Derivative-free Riemannian optimization (DFRO) aims to minimize an objective function using only function evaluations, under the constraint that the decision variables lie on a Riemannian manifold. The rapid increase in problem dimensions over the years calls for computationally cheap DFRO algorithms, that is, algorithms requiring as few function evaluations and retractions as possible. We propose … Read more
In this work, we propose a method for minimizing non-convex functions with Lipschitz continuous \(p\)th-order derivatives, starting from \(p \geq 1\). The method, however, only requires derivative information up to order \((p-1)\), since the \(p\)th-order derivatives are approximated via finite differences. To ensure oracle efficiency, instead of computing finite-difference approximations at every iteration, we reuse … Read more
We consider the problem of accurate computation of the finite difference $f(\x+\s)-f(\x)$ when $\Vert\s\Vert$ is very small. Direct evaluation of this difference in floating point arithmetic succumbs to cancellation error and yields 0 when $\s$ is sufficiently small. Nonetheless, accurate computation of this finite difference is required by many optimization algorithms for a “sufficient decrease” … Read more