This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to 0. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem.

翻译：本文研究在里曼尼框架范围内的固定步骤尺寸近似(SA)计划,包括随机梯度计划,它受到若干应用的推动,在这些应用中,大地测量学可以明确计算,其使用加速了粗的欧clidean方法。一个固定的梯度计划定义了一个时间-均匀的马尔科夫链式家庭,通过分级大小加以平衡。在这里,使用这种配方,在Lyapunov条件下可以得出非预防性的性能界限。然后,对于任何分级大小,相应的马可夫链被证明可以接受独特的固定分布,并且具有几何偏差性。这一结果产生了按分大小指数指数的固定分布式系列,这进一步显示,随着分级大小到0,集中在手边解决问题的Dirac测量上。最后,通过偏差的微调扩张和中央定界,确定了这种趋同的无症状速度。