A variable metric forward--backward method with extrapolation

Forward-backward methods are a very useful tool for the minimization of a functional given by the sum of a differentiable term and a nondifferentiable one and their investigation has experienced several efforts from many researchers in the last decade. In this paper we focus on the convex case and, inspired by recent approaches for accelerating first-order iterative schemes, we develop a scaled inertial forward-backward algorithm which is based on a metric changing at each iteration and on a suitable extrapolation step. Unlike standard forward-backward methods with extrapolation, our scheme is able to handle functions whose domain is not the entire space. Both {an ${\mathcal O}(1/k^2)$ convergence rate estimate on the objective function values and the convergence of the sequence of the iterates} are proved. Numerical experiments on several {test problems arising from image processing, compressed sensing and statistical inference} show the {effectiveness} of the proposed method in comparison to well performing {state-of-the-art} algorithms

A variable metric proximal stochastic gradient method: An application to classification problems

Due to the continued success of machine learning and deep learning in particular, supervised classification problems are ubiquitous in numerous scientific fields. Training these models typically involves the minimization of the empirical risk over large data sets along with a possibly non-differentiable regularization. In this paper, we introduce a stochastic gradient method for the considered classification problem. To control the variance of the objective's gradients, we use an automatic sample size selection along with a variable metric to precondition the stochastic gradient directions. Further, we utilize a non -monotone line search to automatize step size selection. Convergence results are provided for both convex and non-convex objective functions. Extensive numerical experiments verify that the suggested approach performs on par with stateof-the-art methods for training both statistical models for binary classification and artificial neural networks for multi-class image classification. The code is publicly available at https:// github .com /koblererich /lisavm

An introduction to continuous optimization for imaging

International audienceA large number of imaging problems reduce to the optimization of a cost function , with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification