  This paper focus on investigating the distributed Riemannian stochastic
optimization problem on the Stiefel manifold for multi-agent systems, where all
the agents work collaboratively to optimize a function modeled by the average
of their expectation-valued local costs. Each agent only processes its own
local cost function and communicate with neighboring agents to achieve optimal
results while ensuring consensus. Since the local Riemannian gradient in
stochastic regimes cannot be directly calculated, we will estimate the gradient
by the average of a variable number of sampled gradient, which however brings
about noise to the system. We then propose a distributed Riemannian stochastic
optimization algorithm on the Stiefel manifold by combining the variable sample
size gradient approximation method with the gradient tracking dynamic. It is
worth noticing that the suitably chosen increasing sample size plays an
important role in improving the algorithm efficiency, as it reduces the noise
variance. In an expectation-valued sense, the iterates of all agents are proved
to converge to a stationary point (or neighborhood) with fixed step sizes. We
further establish the convergence rate of the iterates for the cases when the
sample size is exponentially increasing, polynomial increasing, or a constant,
respectively. Finally, numerical experiments are implemented to demonstrate the
theoretical results.
