Before revision:  Our approach is based on estimating the Riemannian gradient from the objective function evaluations . We consider three settings for the objective function: (i) deterministic and smooth , (ii) stochastic and smooth , and (iii) composition of smooth and non-smooth parts.
After revision:  Our approach is based on estimating the Riemannian gradient from the objective function evaluations , based on a Riemannian version of the Gaussian smoothing technique. In particular, we consider the following settings for the objective function: (i) deterministic and smooth , (ii) stochastic and smooth , and (iii) composition of smooth and non-smooth parts.