Acceleration via silver step-size on Riemannian manifolds with applications to Wasserstein space

There is extensive literature on accelerating first-order optimization methods in a Euclidean setting. Under which conditions such acceleration is feasible in Riemannian optimization problems is an active area of research. Motivated by the recent success of dynamic stepsize methods in the Euclidean setting, we undertake a study of such algorithms in the Riemannian setting. We provide the new class of algorithms determined by the choice of vector transport that allows the dynamic stepsize acceleration on Riemannian manifolds for the function classes associated with the corresponding vector transport. As a core application, we show our algorithm recovers the standard Wasserstein gradient descent on 2-Wasserstein space, and as a result provides the first provable accelerated gradient method in Wasserstein space. In addition, we validate the numerical strength of the algorithm for standard benchmark tasks on the space of symmetric positive definite matrices.