From Nesterov's Estimate Sequence to Riemannian Acceleration
This enables faster optimization for problems in machine learning and other fields where data naturally lies on curved spaces like spheres or manifolds.
The authors developed the first globally accelerated gradient method for optimization on Riemannian manifolds, achieving an O(1/k^2) convergence rate that matches the optimal Euclidean rate.
We propose the first global accelerated gradient method for Riemannian manifolds. Toward establishing our result we revisit Nesterov's estimate sequence technique and develop an alternative analysis for it that may also be of independent interest. Then, we extend this analysis to the Riemannian setting, localizing the key difficulty due to non-Euclidean structure into a certain ``metric distortion.'' We control this distortion by developing a novel geometric inequality, which permits us to propose and analyze a Riemannian counterpart to Nesterov's accelerated gradient method.