Two-step inertial Bregman alternating minimization algorithm for nonconvex and nonsmooth problems

Zhao, Jing; Dong, Qiao-Li; Rassias, Michael Th.; Wang, Fenghui

doi:10.1007/s10898-022-01176-6

Cited by 6 publications

(2 citation statements)

References 53 publications

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“…Furthermore, Wang et al (Wang et al 2022) investigated a Bregman and inertial extension of the forward-reflectedbackward algorithm (Malitsky and Tam 2020) under relative smoothness conditions. Numerous other works have explored the Bregman gradient method framework to tackle the absence of global Lipschitz continuous gradients, including (Reem, Reich, and Pierro 2019;Zhao et al 2022;Zhu et al 2021;Dragomir et al 2022).…”

Section: Introductionmentioning

confidence: 99%

A Bregman Proximal Stochastic Gradient Method with Extrapolation for Nonconvex Nonsmooth Problems

Wang,

Liu,

Cui

et al. 2024

AAAI

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In this paper, we explore a specific optimization problem that involves the combination of a differentiable nonconvex function and a nondifferentiable function. The differentiable component lacks a global Lipschitz continuous gradient, posing challenges for optimization. To address this issue and accelerate the convergence, we propose a Bregman proximal stochastic gradient method with extrapolation (BPSGE), which only requires smooth adaptivity of the differentiable part. Under variance reduction framework, we not only analyze the subsequential and global convergence of the proposed algorithm under certain conditions, but also analyze the sublinear convergence rate of the subsequence, and the complexity of the algorithm, revealing that the BPSGE algorithm requires at most O(epsilon\^\,(-2)) iterations in expectation to attain an epsilon-stationary point. To validate the effectiveness of our proposed algorithm, we conduct numerical experiments on three real-world applications: graph regularized nonnegative matrix factorization (NMF), matrix factorization with weakly-convex regularization, and NMF with nonconvex sparsity constraints. These experiments demonstrate that BPSGE is faster than the baselines without extrapolation. The code is available at: https://github.com/nothing2wang/BPSGE-Algorithm.

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Section: Introductionmentioning

confidence: 99%

A Bregman Proximal Stochastic Gradient Method with Extrapolation for Nonconvex Nonsmooth Problems

Wang,

Liu,

Cui

et al. 2024

AAAI

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show abstract

Section: Introductionmentioning

confidence: 99%

History of Mathematics in American Mathematical Textbooks of the Late 19th and Early 20th Centuries

Wang

2021

School Mathematics Textbooks in China

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The logarithmic Schrödinger equation (LogSE) has a logarithmic nonlinearity f (u) = u ln |u| 2 that is not differentiable at u = 0. Compared with its counterpart with a regular nonlinear term, it possesses richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges in both analysis and computation. Among very limited numerical studies, the semi-implicit regularized method via regularising f (u) as u ε ln(ε+|u ε |) 2 to overcome the blowup of ln |u| 2 at u = 0 has been investigated recently in literature. With the understanding of f (0) = 0, we analyze the non-regularized first-order Implicit-Explicit (IMEX) scheme for the LogSE. We introduce some new tools for the error analysis that include the characterization of the Hölder continuity of the logarithmic term, and a nonlinear Grönwall's inequality. We provide ample numerical results to demonstrate the expected convergence. We position this work as the first one to study the direct linearized scheme for the LogSE as far as we can tell.

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A stochastic two-step inertial Bregman proximal alternating linearized minimization algorithm for nonconvex and nonsmooth problems

Guo,

Zhao,

Dong

2023

Numer Algor

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Two-step inertial Bregman alternating minimization algorithm for nonconvex and nonsmooth problems

Cited by 6 publications

References 53 publications

A Bregman Proximal Stochastic Gradient Method with Extrapolation for Nonconvex Nonsmooth Problems

A Bregman Proximal Stochastic Gradient Method with Extrapolation for Nonconvex Nonsmooth Problems

History of Mathematics in American Mathematical Textbooks of the Late 19th and Early 20th Centuries

A stochastic two-step inertial Bregman proximal alternating linearized minimization algorithm for nonconvex and nonsmooth problems

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