Understanding Mode Connectivity via Parameter Space Symmetry

TL;DR

This paper investigates the role of parameter space symmetry in neural network loss landscapes, explaining mode connectivity through symmetry groups and providing explicit connecting curves, with insights into when linear mode connectivity holds.

Contribution

It introduces a novel symmetry-based framework to analyze mode connectivity, deriving conditions and explicit curves that connect minima in neural networks.

Findings

Symmetry groups relate to the topology of minima.

Skip connections reduce the number of connected components.

Explicit connecting curves are derived from symmetry considerations.

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.