Stochastic Zeroth-Order Riemannian Derivative Estimation and Optimization

• 成果类型: Article
• 署名作者: Li, Jiaxiang; Balasubramanian, Krishnakumar; Ma, Shiqian
• 署名单位: University of California System; University of California Davis; University of California System; University of California Davis; Rice University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.1302
• 发表日期: 2023
• 页码: 1183-1211
• 关键词: bayesian optimization; convex-optimization; MANIFOLDS; reduction

摘要

We consider stochastic zeroth-order optimization over Riemannian submanifolds embedded in Euclidean space, where the task is to solve Riemannian optimization problems with only noisy objective function evaluations. Toward this, our main contribution is to propose estimators of the Riemannian gradient and Hessian from noisy objective function evaluations, based on a Riemannian version of the Gaussian smoothing technique. The proposed estimators overcome the difficulty of nonlinearity of the manifold constraint and issues that arise in using Euclidean Gaussian smoothing techniques when the function is defined only over the manifold. We use the proposed estimators to solve Riemannian optimization problems in 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 nonsmooth functions, and (iii) Hessian-Lipschitz. For these settings, we analyze 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 and real-world applications on black-box stiffness control for robotics and black-box attacks to neural networks.