Zeroth-order Optimization on Riemannian Manifolds

Stochastic zeroth-order optimization concerns problems where only noisy function evaluations are available. Such problems arises frequently in many important applications. In this paper, we consider stochastic zeroth-order optimization over Riemannian submanifolds embedded in an Euclidean space, an important but less studied area, and propose four algorithms for solving this class of problems under different settings. Our algorithms are based on estimating the Riemannian gradient and Hessian from noisy objective function evaluations, based on a Riemannian version of the Gaussian smoothing technique. In particular, we consider the following settings for the objective function: (i) stochastic and gradient-Lipschitz (in both nonconvex and geodesic convex settings), (ii) sum of gradient-Lipschitz and non-smooth functions, and (iii) Hessian-Lipschitz. For these settings, we characterize the oracle complexity of our algorithms to obtain appropriately defined notions of $\epsilon$-stationary point or $\epsilon$-approximate local minimizer. Notably, our complexities are independent of the dimension of the ambient Euclidean space and depend only on the intrinsic dimension of the manifold under consideration. We demonstrate the applicability of our algorithms by simulation results.