Near-optimal Closed-loop Method via Lyapunov Damping for Convex Optimization
This addresses the challenge of optimal convergence in convex optimization for researchers and practitioners, offering a novel closed-loop approach that is incremental over existing open-loop methods.
The authors tackled the problem of achieving near-optimal convergence rates in first-order convex optimization by introducing an autonomous system with closed-loop damping, which attains rates arbitrarily close to the optimal one, as supported by numerical experiments.
We introduce an autonomous system with closed-loop damping for first-order convex optimization. While, to this day, optimal rates of convergence are almost exclusively achieved by non-autonomous methods via open-loop damping (e.g., Nesterov's algorithm), we show that our system, featuring a closed-loop damping, exhibits a rate arbitrarily close to the optimal one. We do so by coupling the damping and the speed of convergence of the system via a well-chosen Lyapunov function. By discretizing our system we then derive an algorithm and present numerical experiments supporting our theoretical findings.