Convexity Certificates from HessiansDownload PDF

Published: 31 Oct 2022, Last Modified: 10 Oct 2022NeurIPS 2022 AcceptReaders: Everyone
Keywords: disciplined convex programming, matrix calculus, classical machine learning
TL;DR: We provide, to the best of our knowledge,the first implementation of the Hessian approach for certifying the convexity of functions.
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 the Hessian approach is actually more powerful, that is, it can certify the convexity of a larger class of differentiable functions.
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