A New Complexity Result for Strongly Convex Optimization with Locally $\alpha$-H{\"o}lder Continuous Gradients

Abstract

In this paper, we present a new complexity result for the gradient descent method with an appropriately fixed stepsize for minimizing a strongly convex function with locally $\alpha$-H{\"o}lder continuous gradients ($0 < \alpha \leq 1$). The complexity bound for finding an approximate minimizer with a distance to the true minimizer less than $\varepsilon$ is $O(\log (\varepsilon^{-1}) \varepsilon^{2 \alpha - 2})$, which extends the well-known complexity result for $\alpha = 1$.