Fast dual proximal gradient algorithms with rate $O(1/k^{1.5})$ for convex minimization

TL;DR

This paper introduces a generalized fast dual proximal gradient method that achieves an improved convergence rate of O(1/k^{1.5}) for convex minimization problems involving strongly convex and convex functions, enhancing efficiency over existing methods.

Contribution

The paper proposes a novel generalized FDPG method that guarantees an O(1/k^{1.5}) convergence rate for the primal function, improving upon previous rates.

Findings

Achieves an O(1/k^{1.5}) convergence rate for primal function decrease.

Relates dual proximal gradient norm decrease to primal function convergence.

Provides theoretical guarantees for the proposed generalized FDPG method.

Abstract

We consider minimizing the composite function that consists of a strongly convex function and a convex function. The fast dual proximal gradient (FDPG) method decreases the dual function with a rate $O(1/k^2)$, leading to a rate $O(1/k)$ for decreasing the primal function. We propose a generalized FDPG method that guarantees an $O(1/k^{1.5})$ rate for the dual proximal gradient norm decrease. By relating this to the primal function decrease, the proposed approach decreases the primal function with the improved $O(1/k^{1.5})$ rate.