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

2026-04-08

AI Summary

This work uncovers the continuous dynamical essence of the Difference-of-Convex Algorithm (DCA) by interpreting it as an explicit Euler discretization of a nonlinear autonomous system and proposes a damped variant of DCA. By establishing, for the first time, a connection between DCA and Bregman geometry as well as Hessian-Riemannian gradient flows, the study introduces a geometric criterion—based on the metric induced by the convex component—to assess the quality of DC decompositions, thereby clarifying how decomposition choices govern the trade-off between global and local convergence rates. Leveraging tools from continuous dynamical systems, the Kurdyka–Łojasiewicz (KL) inequality, and a metric DC-PL condition, the paper proves that the damped DCA enjoys monotone descent, asymptotic criticality, KL-type convergence, and global linear convergence rates. Its continuous-time limit satisfies an energy identity and exhibits finite length, singleton convergence, and local exponential convergence.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Difference-of-Convex Algorithm
• Continuous-Time Dynamics
• Hessian-Riemannian Gradient Flow
• Kurdyka-Łojasiewicz Convergence
• Bregman Geometry

Innovation

• difference-of-convex algorithm
• continuous-time dynamics
• Bregman geometry
• Hessian-Riemannian gradient flow
• Kurdyka-Łojasiewicz convergence