{"ID":2866307,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.19682","arxiv_id":"2509.19682","title":"A Riemannian AdaGrad-Norm Method","abstract":"We propose a manifold AdaGrad-Norm method (\\textsc{MAdaGrad}), which extends the norm version of AdaGrad (AdaGrad-Norm) to Riemannian optimization. In contrast to line-search schemes, which may require several exponential map computations per iteration, \\textsc{MAdaGrad} requires only one. Assuming the objective function $f$ has Lipschitz continuous Riemannian gradient, we show that the method requires at most $\\mathcal{O}(\\varepsilon^{-2})$ iterations to compute a point $x$ such that $\\|\\operatorname{grad} f(x)\\|\\leq \\varepsilon$. Under the additional assumptions that $f$ is geodesically convex and the manifold has sectional curvature bounded from below, we show that the method takes at most $\\mathcal{O}(\\varepsilon^{-1})$ to find $x$ such that $f(x)-f_{low}\\leqε$, where $f_{low}$ is the optimal value. Moreover, if $f$ satisfies the Polyak--Łojasiewicz condition globally on the manifold, we establish a complexity bound of $\\mathcal{O}(\\log(\\varepsilon^{-1}))$, provided that the norm of the initial Riemannian gradient is sufficiently large. For the manifold of symmetric positive definite matrices, we construct a family of nonconvex functions satisfying the PL condition. Numerical experiments illustrate the remarkable performance of \\textsc{MAdaGrad} in comparison with Riemannian Steepest Descent equipped with Armijo line-search.","short_abstract":"We propose a manifold AdaGrad-Norm method (\\textsc{MAdaGrad}), which extends the norm version of AdaGrad (AdaGrad-Norm) to Riemannian optimization. In contrast to line-search schemes, which may require several exponential map computations per iteration, \\textsc{MAdaGrad} requires only one. Assuming the objective functi...","url_abs":"https://arxiv.org/abs/2509.19682","url_pdf":"https://arxiv.org/pdf/2509.19682v1","authors":"[\"Glaydston de C. Bento\",\"Geovani N. Grapiglia\",\"Mauricio S. Louzeiro\",\"Daoping Zhang\"]","published":"2025-09-24T01:36:18Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}