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Communication Dans Un Congrès Année : 2021

On Riemannian Stochastic Approximation Schemes with Fixed Step-Size

Pablo Jiménez
  • Fonction : Auteur
Salem Said
  • Fonction : Auteur

Résumé

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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Dates et versions

hal-04396864 , version 1 (12-07-2024)

Identifiants

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Alain Durmus, Pablo Jiménez, Éric Moulines, Salem Said. On Riemannian Stochastic Approximation Schemes with Fixed Step-Size. International Conference on Artificial Intelligence and Statistics, Apr 2021, Online, France. ⟨10.48550/arXiv.2102.07586⟩. ⟨hal-04396864⟩
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