Anytime Acceleration of Gradient Descent

Abstract

This work investigates stepsize-based acceleration of gradient descent with {\em anytime} convergence guarantees. For smooth (non-strongly) convex optimization, we propose a stepsize schedule that allows gradient descent to achieve convergence guarantees of $O(T^{-1.119})$ for any stopping time $T$, where the stepsize schedule is predetermined without prior knowledge of the stopping time. This result provides an affirmative answer to a COLT open problem \citep{kornowski2024open} regarding whether stepsize-based acceleration can yield anytime convergence rates of $o(T^{-1})$. We further extend our theory to yield anytime convergence guarantees of $\exp (-\hat{I}\text{c◯}(T/\hat{I}{º}^{0.893}))$ for smooth and strongly convex optimization, with $\hat{I}º$ being the condition number.

v2: We improve the convergence rate from $O(T^{-1.03})$ to O(T^{-1.119}) through more precise computations