Proximal Gradient Algorithm with Momentum and Flexible Parameter Restart for Nonconvex Optimization
This work addresses convergence issues in nonconvex optimization for researchers and practitioners, but it is incremental as it extends restart schemes from convex to nonconvex settings.
The authors tackled the problem of unclear convergence properties for accelerated gradient algorithms with parameter restart in nonconvex optimization by proposing APG-restart, which achieves global sub-linear convergence and guaranteed convergence to a critical point.
Various types of parameter restart schemes have been proposed for accelerated gradient algorithms to facilitate their practical convergence in convex optimization. However, the convergence properties of accelerated gradient algorithms under parameter restart remain obscure in nonconvex optimization. In this paper, we propose a novel accelerated proximal gradient algorithm with parameter restart (named APG-restart) for solving nonconvex and nonsmooth problems. Our APG-restart is designed to 1) allow for adopting flexible parameter restart schemes that cover many existing ones; 2) have a global sub-linear convergence rate in nonconvex and nonsmooth optimization; and 3) have guaranteed convergence to a critical point and have various types of asymptotic convergence rates depending on the parameterization of local geometry in nonconvex and nonsmooth optimization. Numerical experiments demonstrate the effectiveness of our proposed algorithm.