论文标题
Hessians的凸证书
Convexity Certificates from Hessians
论文作者
论文摘要
可区分凸函数的Hessian是阳性半限定。因此，检查给定功能的Hessian是一种自然证明凸度的方法。但是，实施这种方法并不简单，因为它需要允许其分析的Hessian表示。在这里，我们为一类功能实施了这种方法，这些功能足以支持经典的机器学习。对于这类功能，最近显示了如何计算其Hessians的计算图。我们展示了如何检查这些图表是否有积极的半精力。我们将Hessian方法的实施与良好的纪律凸编程（DCP）方法进行了比较，并证明Hessian方法至少与DCP方法的强大功能一样强大。此外，我们展示了DCP方法的最新实现，对于可区分的功能，Hessian方法实际上更强大。也就是说，它可以证明较大类别可区分功能的凸度。
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.
