Convexity Certificates from Hessians

TL;DR

This paper presents a method to certify convexity of differentiable functions by analyzing Hessian graphs, demonstrating it is at least as powerful as, and often more powerful than, traditional disciplined convex programming methods.

Contribution

The paper introduces a Hessian-based approach for convexity certification that surpasses DCP in power for differentiable functions, with practical implementation and comparison.

Findings

Hessian approach is at least as powerful as DCP for differentiable functions.

Hessian approach can certify convexity for a larger class of functions.

Implementation shows practical viability of the Hessian method.

Abstract

The Hessian of a differentiable convex function is positive semidefinite. Therefore, checking the Hessian of a given function is a natural approach to certify convexity. However, implementing this approach is not straightforward since it requires a representation of the Hessian that allows its analysis. Here, we implement this approach for a class of functions that is rich enough to support classical machine learning. For this class of functions, it was recently shown how to compute computational graphs of their Hessians. We show how to check these graphs for positive semidefiniteness. We compare our implementation of the Hessian approach with the well-established disciplined convex programming (DCP) approach and prove that the Hessian approach is at least as powerful as the DCP approach for differentiable functions. Furthermore, we show for a state-of-the-art implementation of the DCP approach that, for differentiable functions, the Hessian approach is actually more powerful. That is, it can certify the convexity of a larger class of differentiable functions.