Information Geometry of Exponentiated Gradient: Convergence beyond L-Smoothness

Yara Elshiaty; Ferdinand Vanmaele; Stefania Petra

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arXiv · 2504.05136

Information Geometry of Exponentiated Gradient: Convergence beyond L-Smoothness

Yara Elshiaty·Ferdinand Vanmaele·Stefania Petra

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Abstract

We study the minimization of smooth, possibly nonconvex functions over the positive orthant, a key setting in Poisson inverse problems, using the exponentiated gradient (EG) method. Interpreting EG as Riemannian gradient descent (RGD) with the $e$-Exp map from information geometry as a retraction, we prove global convergence under weak assumptions -- without the need for $L$-smoothness -- and finite termination of Riemannian Armijo line search. Numerical experiments, including an accelerated variant, highlight EG's practical advantages, such as faster convergence compared to RGD based on interior-point geometry.

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Yara Elshiaty, Ferdinand Vanmaele, Stefania Petra. 2025-04-07. Information Geometry of Exponentiated Gradient: Convergence beyond L-Smoothness. https://arxiv.org/abs/2504.05136

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