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On the convergence of stochastic gradient descent with bandwidth-based step size

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arXiv 2021年

作者： Wang, Xiaoyu Yuan, Ya-Xiang Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences No.19A Yuquan Road Beijing100049 China State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

We investigate the stochastic gradient descent (SGD) method where the step size lies within a banded region instead of being given by a fixed formula. The optimal convergence rate under mild conditions and large initial step size is proved. Our analysis provides comparable theoretical error bounds for SGD associated with a variety of step sizes. In addition, the convergence rates for some existing step size strategies, e.g., triangular policy and cosine-wave, can be revealed by our analytical framework under the boundary constraints. The bandwidth-based step size provides efficient and flexible step size selection in optimization. We also propose a 1/t up-down policy and give several non-monotonic step sizes. Numerical experiments demonstrate the efficiency and significant potential of the bandwidth-based step-size in many *** Codes 65K05, 65K10, 90C15 Copyright © 2021, The Authors. All rights reserved.

关键词： Bandwidth

A diffuse-interface lattice Boltzmann method for the dendritic growth with thermosolutal convection

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arXiv 2022年

作者： Zhan, Chengjie Chai, Zhenhua Shi, Baochang Jiang, Ping Geng, Shaoning Sun, Dongke School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and Engineering Huazhong University of Science and Technology Wuhan430074 China School of Mechanical Engineering Southeast University Nanjing211189 China

In this work, we proposed a diffuse interface model for the dendritic growth with thermosolutal convection. In this model, the sharp boundary between the fluid and solid dendrite is replaced by a thin but nonzero thickness diffuse interface, which is described by the order parameter governed by the phase-field equation for the dendritic growth. The governing equations for solute and heat transfer are modified such that the previous special treatments for source term can be avoided. To solve the model for the dendritic growth with thermosolutal convection, we also developed a diffuse-interface multi-relaxation-time lattice Boltzmann (LB) method. In this method, the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction, and the treatment on the complex fluid-solid interface can be avoided. In addition, four LB models are developed for the phase-field equation, concentration equation, temperature equation and the Navier-Stokes equations in a unified framework. Finally, to test the present diffuse-interface LB method, we performed some simulations of the dendritic growth, and found that the numerical results are in good agreements with some previous works. © 2022, CC BY-NC-ND.

关键词： Navier Stokes equations

A Constraint Dissolving Approach for Nonsmooth Optimization over the Stiefel Manifold

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arXiv 2022年

作者： Hu, Xiaoyin Xiao, Nachuan Liu, Xin Toh, Kim-Chuan School of Computer and Computing Science Hangzhou City University Hangzhou310015 China The Institute of Operations Research and Analytics National University of Singapore Singapore State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China Department of Mathematics Institute of Operations Research and Analytics National University of Singapore Singapore119076 Singapore

This paper focus on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems over the Stiefel manifold. We propose a framework for developing subgradient-based methods and establish their convergence properties based on prior works. Furthermore, based on our proposed framework, we can develop efficient approaches for optimization over the Stiefel manifold. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach yields efficient and direct implementations of various unconstrained approaches to nonsmooth optimization problems over the Stiefel manifold. Copyright © 2022, The Authors. All rights reserved.

关键词： Optimization

Towards Efficient Large-Scale Network Slicing: An LP Dynamic Rounding-and-Refinement Approach

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arXiv 2021年

作者： Chen, Wei-Kun Liu, Ya-Feng Liu, Fan Dai, Yu-Hong Luo, Zhi-Quan School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology Beijing100081 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Electrical and Electronic Engineering Southern University of Science and Technology Shenzhen518055 China Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen518172 China

In this paper, we propose an efficient algorithm for the network slicing problem which attempts to map multiple customized virtual network requests (also called services) to a common shared network infrastructure and allocate network resources to meet diverse service requirements. The problem has been formulated as a mixed integer linear programming (MILP) formulation in the literature. We first propose a novel linear programming (LP) relaxation of the MILP formulation. We show that compared with a natural LP relaxation of the MILP formulation, the novel LP relaxation is much more compact in terms of smaller numbers of variables and constraints, and much stronger in terms of providing a better LP bound, which makes it particularly suitable to be embedded in an LP relaxation based algorithm. Then we design an efficient two-stage LP dynamic rounding-and-refinement algorithm based on this novel LP relaxation. In the first stage, the proposed algorithm uses an LP dynamic rounding procedure to place the virtual network functions of all services into cloud nodes while taking traffic routing of all services into consideration;in the second stage, the proposed algorithm uses an iterative LP refinement procedure to obtain a solution for traffic routing of all services with their end-to-end delay constraints being satisfied. Compared with the existing algorithms which either have an exponential complexity or return a low-quality solution, our proposed algorithm achieves a better trade-off between the solution quality and the computational complexity. In particular, the worst-case complexity of our proposed algorithm is polynomial, which makes it suitable for solving large-scale problems. Numerical results demonstrate the effectiveness and efficiency of our proposed algorithm. Copyright © 2021, The Authors. All rights reserved.

关键词： Iterative methods

Cell-Free Massive MIMO-OFDM for High-Speed Train Communications

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arXiv 2022年

作者： Zheng, Jiakang Zhang, Jiayi Björnson, Emil Li, Zhetao Ai, Bo The School of Electronic and Information Engineering Beijing Jiaotong University Beijing100044 China The Department of Computer Science KTH Royal Institute of Technology Kista Sweden The Hunan International Scientific and Technological Cooperation Base of Intelligent Network Key Laboratory of Intelligent Computing & Information Processing Ministry of Education Xiangtan University Hunan411105 China The State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing100044 China

Cell-free (CF) massive multiple-input multiple-output (MIMO) systems show great potentials in low-mobility scenarios, due to cell boundary disappearance and strong macro diversity. However, the great Doppler frequency offset (DFO) leads to serious inter-carrier interference in orthogonal frequency division multiplexing (OFDM) technology, which makes it difficult to provide high-quality transmissions for both high-speed train (HST) operation control systems and passengers. In this paper, we focus on the performance of CF massive MIMO-OFDM systems with both fully centralized and local minimum mean square error (MMSE) combining in HST communications. Considering the local maximum ratio (MR) combining, the large-scale fading decoding (LSFD) cooperation and the practical effect of DFO on system performance, exact closed-form expressions for uplink spectral efficiency (SE) expressions are derived. We observe that cooperative MMSE combining achieves better SE performance than uncooperative MR combining. In addition, HST communications with small cell and cellular massive MIMO-OFDM systems are compared in terms of SE. Numerical results reveal that the CF massive MIMO-OFDM system achieves a larger and more uniform SE than the other systems. Finally, the train antenna centric (TA-centric) CF massive MIMO-OFDM system is designed for practical implementation in HST communications, and three power control schemes are adopted to optimize the propagation of TAs for reducing the impact of the DFO. Copyright © 2022, The Authors. All rights reserved.

关键词： MIMO systems

A Communication-Efficient And Privacy-Aware Distributed Algorithm For Sparse PCA

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arXiv 2021年

作者： Wang, Lei Liu, Xin Zhang, Yin State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China School of Data Science The Chinese University of Hong Kong Shenzhen China

Sparse principal component analysis (PCA) improves interpretability of the classic PCA by introducing sparsity into the dimension-reduction process. Optimization models for sparse PCA, however, are generally non-convex, non-smooth and more difficult to solve, especially on large-scale datasets requiring distributed computation over a wide network. In this paper, we develop a distributed and centralized algorithm called DSSAL1 for sparse PCA that aims to achieve low communication overheads by adapting a newly proposed subspace-splitting strategy to accelerate convergence. Theoretically, convergence to stationary points is established for DSSAL1. Extensive numerical results show that DSSAL1 requires far fewer rounds of communication than state-of-the-art peer methods. In addition, we make the case that since messages exchanged in DSSAL1 are well-masked, the possibility of private-data leakage in DSSAL1 is much lower than in some other distributed algorithms. Copyright © 2021, The Authors. All rights reserved.

关键词： Principal component analysis

A Stochastic Galerkin Lattice Boltzmann Method for Incompressible Fluid Flows with Uncertainties

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SSRN

SSRN 2024年

作者： Zhong, Mingliang Xiao, Tianbai Krause, Mathias J. Frank, Martin Simonis, Stephan Scientific Computing Center Karlsruhe Institute of Technology Eggenstein-Leopoldshafen76344 Germany State Key Laboratory of High Temperature Gas Dynamics Centre for Interdisciplinary Research in Fluids Institute of Mechanics Chinese Academy of Sciences Beijing100190 China School of Engineering Science University of Chinese Academy of Sciences Beijing100049 China Institute for Applied and Numerical Mathematics Karlsruhe Institute of Technology Karlsruhe76131 Germany Institute of Mechanical Process Engineering and Mechanics Karlsruhe Institute of Technology Karlsruhe76131 Germany Lattice Boltzmann Research Group Karlsruhe Institute of Technology Karlsruhe76131 Germany

Efficient modeling and simulation of uncertainties in computational fluid dynamics (CFD) remains a crucial challenge. In this paper, we present the first stochastic Galerkin (SG) lattice Boltzmann method (LBM) built upon the generalized polynomial chaos (gPC). The proposed method offers an efficient and accurate approach that depicts the propagation of uncertainties in stochastic incompressible flows. Formal analysis shows that the SG LBM preserves the correct Chapman-Enskog asymptotics and recovers the corresponding macroscopic fluid equations. Numerical experiments, including the Taylor-Green vortex flow, lid-driven cavity flow, and isentropic vortex convection, are presented to validate the solution algorithm. The results demonstrate that the SG LBM achieves the expected spectral convergence and the computational cost is significantly reduced compared to the sampling-based non-intrusive approaches, e.g., the routinely used Monte Carlo method. We obtain a speedup factor of 5.72 compared to Monte Carlo sampling in a randomized two-dimensional Taylor-Green vortex flow test case. By leveraging the accuracy and flexibility of LBM and the efficiency of gPC-based SG, the proposed SG LBM provides a powerful framework for uncertainty quantification in CFD practice. © 2024, The Authors. All rights reserved.

关键词： Computational fluid dynamics

LINEARLY-CONSTRAINED NONSMOOTH OPTIMIZATION FOR TRAINING AUTOENCODERS

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arXiv 2021年

作者： Liu, Wei Liu, Xin Chen, Xiaojun State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong

A regularized minimization model with l1-norm penalty (RP) is introduced for training the autoencoders that belong to a class of two-layer neural networks. We show that the RP can act as an exact penalty model which shares the same global minimizers, local minimizers, and d(irectional)-stationary points with the original regularized model under mild conditions. We construct a bounded box region that contains at least one global minimizer of the RP, and propose a linearly constrained regularized minimization model with l1-norm penalty (LRP) for training autoencoders. A smoothing proximal gradient algorithm is designed to solve the LRP. Convergence of the algorithm to a generalized d-stationary point of the RP and LRP is delivered. Comprehensive numerical experiments convincingly illustrate the efficiency as well as the robustness of the proposed algorithm. Copyright © 2021, The Authors. All rights reserved.

关键词： Constrained optimization

On Minimization of Upper Bound for the Convergence Rate of the QHSS Iteration Method

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Communications on Applied Mathematics and Computation 2019年第2期1卷 263-282页

作者： Wen-Ting Wu State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of Sciences. P.O. Box 2719. Beijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

For an upper bound of the spectral radius of the QHSS (quasi Hermitian and skew-Hermitian splitting) iteration matrix which can also bound the contraction factor of the QHSS iteration method,we give its minimum point under the conditions which guarantee that the upper bound is strictly less than one. This provides a good choice of the involved iteration parameters,so that the convergence rate of the QHSS iteration method can be significantly improved.

关键词： System of linear equations Non-Hermitian matrix QHSS iteration method Convergence rate