Local Convergence of the Heavy-ball Method and iPiano for Non-convex Optimization

TL;DR

This paper proves local convergence for abstract descent methods, including iPiano, showing they are attracted to local minima and outperform non-inertial methods in numerical experiments.

Contribution

It establishes a local convergence theory for iPiano and related methods, revealing their equivalence and exploiting local properties of the objective function.

Findings

iPiano converges locally to minima in non-convex optimization.

Inertial methods outperform non-inertial variants in numerical tests.

Theoretical results apply to a broad class of descent algorithms.

Abstract

A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges within this neighborhood. This result allows algorithms to exploit local properties of the objective function. In particular, the abstract theory in this paper applies to the inertial forward--backward splitting method: iPiano---a generalization of the Heavy-ball method. Moreover, it reveals an equivalence between iPiano and inertial averaged/alternating proximal minimization and projection methods. Key for this equivalence is the attraction to a local minimum within a neighborhood and the fact that, for a prox-regular function, the gradient of the Moreau envelope is locally Lipschitz continuous and expressible in terms of the proximal mapping. In a numerical feasibility problem, the inertial alternating projection method significantly outperforms its non-inertial variants.