arXiv subjects

Peter Ochs

Publications and source records attributed to Peter Ochs.

40 records · Page 3Linked to original sources

arXiv2018

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

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.

math.OC↗

arXiv2017

Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano

An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to possibly non-smooth and non-convex lower semi-continuous functions that satisfy the Kurdyka--Lojasiewicz (KL) inequality, which comprises a huge class of problems. Most of the recent algorithms that explicitly prove convergence using the KL inequality can cast into the abstract framework in this paper and, therefore, the generated sequence converges to a stationary point of the objective function. Additional flexibility compared to related approaches is gained by a descent property that is formulated with respect to a function that is allowed to change along the iterations, a generic distance measure, and an explicit/implicit relative error condition with respect to finite linear combinations of distance terms. As an application of the gained flexibility, the convergence of a block coordinate variable metric version of iPiano (an inertial forward--backward splitting algorithm) is proved, which performs favorably on an inpainting problem with a Mumford--Shah-like regularization from image processing.

math.OC↗

arXiv2016

Techniques for Gradient Based Bilevel Optimization with Nonsmooth Lower Level Problems

We propose techniques for approximating bilevel optimization problems with non-smooth lower level problems that can have a non-unique solution. To this end, we substitute the expression of a minimizer of the lower level minimization problem with an iterative algorithm that is guaranteed to converge to a minimizer of the problem. Using suitable non-linear proximal distance functions, the update mappings of such an iterative algorithm can be differentiable, notwithstanding the fact that the minimization problem is non-smooth.

math.OC↗

arXiv2014

iPiano: Inertial Proximal Algorithm for Non-Convex Optimization

In this paper we study an algorithm for solving a minimization problem composed of a differentiable (possibly non-convex) and a convex (possibly non-differentiable) function. The algorithm iPiano combines forward-backward splitting with an inertial force. It can be seen as a non-smooth split version of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for the proposed class of problems yields global convergence of the function values and the arguments. This makes the algorithm robust for usage on non-convex problems. The convergence result is obtained based on the \KL inequality. This is a very weak restriction, which was used to prove convergence for several other gradient methods. First, an abstract convergence theorem for a generic algorithm is proved, and, then iPiano is shown to satisfy the requirements of this theorem. Furthermore, a convergence rate is established for the general problem class. We demonstrate iPiano on computer vision problems: image denoising with learned priors and diffusion based image compression.

cs.CV↗