An Investigation into Neural Net Optimization via Hessian Eigenvalue Density
This work addresses the challenge of optimizing deep neural networks for researchers and practitioners, providing insights into structural features that affect training speed, though it is incremental in building on existing literature on Hessian analysis.
The paper tackled the problem of understanding optimization dynamics in deep neural networks by analyzing the evolution of the Hessian spectrum, revealing that non-batch normalized networks develop large isolated eigenvalues and gradient concentration, which slow optimization, while batch normalized networks avoid these issues, leading to faster convergence.
To understand the dynamics of optimization in deep neural networks, we develop a tool to study the evolution of the entire Hessian spectrum throughout the optimization process. Using this, we study a number of hypotheses concerning smoothness, curvature, and sharpness in the deep learning literature. We then thoroughly analyze a crucial structural feature of the spectra: in non-batch normalized networks, we observe the rapid appearance of large isolated eigenvalues in the spectrum, along with a surprising concentration of the gradient in the corresponding eigenspaces. In batch normalized networks, these two effects are almost absent. We characterize these effects, and explain how they affect optimization speed through both theory and experiments. As part of this work, we adapt advanced tools from numerical linear algebra that allow scalable and accurate estimation of the entire Hessian spectrum of ImageNet-scale neural networks; this technique may be of independent interest in other applications.