Series of Hessian-Vector Products for Tractable Saddle-Free Newton Optimisation of Neural Networks

Elre Talea Oldewage, Ross M Clarke, José Miguel Hernández-Lobato

Abstract: Despite their popularity in the field of continuous optimisation, second-order quasi-Newton methods are challenging to apply in machine learning, as the Hessian matrix is intractably large. This computational burden is exacerbated by the need to address non-convexity, for instance by modifying the Hessian's eigenvalues as in Saddle-Free Newton methods. We propose an optimisation algorithm which addresses both of these concerns – to our knowledge, the first efficiently-scalable optimisation algorithm to asymptotically use the exact inverse Hessian with absolute-value eigenvalues. Our method frames the problem as a series which principally square-roots and inverts the squared Hessian, then uses it to precondition a gradient vector, all without explicitly computing or eigendecomposing the Hessian. A truncation of this infinite series provides a new optimisation algorithm which is scalable and comparable to other first- and second-order optimisation methods in both runtime and optimisation performance. We demonstrate this in a variety of settings, including a ResNet-18 trained on CIFAR-10.

Submission Length: Regular submission (no more than 12 pages of main content)

Code: https://github.com/rmclarke/SeriesOfHessianVectorProducts

Assigned Action Editor: ~Robert_M._Gower1

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Submission Number: 1719