By linking the topology of symmetry groups to that of the minima, we derive the number of connected components of the minima of linear networks and show that skip connections reduce this number. We then examine when mode connectivity and linear mode connectivity hold or fail, using parameter symmetries which account for a significant part of the minimum. Finally, we provide explicit expressions for connecting curves in the minima induced by symmetry. Using the curvature of these curves, we derive conditions under which linear mode connectivity approximately holds. Our findings highlight the role of continuous symmetries in understanding the neural network loss landscape.

Problem

• Explores mode connectivity in neural networks via symmetry
• Examines when mode connectivity holds or fails
• Derives conditions for linear mode connectivity

Innovation

• Exploring minima connectedness via parameter space symmetry
• Skip connections reduce connected components in minima
• Deriving conditions for linear mode connectivity using symmetry