论文标题
Riemannian随机固定点优化算法
Riemannian Stochastic Fixed Point Optimization Algorithm
论文作者
论文摘要
本文考虑了在riemannian歧管上的quasinonexpansive映射的固定点集中的随机优化问题。该问题使我们能够考虑复杂集的Riemannian层次优化问题，例如许多封闭的凸集集合的相交，非平滑凸功能的所有最小化器的集合以及无量子凸出功能的Sublevel集合的相交。我们专注于自适应学习率优化算法，该算法适应了步骤尺寸（在机器学习领域中称为学习率），以迅速找到最佳解决方案。然后，我们提出了一种Riemannian随机固定点优化算法，该算法将Riemannian歧管上的固定点近似方法与自适应学习率优化算法结合在一起。我们还提供了对非平滑凸和平滑非凸优化的建议算法的收敛分析。分析结果表明，使用恒定的步长尺寸，提出的算法近似于该问题的解决方案。考虑到阶数序列减少的情况，表明所提出的算法通过保证的收敛速率解决了问题。本文还提供了数值比较，这些比较证明了基于自适应学习率优化算法（例如Adam和Amsgrad）的公式使用公式的提议算法的有效性。
This paper considers a stochastic optimization problem over the fixed point sets of quasinonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian hierarchical optimization problems over complicated sets, such as the intersection of many closed convex sets, the set of all minimizers of a nonsmooth convex function, and the intersection of sublevel sets of nonsmooth convex functions. We focus on adaptive learning rate optimization algorithms, which adapt step-sizes (referred to as learning rates in the machine learning field) to find optimal solutions quickly. We then propose a Riemannian stochastic fixed point optimization algorithm, which combines fixed point approximation methods on Riemannian manifolds with the adaptive learning rate optimization algorithms. We also give convergence analyses of the proposed algorithm for nonsmooth convex and smooth nonconvex optimization. The analysis results indicate that, with small constant step-sizes, the proposed algorithm approximates a solution to the problem. Consideration of the case in which step-size sequences are diminishing demonstrates that the proposed algorithm solves the problem with a guaranteed convergence rate. This paper also provides numerical comparisons that demonstrate the effectiveness of the proposed algorithms with formulas based on the adaptive learning rate optimization algorithms, such as Adam and AMSGrad.
