  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.
