The exact worst-case asymptotic rate of Nesterov acceleration for strongly convex optimization

Abstract

We study Nesterov's accelerated gradient method with its standard constant parameters for smooth strongly convex optimization. We characterize the exact worst-case asymptotic factor for the squared distance to the minimizer. To this end, we construct candidate worst-case trajectories by combining planar spirals with a common contraction factor that satisfy a self-similarity condition. Each component rotates by a fixed angle at every iteration. The exact factor is the optimal value of a linear optimization problem over probability measures on a fixed disk, with constraints derived from smooth strongly convex interpolation. Every optimal measure has a common contraction factor, which is also the largest factor for which the angular interpolation conditions are feasible. We also show that this worst-case asymptotic rate can be realized by a \emph{single} function in a finite-dimensional space whose iterates attain this factor. A matching upper bound holds uniformly over all functions in the class and all finite dimensions. Consequently, allowing the function, dimension, and initial point to vary with the iteration horizon does not increase the worst-case asymptotic factor. We also show that the exact factor is $1-4/(3\sqrtκ)+o(κ^{-1/2})$ as $κ\to\infty$, where $κ=L/μ$ is the ratio of the gradient Lipschitz constant to the strong convexity parameter.