On the rate of convergence of the Difference-of-Convex Algorithm (DCA)

TL;DR

This paper analyzes the convergence rates of the Difference-of-Convex Algorithm (DCA), providing new bounds for smooth and nonsmooth cases, and employing semidefinite programming for performance estimation.

Contribution

It introduces new convergence rate results for DCA, including $O(1/\sqrt{N})$, $O(1/N)$, and linear rates under specific conditions, using semidefinite programming techniques.

Findings

Convergence rate of $O(1/\sqrt{N})$ for certain DC problems.

Improved $O(1/N)$ rate with a second termination criterion.

Linear convergence under Polyak-ojasiewicz inequality.

Abstract

In this paper, we study the convergence rate of the DCA (Difference-of-Convex Algorithm), also known as the convex-concave procedure, with two different termination criteria that are suitable for smooth and nonsmooth decompositions respectively. The DCA is a popular algorithm for difference-of-convex (DC) problems, and known to converge to a stationary point of the objective under some assumptions. We derive a worst-case convergence rate of $O(1/\sqrt{N})$ after $N$ iterations of the objective gradient norm for certain classes of DC problems, without assuming strong convexity in the DC decomposition, and give an example which shows the convergence rate is exact. We also provide a new convergence rate of $O(1/N)$ for the DCA with the second termination criterion. %In addition, we investigate the DCA with regularization. Moreover, we derive a new linear convergence rate result for the DCA under the assumption of the Polyak-\L ojasiewicz inequality. The novel aspect of our analysis is that it employs semidefinite programming performance estimation.