A Nonsmooth Dynamical Systems Perspective on Accelerated Extensions of ADMM
This work addresses optimization challenges in machine learning and engineering by offering theoretical insights into accelerated methods for constrained problems, though it appears incremental as it builds on existing dynamical systems frameworks.
The paper tackled the problem of extending continuous-time dynamical systems perspectives to nonsmooth and constrained optimization by developing accelerated variants of ADMM, deriving convergence rates through Lyapunov analysis that show tradeoffs between damping strategies, and providing extensions to nonlinear constraints.
Recently, there has been great interest in connections between continuous-time dynamical systems and optimization methods, notably in the context of accelerated methods for smooth and unconstrained problems. In this paper we extend this perspective to nonsmooth and constrained problems by obtaining differential inclusions associated to novel accelerated variants of the alternating direction method of multipliers (ADMM). Through a Lyapunov analysis, we derive rates of convergence for these dynamical systems in different settings that illustrate an interesting tradeoff between decaying versus constant damping strategies. We also obtain modified equations capturing fine-grained details of these methods, which have improved stability and preserve the leading order convergence rates. An extension to general nonlinear equality and inequality constraints in connection with singular perturbation theory is provided.