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

Abstract

Two accelerated first-order methods, HNAG$^+$ and HNAG$^{++}$, are presented for smooth strongly convex optimization. By optimizing the coercivity constant of the HNAG flow and using a refined Lyapunov analysis, it is shown that HNAG$^+$ achieves the optimal global rate $1-2/\sqrtκ$, matching the information-theoretic lower bound for strongly convex optimization. For functions with Local Asymptotic Symmetry at the minimizer, HNAG$^{++}$ is shown to achieve the asymptotic rate $1-2\sqrt{2/κ}$, matching the best known asymptotic rate under $\mathcal C^2$ regularity, while applying to a broader local function class. Numerical experiments on linear and nonlinear examples show that the proposed methods are competitive with existing accelerated schemes.