Convergence Analysis of Riemannian Stochastic Approximation Schemes - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2024

Convergence Analysis of Riemannian Stochastic Approximation Schemes

Pablo Jiménez
  • Fonction : Auteur
Salem Said
  • Fonction : Auteur
Hoi-To Wai
  • Fonction : Auteur

Résumé

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.

Dates et versions

hal-04396909 , version 1 (16-01-2024)

Identifiants

Citer

Alain Durmus, Pablo Jiménez, Éric Moulines, Salem Said, Hoi-To Wai. Convergence Analysis of Riemannian Stochastic Approximation Schemes. 2024. ⟨hal-04396909⟩
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