HNAG$^{++}$: An Accelerated Gradient Method with a Refined Asymptotic Rate for Strongly Convex Optimization

Long Chen; Zeyi Xu

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arXiv · 2510.16680

HNAG$^{++}$: An Accelerated Gradient Method with a Refined Asymptotic Rate for Strongly Convex Optimization

Long Chen·Zeyi Xu

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Abstract

Two accelerated first-order methods, HNAG$^+$ and HNAG$^{++}$, are introduced for smooth strongly convex optimization. They are derived from the Hessian-driven Nesterov Accelerated Gradient (HNAG) flow by optimizing the coercivity of shifted Lyapunov functions. Let $\kappa=L/\mu$, where $\mu$ is the strong-convexity constant and $L$ is the gradient Lipschitz constant. HNAG$^+$ attains the optimal global rate $1-2/\sqrt{\kappa}$, matching the information-theoretic lower bound. For functions with local asymptotic symmetry at the minimizer, HNAG$^{++}$ attains the asymptotic rate $1-2\sqrt{2/\kappa}$. This matches the best known asymptotic rate under $\mathcal C^2$ regularity, while applying to a broader function class. Numerical experiments confirm the predicted rates and show favorable performance against existing accelerated schemes.

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BibTeX RIS

Long Chen, Zeyi Xu. 2025-10-19. HNAG$^{++}$: An Accelerated Gradient Method with a Refined Asymptotic Rate for Strongly Convex Optimization. https://arxiv.org/abs/2510.16680

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