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We show that \\texttt{Zo-RASA} achieves optimal sample complexities for generating $ε$-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.","title":"Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms","url":"https://arxiv.org/pdf/2309.14506v1","vendor":"arxiv"},"summary":"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 $ε$-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. 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