Before revision:  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:  We consider three settings for the objective function: (i) stochastic and gradient-Lipschitz (in both nonconvex and geodesic convex settings) , (ii) stochastic and smooth , and (iii) composition of smooth and non-smooth parts.