Stochastic gradient descent on Riemannian manifolds

Details Stochastic gradient descent on Riemannian manifolds Stochastic gradient descent on Riemannian manifolds

2011-11-22

arxiv.org/abs/1111.5280v2

Mathematics

Optimization and Control

Submitted

Scientific paper

Stochastic gradient descent is a simple appproach to find the local minima of a function whose evaluations are corrupted by noise. In this paper, mostly motivated by machine learning applications, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and we show several well-known algorithms can be cast in our versatile geometric framework. We also address the gain tuning issue in connection with the tools of the recent theory of symmetry-preserving observers.

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