Riemannian Stochastic Gradient Method for Nested Composition Optimization

This work considers the optimization of the composition of functions in a nested form over Riemannian manifolds where each function contains an expectation. This problem type is gaining popularity in applications such as policy evaluation in reinforcement learning or model customization in meta-learning. The standard Riemannian stochastic gradient methods for non-compositional optimization cannot be directly applied as the stochastic approximation of the inner functions creates biases in the gradients of the outer functions. For two-level composition optimization, we present a Riemannian Stochastic Composition Gradient Descent (R-SCGD) method that finds an approximate stationary point, with expected squared Riemannian gradient smaller than $\epsilon$, in $\mathcal{O}\left(\epsilon^{-2}\right)$ calls to the stochastic gradient oracle of the outer function and stochastic function and gradient oracles of the inner function.