Submitted

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

Long Chen and Zeyi Xu

Submitted

arXiv   Bibtex

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.