A class of generalized Nesterov's accelerated gradient method from dynamical perspective
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911109909315584 |
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| author | Cheng, Xu Liu, Jiaqi Shang, Zaijiu |
| author_facet | Cheng, Xu Liu, Jiaqi Shang, Zaijiu |
| contents | We propose a class of \textit{Euler-Lagrange} equations indexed by a pair of parameters ($α,r$) that generalizes Nesterov's accelerated gradient methods for convex ($α=1$) and strongly convex ($α=0$) functions from a continuous-time perspective. This class of equations also serves as an interpolation between the two Nesterov's schemes. The corresponding \textit{Hamiltonian} systems can be integrated via the symplectic Euler scheme with a fixed step-size. Furthermore, we can obtain the convergence rates for these equations ($0<α<1$) that outperform Nesterov's when time is sufficiently large for $μ$-strongly convex functions, without requiring a priori knowledge of $μ$. We demonstrate this by constructing a class of Lyapunov functions that also provide a unified framework for Nesterov's schemes for convex and strongly convex functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A class of generalized Nesterov's accelerated gradient method from dynamical perspective Cheng, Xu Liu, Jiaqi Shang, Zaijiu Optimization and Control Dynamical Systems We propose a class of \textit{Euler-Lagrange} equations indexed by a pair of parameters ($α,r$) that generalizes Nesterov's accelerated gradient methods for convex ($α=1$) and strongly convex ($α=0$) functions from a continuous-time perspective. This class of equations also serves as an interpolation between the two Nesterov's schemes. The corresponding \textit{Hamiltonian} systems can be integrated via the symplectic Euler scheme with a fixed step-size. Furthermore, we can obtain the convergence rates for these equations ($0<α<1$) that outperform Nesterov's when time is sufficiently large for $μ$-strongly convex functions, without requiring a priori knowledge of $μ$. We demonstrate this by constructing a class of Lyapunov functions that also provide a unified framework for Nesterov's schemes for convex and strongly convex functions. |
| title | A class of generalized Nesterov's accelerated gradient method from dynamical perspective |
| topic | Optimization and Control Dynamical Systems |
| url | https://arxiv.org/abs/2508.12816 |