This paper analyzes the convergence for a large class of Riemannian stochastic approximation (SA) schemes, which aim at tackling stochastic optimization problems. 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. First, all our analysis are global as we do not assume iterates to be a-priori bounded. Second, we study biased SA schemes. To be more specific, we consider the case where the mean-field function can only be estimated up to a small bias, and/or the case in which the samples are drawn from a controlled Markov chain. Third, the conditions on retractions required to ensure convergence of the related SA schemes are weak and hold for well-known examples. We illustrate our results on three machine learning problems.

翻译：本文分析了一大批里曼尼近似( SA) 的趋同性( Riemannian stochatstic pressive (SA) 机制的趋同性( SA) ) 。 我们研究的Riemannian SA 机制的递归性( SA) 旨在解决随机优化问题 。 特别是, 我们研究的Riemannian SA 方案在 指数映射地图中, 或使用 更一般的撤回功能( tractition program) 的指数图( tracision program) 。 这种近似性( tractical program) 在 $\ mathcal { ( O{ ( b ⁇ infty +\\\\\\\ sqrt}} ( retracripormation production) 中 。 这种近似的近似性( ) 在 $\ macalfty (n) plical) roductions press review press relist press.