{"ID":2881077,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.12816","arxiv_id":"2508.12816","title":"A class of generalized Nesterov's accelerated gradient method from dynamical perspective","abstract":"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 ($0\u003cα\u003c1$) 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.","short_abstract":"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 Ne...","url_abs":"https://arxiv.org/abs/2508.12816","url_pdf":"https://arxiv.org/pdf/2508.12816v1","authors":"[\"Xu Cheng\",\"Jiaqi Liu\",\"Zaijiu Shang\"]","published":"2025-08-18T10:54:09Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.DS\"]","methods":"[]","has_code":false}