- Activity Identification and Local Linear Convergence of Forward--Backward-type methods Jingwei Liang (jingwei.liangensicaen.fr) Jalal Fadili (jalal.Fadiliensicaen.fr) Gabriel Peyré (peyreceremade.dauphine.fr) Abstract: In this paper, we consider a class of Forward--Backward (FB) splitting methods that includes several variants (e.g. inertial schemes, FISTA) for minimizing the sum of two proper convex and lower semi-continuous functions, one of which has a Lipschitz continuous gradient, and the other is partly smooth relatively to a smooth active manifold $\mathcal{M}$. We propose a unified framework, under which we show that, this class of FB-type algorithms (i) correctly identifies the active manifolds in a finite number of iterations (finite activity identification), and (ii) then enters a local linear convergence regime, which we characterize precisely in terms of the structure of the underlying active manifolds. For simpler problems involving polyhedral functions, we show finite termination. We also establish and explain why FISTA (with convergent sequences) can be locally slower than FB. These results may have numerous applications including in signal/image processing, sparse recovery and machine learning. Indeed, the obtained results explain the typical behaviour that has been observed numerically for many problems in these fields such as the Lasso, the group Lasso, the fused Lasso and the nuclear norm regularization to name only a few. Keywords: Forward--Backward, Inertial Methods, ISTA/FISTA, Partial Smoothness, Local Linear Convergence. Category 1: Convex and Nonsmooth Optimization Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Citation: Download: [PDF]Entry Submitted: 03/13/2015Entry Accepted: 03/13/2015Entry Last Modified: 01/03/2018Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.