Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms

• URL: http://arxiv.org/abs/2309.14506v1
• Date: Mon, 25 Sep 2023 20:13:36 GMT
• Title: Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms
• Authors: Jiaxiang Li, Krishnakumar Balasubramanian and Shiqian Ma
• Abstract summary: We show that textttZo-RASA achieves optimal sample complexities for generating $epsilon$-approximation first-order stationary solutions. We improve the algorithm's practicality by using geometrics and vector transport, instead of exponential mappings and parallel transports.
• Score: 19.99781875916751
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present Zeroth-order Riemannian Averaging Stochastic Approximation (\texttt{Zo-RASA}) algorithms for stochastic optimization on Riemannian manifolds. We show that \texttt{Zo-RASA} achieves optimal sample complexities for generating $\epsilon$-approximation first-order stationary solutions using only one-sample or constant-order batches in each iteration. Our approach employs Riemannian moving-average stochastic gradient estimators, and a novel Riemannian-Lyapunov analysis technique for convergence analysis. We improve the algorithm's practicality by using retractions and vector transport, instead of exponential mappings and parallel transports, thereby reducing per-iteration complexity. Additionally, we introduce a novel geometric condition, satisfied by manifolds with bounded second fundamental form, which enables new error bounds for approximating parallel transport with vector transport.

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