The Hessian of tall-skinny networks is easy to invert
This addresses computational bottlenecks for researchers and practitioners in deep learning who need to invert Hessians for optimization or analysis.
The paper tackles the problem of efficiently solving linear systems involving the Hessian of deep neural networks, presenting an exact algorithm that computes Hessian-inverse-vector products with linear scaling in layers, avoiding quadratic storage and cubic time costs of naive methods.
We describe an exact algorithm to solve linear systems of the form $Hx=b$ where $H$ is the Hessian of a deep net. The method computes Hessian-inverse-vector products without storing the Hessian or its inverse. It requires time and storage that scale linearly in the number of layers. This is in contrast to the naive approach of first computing the Hessian, then solving the linear system, which takes storage and time that are respectively quadratic and cubic in the number of layers. The Hessian-inverse-vector product method scales roughly like Pearlmutter's algorithm for computing Hessian-vector products.