Understanding Mode Connectivity via Parameter Space Symmetry
This work provides theoretical insights into neural network loss landscapes, which could aid in model merging and fine-tuning, but it is incremental as it builds on existing concepts of mode connectivity.
The paper tackled the problem of explaining mode connectivity in neural networks by linking it to parameter space symmetry, showing that skip connections reduce the number of connected components in minima and deriving conditions for when linear mode connectivity holds.
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.