Local Linear Convergence of Forward--Backward under Partial Smoothness

Jingwei Liang (1) , Jalal M. Fadili (1) , Gabriel Peyré (2)

1 Equipe Image - Laboratoire GREYC - UMR6072
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper, we consider the Forward--Backward proximal splitting algorithm to minimize the sum of two proper convex functions, one of which having a Lipschitz continuous gradient and the other being partly smooth relative to an active manifold M. We propose a generic framework under which we show that the Forward--Backward (i) correctly identifies the active manifold M in a finite number of iterations, and then (ii) enters a local linear convergence regime that we characterize precisely. This gives a grounded and unified explanation to the typical behaviour that has been observed numerically for many problems encompassed in our framework, including the Lasso, the group Lasso, the fused Lasso and the nuclear norm regularization to name a few. These results may have numerous applications including in signal/image processing processing, sparse recovery and machine learning.

Domaines: Traitement des images [eess.IV]

Jingwei Liang, Jalal M. Fadili, Gabriel Peyré. Local Linear Convergence of Forward--Backward under Partial Smoothness. Advances in Neural Information Processing Systems (NIPS), Dec 2014, Montréal, Canada. ⟨hal-01150460⟩