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In particular, the recursions we study use either the exponential map of the considered manifold (geodesic schemes) or more general retraction functions (retraction schemes) used as a proxy for the exponential map. Such approximations are of great interest since they are low complexity alternatives to geodesic schemes. Under the assumption that the mean field of the SA is correlated with the gradient of a smooth Lyapunov function (possibly non-convex), we show that the above Riemannian SA schemes find an ${\\mathcal{O}}(b_\\infty + \\log n / \\sqrt{n})$-stationary point (in expectation) within ${\\mathcal{O}}(n)$ iterations, where $b_\\infty \\geq 0$ is the asymptotic bias. Compared to previous works, the conditions we derive are considerably milder. 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Under the assumption that the mean field of the SA is correlated with the gradient of a smooth Lyapunov function (possibly non-convex), we show that the above Riemannian SA schemes find an ${\\mathcal{O}}(b_\\infty + \\log n / \\sqrt{n})$-stationary point (in expectation) within ${\\mathcal{O}}(n)$ iterations, where $b_\\infty \\geq 0$ is the asymptotic bias. Compared to previous works, the conditions we derive are considerably milder. 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