The Hessian of tall-skinny networks is easy to invert

TL;DR

This paper introduces an efficient exact algorithm for solving linear systems involving the Hessian of deep networks, enabling Hessian-inverse-vector products computation without full Hessian storage, scaling linearly with network layers.

Contribution

The paper presents a novel algorithm that computes Hessian-inverse-vector products efficiently, avoiding full Hessian computation and storage, with linear scaling in the number of layers.

Findings

The method computes Hessian-inverse-vector products without storing the Hessian.

It scales linearly with the number of layers, unlike naive approaches.

The approach is comparable in efficiency to Pearlmutter's Hessian-vector product algorithm.

Abstract

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.