论文标题
无启示录的一阶低级优化算法，最多可以降低迭代级别的降低尝试
An apocalypse-free first-order low-rank optimization algorithm with at most one rank reduction attempt per iteration
论文作者
论文摘要
我们考虑了在实际决定性品种上使用局部Lipschitz连续梯度最小化可区分功能的问题，并提出了一种旨在找到该问题的固定点的一阶算法。该算法应用了Schneider和Uschmajew（2015）提出的无缩回下降方法的步骤，同时考虑了数值等级来尝试降级。我们证明该算法会产生一系列迭代的静态累积点，因此不遵循Levin，Kileel和Boumal（2022）所描述的所谓启示录。此外，该算法的排名降低机制最多需要每次迭代的排名降低尝试，与$ \ mathrm {p}^2 \ mathrm {gdrm {gdr} $ algorithm之一相反，奥尔基尔，加利维尔，加利维尔（Gallivan），加利维尔（Gallivan）和absil（2022）需要等级的数量，即划分数量的数量。
We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient over the real determinantal variety, and present a first-order algorithm designed to find stationary points of that problem. This algorithm applies steps of a retraction-free descent method proposed by Schneider and Uschmajew (2015), while taking the numerical rank into account to attempt rank reductions. We prove that this algorithm produces a sequence of iterates the accumulation points of which are stationary, and therefore does not follow the so-called apocalypses described by Levin, Kileel, and Boumal (2022). Moreover, the rank reduction mechanism of this algorithm requires at most one rank reduction attempt per iteration, in contrast with the one of the $\mathrm{P}^2\mathrm{GDR}$ algorithm introduced by Olikier, Gallivan, and Absil (2022) which can require a number of rank reduction attempts equal to the rank of the iterate in the worst-case scenario.
