Accelerated gradient sliding for structured convex optimization

TL;DR

This paper introduces an accelerated gradient sliding method that reduces the number of gradient computations needed for certain structured convex problems, especially when components have different Lipschitz constants, improving efficiency.

Contribution

The paper presents a novel accelerated gradient sliding algorithm that skips gradient evaluations for one component without losing convergence speed, and applies it to saddle point problems with significant computational savings.

Findings

Reduces gradient computations for one component in structured convex problems.

Achieves optimal convergence rates while skipping certain gradient evaluations.

Provides significant computational savings for strongly convex and saddle point problems.

Abstract

Our main goal in this paper is to show that one can skip gradient computations for gradient descent type methods applied to certain structured convex programming (CP) problems. To this end, we first present an accelerated gradient sliding (AGS) method for minimizing the summation of two smooth convex functions with different Lipschitz constants. We show that the AGS method can skip the gradient computation for one of these smooth components without slowing down the overall optimal rate of convergence. This result is much sharper than the classic black-box CP complexity results especially when the difference between the two Lipschitz constants associated with these components is large. We then consider an important class of bilinear saddle point problem whose objective function is given by the summation of a smooth component and a nonsmooth one with a bilinear saddle point structure. Using the aforementioned AGS method for smooth composite optimization and Nesterov's smoothing technique, we show that one only needs ${\cal O}(1/\sqrt{\epsilon})$ gradient computations for the smooth component while still preserving the optimal ${\cal O}(1/\epsilon)$ overall iteration complexity for solving these saddle point problems. We demonstrate that even more significant savings on gradient computations can be obtained for strongly convex smooth and bilinear saddle point problems.