论文标题
具有不精确子问题解决方案和梯度聚集的梯度抽样方法
Gradient Sampling Methods with Inexact Subproblem Solutions and Gradient Aggregation
论文作者
论文摘要
事实证明，梯度采样（GS）是一种有效的方法，用于最小化可能是非凸和/或非滑动的目标函数。当代GS方法中最昂贵的计算成分是需要在每次迭代中解决凸二次子问题。在本文中，提出了一种策略，该策略允许使用这些子问题的不精确解决方案，如本文所证明的那样，可以在不丢失理论融合保证的情况下进行合并。数值实验表明，通过利用不精确的子问题解决方案，人们可以始终如一地减少GS方法所需的计算工作。此外，提出了一种策略，用于在解决子问题（可能不确定）之后汇总梯度信息，这是在捆绑方法中利用的，以进行非平滑优化。证明可以在没有理论融合保证的情况下引入聚合方案。数值实验表明，结合这种梯度聚集方法可大大减少GS方法所需的计算工作。
Gradient sampling (GS) has proved to be an effective methodology for the minimization of objective functions that may be nonconvex and/or nonsmooth. The most computationally expensive component of a contemporary GS method is the need to solve a convex quadratic subproblem in each iteration. In this paper, a strategy is proposed that allows the use of inexact solutions of these subproblems, which, as proved in the paper, can be incorporated without the loss of theoretical convergence guarantees. Numerical experiments show that by exploiting inexact subproblem solutions, one can consistently reduce the computational effort required by a GS method. Additionally, a strategy is proposed for aggregating gradient information after a subproblem is solved (potentially inexactly), as has been exploited in bundle methods for nonsmooth optimization. It is proved that the aggregation scheme can be introduced without the loss of theoretical convergence guarantees. Numerical experiments show that incorporating this gradient aggregation approach substantially reduces the computational effort required by a GS method.
