Exploring the Geometry and Topology of Neural Network Loss Landscapes

TL;DR

This paper introduces a new method combining a 'jump and retrain' sampling technique with non-linear dimensionality reduction and topological analysis to better understand neural network loss landscapes and their relation to generalization.

Contribution

It proposes a novel sampling procedure and applies advanced visualization and topological tools to analyze loss landscapes, improving insights into neural network generalization.

Findings

The jump and retrain method captures meaningful loss landscape features.

PHATE visualization reveals differences between well and poorly generalizing networks.

Topological analysis quantifies trajectory differences in the loss landscape.

Abstract

Recent work has established clear links between the generalization performance of trained neural networks and the geometry of their loss landscape near the local minima to which they converge. This suggests that qualitative and quantitative examination of the loss landscape geometry could yield insights about neural network generalization performance during training. To this end, researchers have proposed visualizing the loss landscape through the use of simple dimensionality reduction techniques. However, such visualization methods have been limited by their linear nature and only capture features in one or two dimensions, thus restricting sampling of the loss landscape to lines or planes. Here, we expand and improve upon these in three ways. First, we present a novel "jump and retrain" procedure for sampling relevant portions of the loss landscape. We show that the resulting sampled data holds more meaningful information about the network's ability to generalize. Next, we show that non-linear dimensionality reduction of the jump and retrain trajectories via PHATE, a trajectory and manifold-preserving method, allows us to visualize differences between networks that are generalizing well vs poorly. Finally, we combine PHATE trajectories with a computational homology characterization to quantify trajectory differences.