Convergence Analysis of Riemannian Stochastic Approximation Schemes

• URL: http://arxiv.org/abs/2005.13284v3
• Date: Wed, 19 May 2021 14:25:42 GMT
• Title: Convergence Analysis of Riemannian Stochastic Approximation Schemes
• Authors: Alain Durmus, Pablo Jim\'enez, \'Eric Moulines, Salem Said, Hoi-To Wai
• Abstract summary: This paper analyzes a class of correlated approximation (SA) schemes to tackle optimization problems. We show that the conditions we derive are considerably milder than previous works. Third, 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 chain.
• Score: 39.32179384256228
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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