Visualizing high-dimensional loss landscapes with Hessian directions

TL;DR

This paper investigates how high-dimensional loss landscapes of neural networks can be visualized and understood through the lens of differential geometry and random projections, revealing limitations of random projections and proposing Hessian-based methods.

Contribution

It introduces a novel analysis linking curvature in high-dimensional loss spaces to lower-dimensional projections and proposes Hessian-based projections for better landscape visualization.

Findings

Random projections often misidentify saddle points in loss landscapes.

Hessian-based projections better capture true curvature directions.

The methods scale to models with millions of parameters.

Abstract

Analyzing geometric properties of high-dimensional loss functions, such as local curvature and the existence of other optima around a certain point in loss space, can help provide a better understanding of the interplay between neural network structure, implementation attributes, and learning performance. In this work, we combine concepts from high-dimensional probability and differential geometry to study how curvature properties in lower-dimensional loss representations depend on those in the original loss space. We show that saddle points in the original space are rarely correctly identified as such in expected lower-dimensional representations if random projections are used. The principal curvature in the expected lower-dimensional representation is proportional to the mean curvature in the original loss space. Hence, the mean curvature in the original loss space determines if saddle points appear, on average, as either minima, maxima, or almost flat regions. We use the connection between expected curvature in random projections and mean curvature in the original space (i.e., the normalized Hessian trace) to compute Hutchinson-type trace estimates without calculating Hessian-vector products as in the original Hutchinson method. Because random projections are not suitable to correctly identify saddle information, we propose to study projections along dominant Hessian directions that are associated with the largest and smallest principal curvatures. We connect our findings to the ongoing debate on loss landscape flatness and generalizability. Finally, for different common image classifiers and a function approximator, we show and compare random and Hessian projections of loss landscapes with up to about $7\times 10^6$ parameters.