Exploring the Geometry and Topology of Neural Network Loss Landscapes
This work addresses the challenge of interpreting neural network generalization for researchers, though it is incremental as it builds on existing visualization techniques.
The paper tackled the problem of visualizing neural network loss landscapes to understand generalization, by introducing a 'jump and retrain' sampling method and using non-linear dimensionality reduction with PHATE to differentiate between well- and poorly-generalizing networks, showing that the sampled data provides more meaningful insights.
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.