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

TL;DR

This paper analyzes the continuous-time dynamics of the difference-of-convex algorithm (DCA), introducing a damped scheme and exploring convergence properties, geometric interpretations, and the impact of different decompositions.

Contribution

It provides a novel continuous-time perspective on DCA, introduces a damped variant with convergence guarantees, and links DCA dynamics to Bregman geometry and decomposition quality.

Findings

Damped DCA scheme exhibits monotone descent and convergence properties.

The limiting flow satisfies an exact energy identity and converges under KL assumptions.

Different DC decompositions induce distinct continuous dynamics, linking geometry and decomposition quality.

Abstract

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.