In this study, the authors investigate stochastic optimization on Riemannian manifolds, introducing loopless variance reduction methods for improved efficiency. The methods proposed replace inner loops with probabilistic gradient computations triggered by a coin flip, simplifying proofs and ensuring rapid convergence. The research showcases applicability to various important settings in non-convex distributed optimization over Riemannian manifolds.
The study addresses challenges in traditional approaches involving alternating between optimization in Euclidean space and projecting onto the manifold. By directly interacting with the specific manifold under consideration, Riemannian optimization eliminates the need for projections, offering insights into problem geometry and facilitating more efficient algorithms.
The content discusses key concepts such as geodesic convexity, strong convexity, smoothness, and curvature-driven terms essential for theoretical analysis of Riemannian optimization problems. Various assumptions are made to ensure mathematical rigor and practical applicability of the proposed methods.
Experimental results support theoretical findings regarding accelerated convergence rates compared to traditional gradient descent algorithms across different scenarios. The study also explores distributed learning scenarios incorporating communication compression and variance reduction techniques for enhanced efficiency.
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by Yury... kl. arxiv.org 03-12-2024
https://arxiv.org/pdf/2403.06677.pdfDybere Forespørgsler