Optimal local linear convergence of Nesterov's accelerated gradient method for $C^2$ functions under the Polyak--{\L}ojasiewicz inequality

In this work, we establish that Nesterov's accelerated gradient method, applied to $C^2$ functions satisfying the Polyak--{\L}ojasiewicz inequality around local minimizers, achieves the optimal local linear convergence rate $\rho=\frac{\sqrt{3L+\mu}-2\sqrt{\mu}}{\sqrt{3L+\mu}}+\varepsilon$, where $\varepsilon$ is an arbitrarily small constant. Our analysis requires neither higher-order smoothness beyond $C^2$ of the objective function nor any additional geometric regularity of the submanifold of local minimizers. The key novelty lies in a two-stage argument: we first establish a coarse yet valid local linear convergence rate and then, building upon this a priori convergence guarantee, obtain a refined characterization of the linearized iteration operator, which yields the optimal rate. As a result, we only need to slightly strengthen the standard $C^{1,1}$ assumption, which is commonly required in theoretical analyses of linear convergence for first-order methods, to $C^2$ smoothness. Moreover, the same analytical framework allows us to recover, under identical conditions, the optimal local exponential convergence rate $\sqrt{\mu}$ for the continuous-time Heavy Ball dynamics. Finally, a representative numerical experiment corroborates our theoretical findings.