Optimal local linear convergence of Nesterov's accelerated gradient method for C2 functions under the Polyak--Lojasiewicz inequality

In this work, we establish that Nesterov's accelerated gradient method, applied to C2 functions satisfying the Polyak--Lojasiewicz inequality around local minimizers, achieves the optimal local linear convergence rate ρ=frac√3L+μ-2√μ√3L+μ+varepsilon, where varepsilon is an arbitrarily small constant. Our analysis requires neither higher-order smoothness beyond C2 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 C1,1 assumption, which is commonly required in theoretical analyses of linear convergence for first-order methods, to C2 smoothness. Moreover, the same analytical framework allows us to recover, under identical conditions, the optimal local exponential convergence rate √μ for the continuous-time Heavy Ball dynamics. Finally, a representative numerical experiment corroborates our theoretical findings.