1,307 research outputs found
Faster First-Order Methods for Stochastic Non-Convex Optimization on Riemannian Manifolds
SPIDER (Stochastic Path Integrated Differential EstimatoR) is an efficient
gradient estimation technique developed for non-convex stochastic optimization.
Although having been shown to attain nearly optimal computational complexity
bounds, the SPIDER-type methods are limited to linear metric spaces. In this
paper, we introduce the Riemannian SPIDER (R-SPIDER) method as a novel
nonlinear-metric extension of SPIDER for efficient non-convex optimization on
Riemannian manifolds. We prove that for finite-sum problems with
components, R-SPIDER converges to an -accuracy stationary point
within
stochastic gradient evaluations, which is sharper in magnitude than the prior
Riemannian first-order methods. For online optimization, R-SPIDER is shown to
converge with complexity which is,
to the best of our knowledge, the first non-asymptotic result for online
Riemannian optimization. Especially, for gradient dominated functions, we
further develop a variant of R-SPIDER and prove its linear convergence rate.
Numerical results demonstrate the computational efficiency of the proposed
methods
Riemannian Optimization via Frank-Wolfe Methods
We study projection-free methods for constrained Riemannian optimization. In
particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze
non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex
problems, and to a critical point for nonconvex objectives. We also present a
practical setting under which RFW can attain a linear convergence rate. As a
concrete example, we specialize Rfw to the manifold of positive definite
matrices and apply it to two tasks: (i) computing the matrix geometric mean
(Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter.
Both tasks involve geodesically convex interval constraints, for which we show
that the Riemannian "linear oracle" required by RFW admits a closed-form
solution; this result may be of independent interest. We further specialize RFW
to the special orthogonal group and show that here too, the Riemannian "linear
oracle" can be solved in closed form. Here, we describe an application to the
synchronization of data matrices (Procrustes problem). We complement our
theoretical results with an empirical comparison of Rfw against
state-of-the-art Riemannian optimization methods and observe that RFW performs
competitively on the task of computing Riemannian centroids.Comment: Under Review. Largely revised version, including an extended
experimental section and an application to the special orthogonal group and
the Procrustes proble
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