Accelerated gradient methods on the Grassmann and Stiefel manifolds

In this paper we extend a nonconvex version of Nesterov’s accelerated gradient (AG) method to optimization over the Grassmann and Stiefel manifolds. We propose an exponential-based AG algorithm for the Grassmann manifold and a retraction-based AG algorithm that exploits the Cayley transform for both of the Grassmann and Stiefel manifolds. Under some mild assumptions, we … Read more

OFFO minimization algorithms for second-order optimality and their complexity

An Adagrad-inspired class of algorithms for smooth unconstrained optimization is presented in which the objective function is never evaluated and yet the gradient norms decrease at least as fast as O(1/\sqrt{k+1}) while second-order optimality measures converge to zero at least as fast as O(1/(k+1)^{1/3}). This latter rate of convergence is shown to be essentially sharp … Read more

On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming

We estimate the worst-case complexity of minimizing an unconstrained, nonconvex composite objective with a structured nonsmooth term by means of some first-order methods. We find that it is unaffected by the nonsmoothness of the objective in that a first-order trust-region or quadratic regularization method applied to it takes at most O($\epsilon^{-2}$) function-evaluations to reduce the … Read more