Recently, a new class of non-convex optimization problems motivated by the statistical problem of learning an acyclic directed graphical model from data has attracted significant interest. While existing work uses standard first-order optimization schemes to solve this problem, proving the global optimality of such approaches has proven elusive. The difficulty lies in the fact that unlike other non-convex problems in the literature, this problem is not "benign", and possesses multiple spurious solutions that standard approaches can easily get trapped in. In this paper, we prove that a simple path-following optimization scheme globally converges to the global minimum of the population loss in the bivariate setting.

翻译：近期，一类由从数据中学习有向无环图模型的统计问题所催生的非凸优化问题引起了广泛关注。尽管现有工作采用标准一阶优化方案来解决该问题，但这些方法的全局最优性证明却始终难以实现。其难点在于：与文献中其他非凸问题不同，该问题并非"良性"，存在多个虚假最优解，标准方法极易陷入其中。本文证明，在双变量设定下，一种简单的路径跟踪优化方案能够全局收敛至总体损失的全局最小值。