# Continuous-Time Dynamics of the Difference-of-Convex Algorithm

arXiv: https://arxiv.org/abs/2604.06926 · HTML: https://mlsanity.com/paper/2604.06926/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 19 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 5 of 5)

- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [On the rate of convergence of the Difference-of-Convex Algorithm (DCA)](https://mlsanity.com/paper/2109.13566/) (2021)
- [On the Convergence Analysis of DCA](https://mlsanity.com/paper/2211.10942/) (2022)
- [A general double-proximal gradient algorithm for d.c. programming](https://mlsanity.com/paper/1610.06538/) (2016)

## Built on by (most cited first, 2 of 2)

- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)
- [K"ahler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold](https://mlsanity.com/paper/2608.19584/) (2026)

## Abstract (arXiv)

We study the continuous-time structure of the difference-of-convex algorithm (DCA) for smooth DC decompositions with a strongly convex component. In dual coordinates, classical DCA is exactly the full-step explicit Euler discretization of a nonlinear autonomous system. This viewpoint motivates a damped DCA scheme, which is also a Bregman-regularized DCA variant, and whose vanishing-step limit yields a Hessian-Riemannian gradient flow generated by the convex part of the decomposition. For the damped scheme we prove monotone descent, asymptotic criticality, Kurdyka-Lojasiewicz convergence under boundedness, and a global linear rate under a metric DC-PL inequality. For the limiting flow we establish an exact energy identity, asymptotic criticality of bounded trajectories, explicit global rates under metric relative error bounds, finite-length and single-point convergence under a Kurdyka-Lojasiewicz hypothesis, and local exponential convergence near nondegenerate local minima. The analysis also reveals a global-local tradeoff: the half-relaxed scheme gives the best provable global guarantee in our framework, while the full-step scheme is locally fastest near a nondegenerate minimum. Finally, we show that different DC decompositions of the same objective induce different continuous dynamics through the metric generated by the convex component, providing a geometric criterion for decomposition quality and linking DCA with Bregman geometry.