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arXiv · 2102.07586

On Riemannian Stochastic Approximation Schemes with Fixed Step-Size

Alain Durmus·Pablo Jiménez·Éric Moulines·Salem Said
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Abstract

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.

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BibTeXRIS

Alain Durmus, Pablo Jiménez, Éric Moulines, Salem Said. 2021-02-15. On Riemannian Stochastic Approximation Schemes with Fixed Step-Size. https://arxiv.org/abs/2102.07586

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