Approximate message-passing for convex optimization with non-separable penalties

Andre Manoel (1, 2) , Florent Krzakala (3) , Bertrand Thirion (2) , Gaël Varoquaux (2) , Lenka Zdeborová (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 PARIETAL - Modelling brain structure, function and variability based on high-field MRI data
3 LPS - Laboratoire de Physique Statistique de l'ENS

We introduce an iterative optimization scheme for convex objectives consisting of a linear loss and a non-separable penalty, based on the expectation-consistent approximation and the vector approximate message-passing (VAMP) algorithm. Specifically, the penalties we approach are convex on a linear transformation of the variable to be determined, a notable example being total variation (TV). We describe the connection between message-passing algorithms-typically used for approximate inference-and proximal methods for optimization, and show that our scheme is, as VAMP, similar in nature to the Peaceman-Rachford splitting, with the important difference that stepsizes are set adaptively. Finally, we benchmark the performance of our VAMP-like iteration in problems where TV penalties are useful, namely classification in task fMRI and reconstruction in tomography, and show faster convergence than that of state-of-the-art approaches such as FISTA and ADMM in most settings.

Domaines: Physique [physics]

Andre Manoel, Florent Krzakala, Bertrand Thirion, Gaël Varoquaux, Lenka Zdeborová. Approximate message-passing for convex optimization with non-separable penalties. 2018. ⟨cea-01932983⟩