论文标题
一种自适应高阶方法，用于查找非convex优化的三阶关键点
An Adaptive High Order Method for Finding Third-Order Critical Points of Nonconvex Optimization
论文作者
论文摘要
众所周知，对于非convex优化，找到全球最佳量极具挑战性。关于{Anandkumar2016效率，Cartis2018秒，Cartis2020Sharp，Chen2019high}的一些最近的努力{Anandkumar2016效率，用于计算高阶临界点的优化方法，这些方法可以排除所谓的日益鞍点，并具有更好质量的解决方案。 \ cite {Anandkumar2016效率，Cartis2018秒，Cartis2020Sharp，Chen2019high}中的Desipte理论发展，相应的数值实验丢失了。在本文中，我们提出了一种可实施的高级方法，旨在找到三阶关键点，该方法称为自适应高阶方法（AHOM）。这是通过求解``更轻松''子问题的方法来实现的，并将参数调整的自适应策略纳入算法的每种迭代中。建立了所提出方法的迭代复杂性。提供了一些初步的数值结果，以表明AHOM能够逃脱退化的鞍点，其中二阶方法可能会被卡住。
It is well known that finding a global optimum is extremely challenging for nonconvex optimization. There are some recent efforts \cite{anandkumar2016efficient, cartis2018second, cartis2020sharp, chen2019high} regarding the optimization methods for computing higher-order critical points, which can exclude the so-called degenerate saddle points and reach a solution with better quality. Desipte theoretical development in \cite{anandkumar2016efficient, cartis2018second, cartis2020sharp, chen2019high}, the corresponding numerical experiments are missing. In this paper, we propose an implementable higher-order method, named adaptive high order method (AHOM), that aims to find the third-order critical points. This is achieved by solving an ``easier'' subproblem and incorporating the adaptive strategy of parameter-tuning in each iteration of the algorithm. The iteration complexity of the proposed method is established. Some preliminary numerical results are provided to show AHOM is able to escape the degenerate saddle points, where the second-order method could possibly get stuck.
