Exploring the Geometry and Topology of Neural Network Loss Landscapes (2102.00485v2)

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.

• The paper introduces a jump and retrain procedure that samples loss landscapes to provide deeper insight into neural network generalization.
• It employs PHATE for non-linear dimensionality reduction, enabling clear visualization of differences between networks with varying performance.
• It integrates computational homology to quantitatively analyze training trajectories, offering a framework to enhance neural network design.

The paper "Exploring the Geometry and Topology of Neural Network Loss Landscapes" explores the intriguing connection between neural network generalization performance and the geometry of the loss landscape near local minima. The general idea is that the characteristics of these loss landscapes can provide significant insight into how well neural networks will perform on unseen data.

Key Contributions

1. Jump and Retrain Procedure: The authors introduce a new "jump and retrain" method to sample the loss landscape more effectively. This method aims to capture portions of the loss landscape that are more informative about the neural network's generalization abilities. By sampling in this manner, the process extends beyond traditional linear sampling, which is often constrained to lines or planes, thus providing richer data.
2. Non-linear Dimensionality Reduction: The paper utilizes PHATE (Potential of Heat-diffusion for Affinity-based Transition Embedding), a sophisticated tool for trajectory and manifold preservation, to achieve non-linear dimensionality reduction. The application of PHATE to the "jump and retrain" trajectories allows for improved visualization, making it easier to discern the differences between networks that generalize well compared to those that don't. This approach moves beyond the limitations of traditional linear methods, opening up more dimensions for analysis.
3. Computational Homology: The paper integrates computational homology with PHATE visualizations to measure differences in the loss landscape trajectories. This technique provides a quantitative framework to analyze and compare the topological features of the paths taken by networks during training, offering a deeper understanding of why some networks may be performing better than others.

Insights and Implications

The paper's methodologies and findings contribute significant insights into understanding neural networks' generalization capabilities. By employing more sophisticated sampling and visualization techniques, the researchers aim to uncover and analyze features of the loss landscape that are closely linked to network performance, offering a potentially impactful direction for future research in neural network training and evaluation.

Through its novel approaches, the paper not only enhances visualization but also introduces quantitative measures that could guide improvements in neural network design and training processes, ultimately leading to more robust generalization.