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

arXiv: https://arxiv.org/abs/2109.13566 · HTML: https://mlsanity.com/paper/2109.13566/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2021.

## Built on by (most cited first, 9 of 9)

- [Large-scale Optimization of Partial AUC in a Range of False Positive Rates](https://mlsanity.com/paper/2203.01505/) (2022)
- [Difference of Submodular Minimization via DC Programming](https://mlsanity.com/paper/2305.11046/) (2023)
- [Revisiting Frank-Wolfe for Structured Nonconvex Optimization](https://mlsanity.com/paper/2503.08921/) (2025)
- [Enforcing Fair Predicted Scores on Intervals of Percentiles by Difference-of-Convex Constraints](https://mlsanity.com/paper/2505.12530/) (2025)
- [Continuous-Time Dynamics of the Difference-of-Convex Algorithm](https://mlsanity.com/paper/2604.06926/) (2026)
- [Scalable DC Optimization via Adaptive Frank-Wolfe Algorithms](https://mlsanity.com/paper/2507.17545/) (2025)
- [A Novel Plug-and-Play Approach for Adversarially Robust Generalization](https://mlsanity.com/paper/2208.09449/) (2022)
- [Green Robotic Mixed Reality with Gaussian Splatting](https://mlsanity.com/paper/2504.13697/) (2025)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

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/√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.