A Simple Adaptive Proximal Gradient Method for Nonconvex Optimization
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911196119040000 |
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| author | Ye, Zilong Ma, Shiqian Yang, Junfeng Zhou, Danqing |
| author_facet | Ye, Zilong Ma, Shiqian Yang, Junfeng Zhou, Danqing |
| contents | Consider composite nonconvex optimization problems where the objective function consists of a smooth nonconvex term (with Lipschitz-continuous gradient) and a convex (possibly nonsmooth) term. Existing parameter-free methods for such problems often rely on complex multi-loop structures, require line searches, or depend on restrictive assumptions (e.g., bounded iterates). To address these limitations, we introduce a novel adaptive proximal gradient method (referred to as AdaPGNC) that features a simple single-loop structure, eliminates the need for line searches, and only requires the gradient's Lipschitz continuity to ensure convergence. Furthermore, AdaPGNC achieves the theoretically optimal iteration/gradient evaluation complexity of $\mathcal{O}(\varepsilon^{-2})$ for finding an $\varepsilon$-stationary point. Our core innovation lies in designing an adaptive step size strategy that leverages upper and lower curvature estimates. A key technical contribution is the development of a novel Lyapunov function that effectively balances the function value gap and the norm-squared of consecutive iterate differences, serving as a central component in our convergence analysis. Preliminary experimental results indicate that AdaPGNC demonstrates competitive performance on several benchmark nonconvex (and convex) problems against state-of-the-art parameter-free methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06079 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Simple Adaptive Proximal Gradient Method for Nonconvex Optimization Ye, Zilong Ma, Shiqian Yang, Junfeng Zhou, Danqing Optimization and Control Consider composite nonconvex optimization problems where the objective function consists of a smooth nonconvex term (with Lipschitz-continuous gradient) and a convex (possibly nonsmooth) term. Existing parameter-free methods for such problems often rely on complex multi-loop structures, require line searches, or depend on restrictive assumptions (e.g., bounded iterates). To address these limitations, we introduce a novel adaptive proximal gradient method (referred to as AdaPGNC) that features a simple single-loop structure, eliminates the need for line searches, and only requires the gradient's Lipschitz continuity to ensure convergence. Furthermore, AdaPGNC achieves the theoretically optimal iteration/gradient evaluation complexity of $\mathcal{O}(\varepsilon^{-2})$ for finding an $\varepsilon$-stationary point. Our core innovation lies in designing an adaptive step size strategy that leverages upper and lower curvature estimates. A key technical contribution is the development of a novel Lyapunov function that effectively balances the function value gap and the norm-squared of consecutive iterate differences, serving as a central component in our convergence analysis. Preliminary experimental results indicate that AdaPGNC demonstrates competitive performance on several benchmark nonconvex (and convex) problems against state-of-the-art parameter-free methods. |
| title | A Simple Adaptive Proximal Gradient Method for Nonconvex Optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2510.06079 |