Global rates of convergence for nonconvex optimization on manifolds
Provides foundational convergence guarantees for nonconvex optimization on manifolds, benefiting optimization theory and applications in machine learning and robotics.
The paper proves that Riemannian gradient descent and Riemannian trust regions achieve convergence rates of O(1/ε²) for first-order optimality and O(1/ε³) for second-order optimality on manifolds, matching unconstrained rates. These are the first deterministic global convergence rates for such problems.
We consider the minimization of a cost function $f$ on a manifold $M$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance $\varepsilon$. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of $f$ to the tangent spaces of $M$, both of these algorithms produce points with Riemannian gradient smaller than $\varepsilon$ in $O(1/\varepsilon^2)$ iterations. Furthermore, RTR returns a point where also the Riemannian Hessian's least eigenvalue is larger than $-\varepsilon$ in $O(1/\varepsilon^3)$ iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy $\varepsilon$ (up to constants) and hence are sharp in that sense. These are the first deterministic results for global rates of convergence to approximate first- and second-order Karush-Kuhn-Tucker points on manifolds. They apply in particular for optimization constrained to compact submanifolds of $\mathbb{R}^n$, under simpler assumptions.