Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties

Understand

We propose a globally-accelerated, first-order method for the optimization of smooth and (strongly or not) geodesically-convex functions in a wide class of Hadamard manifolds.

• We achieve the same convergence rates as Nesterov's accelerated gradient descent, up to a multiplicative geometric penalty and log factors.
• Crucially, we can enforce our method to stay within a compact set we define.
• Prior fully accelerated works \emph{resort to assuming} that the iterates of their algorithms stay in some pre-specified compact set, except for two previous methods of limited applicability.