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Conference on Wavelets and Sparsity XVII

作者： Goossens, Bart Luong, Hiep Philips, Wilfried Univ Ghent Dept Telecommun & Informat Proc Imec IPI UGent Sint Pietersnieuwstr 41 B-9000 Ghent Belgium

ISBN: (数字)9781510612464

ISBN: (纸本)9781510612464;9781510612457

Many inverse problems (e.g., demosaicking, deblurring, denoising, image fusion, HDR synthesis) share various similarities: degradation operators are often modeled by a specific data fitting function while image prior knowledge (e.g., sparsity) is incorporated by additional regularization terms. In this paper, we investigate automatic algorithmic techniques for evaluating proximal operators. These algorithmic techniques also enable efficient calculation of adjoints from linear operators in a general matrix-free setting. In particular, we study the simultaneous-direction method of multipliers (SDMM) and the parallel proximal algorithm (PPXA) solvers and show that the automatically derived implementations are well suited for both single-GPU and multi-GPU processing. We demonstrate this approach for an Electron Microscopy (EM) deconvolution problem.

关键词： convex optimization automatic differentiation inverse problems proximal algorithms

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Delay-tolerant distributed Bregman proximal algorithms

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OPTIMIZATION METHODS & SOFTWARE 2024年

作者： Chraibi, S. Iutzeler, F. Malick, J. Rogozin, A. Univ Grenoble Alpes LJK Grenoble France Univ Grenoble Alpes CNRS Grenoble INP LJK Grenoble France Moscow Inst Phys & Technol Moscow Russia Univ Grenoble Alpes LJK F-38000 Grenoble France

Many problems in machine learning write as the minimization of a sum of individual loss functions over the training examples. These functions are usually differentiable but, in some cases, their gradients are not Lipschitz continuous, which compromises the use of (proximal) gradient algorithms. Fortunately, changing the geometry and using Bregman divergences can alleviate this issue in several applications, such as for Poisson linear inverse problems. However, the Bregman operation makes the aggregation of several points and gradients more involved, hindering the distribution of computations for such problems. In this paper, we propose an asynchronous variant of the Bregman proximal-gradient method, able to adapt to any centralized computing system. In particular, we prove that the algorithm copes with arbitrarily long delays and we illustrate its behaviour on distributed Poisson inverse problems.

关键词： Distributed optimization asynchronous methods proximal algorithms Bregman divergence Poisson inverse problems

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Ptychographic phase retrieval by proximal algorithms

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NEW JOURNAL OF PHYSICS 2020年第2期22卷

作者： Yan, Hanfei Brookhaven Natl Lab Natl Synchrotron Light Source 2 Upton NY 11973 USA

We derive a set of ptychography phase-retrieval iterative engines based on proximal algorithms originally developed in convex optimization theory, and discuss their connections with existing ones. The use of proximal operator creates a simple frame work that allows us to incorporate the effect of noise from a maximum-likelihood (ML) principle. We focus on three particular algorithms, namely proximal minimization, alternating direction method of multiplier and accelerated proximal gradient (APG). We benchmark their performance with numerical simulations, and discuss their optimal conditions for convergence and accuracy. An experimental dataset is used to demonstrate their effectiveness as well, in which case an array of cubic Au nanoparticles with a size of 50 nm is imaged. We show that with the presence of Poisson noise, a dataset with photon counts up to 10(4) at one detector pixel already requires ML-based methods to achieve a stable convergence. Among the three algorithms derived in this work, APG method is reported first time for its application in ptychographic reconstruction and shows superior performance in terms of both accuracy and convergence rate with a noisy dataset.

关键词： ptychographic phase-retrieval proximal algorithms x-ray imaging maximum-likelihood

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A new class of alternating proximal minimization algorithms with costs-to-move

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SIAM JOURNAL ON OPTIMIZATION 2007年第3期18卷 1061-1081页

作者： Attouch, H. Redont, P. Soubeyran, A. Univ Montpellier 2 Inst Math & Modelisat Montpellier CNRS UMR 5149 F-34095 Montpellier France Univ Aix Marseille 2 CNRS GREQAM UMR 6579 F-13290 Les Milles France

Given two objective functions f : X -> R boolean OR{+infinity} and g : Y -> R boolean OR{+infinity} on abstract spaces X and Y, and a coupling function c : X x Y -> R+, we introduce and study alternative minimization algorithms of the following type: ( x(0), y(0)) is an element of X x Y given;(x(n), y(n)) -> ( x(n+1), y(n)) -> (x(n+1), y(n+1)) as follows: x(n+1) is an element of argmin{f(xi) + beta(n)c(xi, y(n)) + alpha(n)h(x(n), xi) : xi is an element of X}, y(n+1) is an element of argmin{g(eta) + mu(n)c(x(n+1), eta) + nu(n)k( y(n), eta) : eta is an element of Y}. Their most original feature is the introduction of the terms h : X x X -> R+ and k : Y x Y -> R+ which are costs to change or to move (distance- like functions, relative entropies) accounting for various inertial, friction, or anchoring e. ects. These algorithms are studied in a general abstract framework. The introduction of the costs to change h and k leads to proximal minimizations with corresponding dissipative effects. As a result, the algorithms enjoy nice convergent properties. Coefficients alpha(n), beta(n), mu(n), nu(n) are nonnegative parameters. When taking alpha(n) = nu(n) = 0 and quadratic costs on a Hilbert space, one recovers the classical alternating minimization algorithm, which itself is a natural extension of the alternating projection algorithm of von Neumann. A number of new signi. cant results hold in general metric spaces. We pay particular attention to the following cases: (1.) (X, d(X)) and (Y, d(Y)) are complete metric spaces and h >= d(X), k >= d(Y) ("high local costs to move");the algorithms then provide sequences that converge to Nash equilibria. (2.) X = Y = H is a Hilbert space, the costs to change are quadratic ("low local costs to move") and the functions f, g : H -> R boolean OR{+infinity} are closed, convex, proper;then some of the classical convergence theorems for alternating convex minimization algorithms, including those of Acker and Prestel, are properly extended wi

关键词： alternating minimization alternating projection proximal algorithms costs to change inertia anchoring effect dynamical game theory Nash equilibria steepest descent dissipative dynamical systems

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A new class of alternating proximal minimization algorithms with costs-to-move

A new class of alternating proximal minimization algorithms ...

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10th Workshop on Well-Posedness of Optimization Problems and Related Topics

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MODULAR-proximal GRADIENT algorithms IN VARIABLE EXPONENT LEBESGUE SPACES

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2022年第6期44卷 A3463-A3489页

作者： Lazzaretti, Marta Calatroni, Luca Estatico, Claudio Univ Cote Azur CNRS Lab I3S F-06900 Sophia Antipolis France Univ Genoa Dipartimento Matemat I-16146 Genoa Italy

We consider structured optimization problems defined in terms of the sum of a smooth and convex function and a proper, lower semicontinuous (l.s.c.), convex (typically nonsmooth) function in reflexive variable exponent Lebesgue spaces Lp(\cdot )(\Omega ). Due to their intrinsic space-variant properties, such spaces can be naturally used as solution spaces and combined with space-variant functionals for the solution of ill-posed inverse problems. For this purpose, we propose and analyze two instances (primal and dual) of proximal-gradient algorithms in Lp(\cdot )(\Omega ), where the proximal step, rather than depending on the natural (nonseparable) Lp(\cdot )(\Omega ) norm, is defined in terms of its modular function, which, thanks to its separability, allows for the efficient computation of algorithmic iterates. Convergence in function values is proved for both algorithms, with convergence rates depending on problem/space smoothness. To show the effectiveness of the proposed modeling, some numerical tests highlighting the flexibility of the space Lp(\cdot )(\Omega ) are shown for exemplar deconvolution and mixed noise removal problems. Finally, a numerical verification of the convergence speed and computational costs of both algorithms in comparison with analogous ones defined in standard Lp(\Omega ) spaces is presented.

关键词： proximal algorithms variable exponent Lebesgue spaces modular proximity sep-arability thresholding functions

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Stagewise Training With Exponentially Growing Training Sets

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2025年第4期36卷 6148-6158页

作者： Gu, Bin Alquabeh, Hilal de Vazelhes, William Huo, Zhouyuan Huang, Heng Jilin Univ Sch Artificial Intelligence Changchun 130012 Peoples R China Mohamed bin Zayed Univ Artificial Intelligence Dept Machine Learning Abu Dhabi U Arab Emirates Univ Pittsburgh Dept Elect & Comp Engn Pittsburgh PA 15260 USA Univ Maryland Dept Comp Sci College Pk MD 20742 USA

In the world of big data, training large-scale machine learning problems has gained considerable attention. Numerous innovative optimization strategies have been presented in recent years to accelerate the large-scale training process. However, the possibility of further accelerating the training process of various optimization algorithms remains an unresolved subject. To begin addressing this difficult problem, we exploit the researched findings that when training data are independent and identically distributed, the learning problem on a smaller dataset is not significantly different from the original one. Upon that, we propose a stagewise training technique that grows the size of the training set exponentially while solving nonsmooth subproblem. We demonstrate that our stagewise training via exponentially growing the size of the training sets (STEGSs) are compatible with a large number of proximal gradient descent and gradient hard thresholding (GHT) techniques. Interestingly, we demonstrate that STEGS can greatly reduce overall complexity while maintaining statistical accuracy or even surpassing the intrinsic error introduced by GHT approaches. In addition, we analyze the effect of the training data growth rate on the overall complexity. The practical results of applying l(2,1) -and l(0) -norms to a variety of large-scale real-world datasets not only corroborate our theories but also demonstrate the benefits of our STEGS framework.

关键词： Training Optimization Computational complexity Upper bound Approximation algorithms Training data Risk minimization Gradient hard thresholding (GHT) algorithms proximal algorithms stagewise training strategy stochastic optimization

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proximal Markov chain Monte Carlo algorithms

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STATISTICS AND COMPUTING 2016年第4期26卷 745-760页

作者： Pereyra, Marcelo Univ Bristol Dept Math Bristol BS8 1TW Avon England

This paper presents a new Metropolis-adjusted Langevin algorithm (MALA) that uses convex analysis to simulate efficiently from high-dimensional densities that are log-concave, a class of probability distributions that is widely used in modern high-dimensional statistics and data analysis. The method is based on a new first-order approximation for Langevin diffusions that exploits log-concavity to construct Markov chains with favourable convergence properties. This approximation is closely related to Moreau-Yoshida regularisations for convex functions and uses proximity mappings instead of gradient mappings to approximate the continuous-time process. The proposed method complements existing MALA methods in two ways. First, the method is shown to have very robust stability properties and to converge geometrically for many target densities for which other MALA are not geometric, or only if the step size is sufficiently small. Second, the method can be applied to high-dimensional target densities that are not continuously differentiable, a class of distributions that is increasingly used in image processing and machine learning and that is beyond the scope of existing MALA and HMC algorithms. To use this method it is necessary to compute or to approximate efficiently the proximity mappings of the logarithm of the target density. For several popular models, including many Bayesian models used in modern signal and image processing and machine learning, this can be achieved with convex optimisation algorithms and with approximations based on proximal splitting techniques, which can be implemented in parallel. The proposed method is demonstrated on two challenging high-dimensional and non-differentiable models related to image resolution enhancement and low-rank matrix estimation that are not well addressed by existing MCMC methodology.

关键词： Bayesian inference Convex analysis High-dimensional statistics Markov chain Monte Carlo proximal algorithms Signal processing

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On the exponential convergence rate of proximal gradient flow algorithms 57

On the exponential convergence rate of proximal gradient flo...

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57th IEEE Conference on Decision and Control (CDC)

作者： Hassan-Moghaddam, Sepideh Jovanovic, Mihailo R. Univ Southern Calif Ming Hsieh Dept Elect Engn Los Angeles CA 90089 USA

ISBN: (纸本)9781538613955

Many modern large-scale and distributed optimization problems can be cast into a form in which the objective function is a sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal gradient method which generalizes standard gradient descent to a nonsmooth setup. In this paper, we leverage the tools from control theory to study global convergence of proximal gradient flow algorithms. We utilize the fact that the proximal gradient algorithm can be interpreted as a variable-metric gradient method on the forward-backward envelope. This continuously differentiable function can be obtained from the augmented Lagrangian associated with the original nonsmooth problem and it enjoys a number of favorable properties. We prove that global exponential convergence can be achieved even in the absence of strong convexity. Moreover, for in-network optimization problems, we provide a distributed implementation of the gradient flow dynamics based on the proximal augmented Lagrangian and prove global exponential stability for strongly convex problems.

关键词： Distributed optimization forward-backward envelope global exponential stability gradient flow large-scale systems proximal algorithms primal-dual method proximal augmented Lagrangian

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Comparison of proximal First-Order Primal and Primal-Dual algorithms via Performance Estimation 32

Comparison of Proximal First-Order Primal and Primal-Dual Al...

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32nd European Signal Processing Conference (EUSIPCO)

作者： Bousselmi, Nizar Pustelnik, Nelly Hendrickx, Julien M. Glineur, Francois UCLouvain ICTEAM Louvain La Neuve Belgium ENS Lyon CNRS Lyon France UCLouvain CORE ICTEAM Louvain La Neuve Belgium

ISBN: (纸本)9789464593617;9798331519773

Selecting the fastest algorithm for a specific signal/image processing task is a challenging question. We propose an approach based on the Performance Estimation Problem framework that numerically and automatically computes the worst-case performance of a given optimization method on a class of functions. We first propose a computer-assisted analysis and comparison of several first-order primal optimization methods, namely, the gradient method, the forward-backward, Peaceman-Rachford, and Douglas-Rachford splittings. We tighten the existing convergence results of these algorithms and extend them to new classes of functions. Our analysis is then extended and evaluated in the context of the primal-dual Chambolle-Pock and Condat-Vu methods.

关键词： Complexity analysis proximal algorithms Primal methods Primal-dual methods

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