Manifold constrained steepest descent
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912858248314880 |
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| author | Yang, Kaiwei Lai, Lexiao |
| author_facet | Yang, Kaiwei Lai, Lexiao |
| contents | Norm-constrained linear minimization oracle (LMO)-based optimizers such as spectral gradient descent and Muon are attractive in large-scale learning, but extending them to manifold-constrained problems is nontrivial and often leads to nested-loop schemes that solve tangent-space subproblems iteratively. We propose \emph{Manifold Constrained Steepest Descent} (MCSD), a single-loop framework for optimization over manifolds that selects a norm-induced steepest-descent direction via an LMO applied to the Riemannian gradient, and then returns to the manifold via projection. Under standard smoothness assumptions, we establish convergence guarantees for MCSD and a stochastic momentum variant. We further introduce \emph{SPEL}, the spectral-norm specialization of MCSD on the Stiefel manifold, which admits scalable implementations via fast matrix sign computations. Experiments on PCA, orthogonality-constrained CNNs, and manifold-constrained LLM adapter tuning demonstrate improved stability and competitive performance relative to standard Riemannian baselines and existing manifold-aware LMO methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21487 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Manifold constrained steepest descent Yang, Kaiwei Lai, Lexiao Optimization and Control Machine Learning Norm-constrained linear minimization oracle (LMO)-based optimizers such as spectral gradient descent and Muon are attractive in large-scale learning, but extending them to manifold-constrained problems is nontrivial and often leads to nested-loop schemes that solve tangent-space subproblems iteratively. We propose \emph{Manifold Constrained Steepest Descent} (MCSD), a single-loop framework for optimization over manifolds that selects a norm-induced steepest-descent direction via an LMO applied to the Riemannian gradient, and then returns to the manifold via projection. Under standard smoothness assumptions, we establish convergence guarantees for MCSD and a stochastic momentum variant. We further introduce \emph{SPEL}, the spectral-norm specialization of MCSD on the Stiefel manifold, which admits scalable implementations via fast matrix sign computations. Experiments on PCA, orthogonality-constrained CNNs, and manifold-constrained LLM adapter tuning demonstrate improved stability and competitive performance relative to standard Riemannian baselines and existing manifold-aware LMO methods. |
| title | Manifold constrained steepest descent |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2601.21487 |