A Brief Overview of Practical Optimization Algorithms in the Context of Relaxation
|
|
|
- Sharon Moody
- 5 years ago
- Views:
Transcription
1 A Brief Overview of Practical Optimization Algorithms in the Context of Relaxation Zhouchen Lin Peking University April 22, 2018
2 Too Many Opt. Problems!
3 Too Many Opt. Algorithms! Zero-th order algorithms: Interpolation Methods Pattern Search Methods Using f(x k ) only First order algorithms: Using f(x k ) & f(x k ) only Coordinate Descent (Stochastic) Gradient/Subgradient Descent Conjugate Gradient Quasi-Newton/Hessian-Free Methods (Augmented) Lagrangian Method of Multipliers
4 Too Many Opt. Algorithms! Second order algorithms: Newton s Method Sequential Quadratic Programming Interior Point Methods Using f(x k ), f(x k ) & H f (x k ) only
5 Questions Can we master optimization algorithms more systematically? Is there a methodology to tweak existing algorithms (without proofs)?
6 Model Optimization Problem & Algorithm relaxing the objective function relaxing the constraint relaxing the update process
7 Relaxing the Objective Function Majorization Minimization globally majorant globally maj. locally maj.
8 Relaxing the Objective Function Majorization Minimization How to choose the majorant function? projected gradient descent gradient descent
9 Relaxing the Objective Function Majorization Minimization How to choose the stepsize α? locally majorant backtracking globally majorant Armijo s rule
10 C. Xu, Z. Lin, and H. Zha, Relaxed Majorization-Minimization for Non-smooth and Non-convex Optimization, AAAI Zhouchen Lin, Chen Xu,and Hongbin Zha, Robust Matrix Factorization by Majorization-Minimization, IEEE TPAMI, Relaxing the Objective Function Majorization Minimization How to choose the majorant function? asymptotic smoothness globally majorant locally majorant Relaxed Majorization Minimization
11 Relaxing the Objective Function Majorization Minimization How to choose the majorant function? convex concave procedure (CCCP) low-rankness regularizer A. L. Yuille, A. Rangarajan, The Concave-Convex Procedure. Neural Computation 15(4): (2003). C. Lu, J. Tang, S. Yan, Z. Lin, Generalized nonconvex nonsmooth low-rank minimization, CVPR C. Lu, J. Tang, S. Yan, Z. Lin, Nonconvex nonsmooth low-rank minimization via iteratively reweighted nuclear norm, IEEE TIP Canyi Lu, Yunchao Wei, Zhouchen Lin, and Shuicheng Yan, Proximal Iteratively Reweighted Algorithm with Multiple Splitting for Nonconvex Sparsity Optimization, AAAI 2014.
12 Relaxing the Objective Function Majorization Minimization How to choose the majorant function? Chen Xu, Zhouchen Lin, and Hongbin Zha, A Unified Convex Surrogate for the Schatten-p Norm, AAAI Fanhua Shang, Yuanyuan Liu, James Cheng, Zhi-Quan Luo, and Zhouchen Lin, Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications, IEEE TPAMI, 2018.
13 Relaxing the Objective Function Majorization Minimization How to choose the majorant function? l p -norm, sparsity regularizer Schatten-p norm, low-rankness regularizer Iteratively Reweighted Least Squares E. Candès, M.B. Wakin, S.P. Boyd, Enhancing sparsity by reweighted l 1 minimization, Journal of Fourier Analysis and Applications 14 (5 6) (2008) C. Lu, Z. Lin, S. Yan, Smoothed low rank and sparse matrix recovery by iteratively reweighted least squared minimization, IEEE TIP, 2015.
14 Relaxing the Objective Function Majorization Minimization How to choose the majorant function? Other choices: Julien Mairal. Incremental majorization-minimization optimization with application to large-scale machine learning. SIAM Journal on Optimization, 25(2): , 2015.
15 Relaxing the Constraints What if no relaxation? Only works for simple constraints
16 Relaxing the Constraints Method of Lagrange Multipliers may not have a solution not easy to choose
17 Relaxing the Constraints Method of Lagrange Multipliers majorant Zhouchen Lin, Risheng Liu, and Huan Li, Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty for Separable Convex Programs in Machine Learning, Machine Learning, 2015
18 Relaxing the Constraints Method of Lagrange Multipliers can be further relaxed if f is still complex proximal operator relatively easy to solve projection operator Zhouchen Lin, Risheng Liu, and Huan Li, Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty for Separable Convex Programs in Machine Learning, Machine Learning, 2015
19 Relaxing the Constraints Method of Lagrange Multipliers More investigations: Zhouchen Lin, Risheng Liu, and Huan Li, Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty for Separable Convex Programs in Machine Learning, Machine Learning, Canyi Lu, Jiashi Feng, Shuicheng Yan, and Zhouchen Lin, A Unified Alternating Direction Method of Multipliers by Majorization Minimization, IEEE Trans. Pattern Analysis and Machine Intelligence, 2018.
20 Relaxing the Update Process Relaxing the location to compute gradient Y. Nesterov, A method of solving a convex programming problem with convergence rate O(1/k 2 ), Soviet Mathematics Doklady 27 (2) (1983) A. Beck,M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences 2 (1) (2009)
21 Relaxing the Update Process Relaxing the location to compute gradient monitor corrector Canyi Lu, Huan Li, Zhouchen Lin, and Shuicheng Yan, Fast Proximal Linearized Alternating Direction Method of Multiplier with Parallel Splitting, pp , AAAI Huan Li and Zhouchen Lin, Accelerated Proximal Gradient Methods for Nonconvex Programming, NIPS 2015.
22 Relaxing the Update Process Relaxing the evaluation of gradient Stochastic gradient descent (SGD) snapshot vector fixed stepsize R. Johnson and T. Zhang, Accelerating stochastic gradient descent using predictive variance reduction. NIPS 2013.
23 Relaxing the Update Process Relaxing the update of variable Asynchronous update R. Johnson and T. Zhang, Accelerating stochastic gradient descent using predictive variance reduction. NIPS Cong Fang and Zhouchen Lin, Parallel Asynchronous Stochastic Variance Reduction for Nonconvex Optimization, AAAI 2017.
24 Relaxing the Update Process Relaxing the update of variable Asynchronous update Shared Memory Locked/ Unlocked Local Memory Atom/ Wild Local Memory Cong Fang and Zhouchen Lin, Parallel Asynchronous Stochastic Variance Reduction for Nonconvex Optimization, AAAI 2017.
25 Conclusions Relaxation is good and even necessary for optimization The same for life!
26 Thanks! Recruitment: PostDocs (540K RMB/year) and Faculties in machine learning related areas Please Google me and visit my webpage!
Graph based Subspace Segmentation. Canyi Lu National University of Singapore Nov. 21, 2013
Graph based Subspace Segmentation Canyi Lu National University of Singapore Nov. 21, 2013 Content Subspace Segmentation Problem Related Work Sparse Subspace Clustering (SSC) Low-Rank Representation (LRR)
Lasso: Algorithms and Extensions
ELE 538B: Sparsity, Structure and Inference Lasso: Algorithms and Extensions Yuxin Chen Princeton University, Spring 2017 Outline Proximal operators Proximal gradient methods for lasso and its extensions
Relaxed Majorization-Minimization for Non-smooth and Non-convex Optimization
Relaxed Majorization-Minimization for Non-smooth and Non-convex Optimization Chen Xu 1, Zhouchen Lin 1,2,, Zhenyu Zhao 3, Hongbin Zha 1 1 Key Laboratory of Machine Perception (MOE), School of EECS, Peking
ECE G: Special Topics in Signal Processing: Sparsity, Structure, and Inference
ECE 18-898G: Special Topics in Signal Processing: Sparsity, Structure, and Inference Sparse Recovery using L1 minimization - algorithms Yuejie Chi Department of Electrical and Computer Engineering Spring
Convex Optimization Algorithms for Machine Learning in 10 Slides
Convex Optimization Algorithms for Machine Learning in 10 Slides Presenter: Jul. 15. 2015 Outline 1 Quadratic Problem Linear System 2 Smooth Problem Newton-CG 3 Composite Problem Proximal-Newton-CD 4 Non-smooth,
Proximal Minimization by Incremental Surrogate Optimization (MISO)
Proximal Minimization by Incremental Surrogate Optimization (MISO) (and a few variants) Julien Mairal Inria, Grenoble ICCOPT, Tokyo, 2016 Julien Mairal, Inria MISO 1/26 Motivation: large-scale machine
Relaxed Majorization-Minimization for Non-smooth and Non-convex Optimization
Relaxed Majorization-Minimization for Non-smooth and Non-convex Optimization MM has been successfully applied to a wide range of problems. Mairal (2013 has given a comprehensive review on MM. Conceptually,
Low-Rank Subspace Clustering
Low-Rank Subspace Clustering Zhouchen Lin 1 & Jiashi Feng 2 1 Peking University 2 National University of Singapore June 26 Outline Representative Models Low-Rank Representation (LRR) Based Subspace Clustering
Optimization methods
Optimization methods Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda /8/016 Introduction Aim: Overview of optimization methods that Tend to
SIAM Conference on Imaging Science, Bologna, Italy, Adaptive FISTA. Peter Ochs Saarland University
SIAM Conference on Imaging Science, Bologna, Italy, 2018 Adaptive FISTA Peter Ochs Saarland University 07.06.2018 joint work with Thomas Pock, TU Graz, Austria c 2018 Peter Ochs Adaptive FISTA 1 / 16 Some
Learning with stochastic proximal gradient
Learning with stochastic proximal gradient Lorenzo Rosasco DIBRIS, Università di Genova Via Dodecaneso, 35 16146 Genova, Italy lrosasco@mit.edu Silvia Villa, Băng Công Vũ Laboratory for Computational and
Linearized Alternating Direction Method with Adaptive Penalty for Low-Rank Representation
Linearized Alternating Direction Method with Adaptive Penalty for Low-Rank Representation Zhouchen Lin Visual Computing Group Microsoft Research Asia Risheng Liu Zhixun Su School of Mathematical Sciences
Fast proximal gradient methods
L. Vandenberghe EE236C (Spring 2013-14) Fast proximal gradient methods fast proximal gradient method (FISTA) FISTA with line search FISTA as descent method Nesterov s second method 1 Fast (proximal) gradient
A Unified Approach to Proximal Algorithms using Bregman Distance
A Unified Approach to Proximal Algorithms using Bregman Distance Yi Zhou a,, Yingbin Liang a, Lixin Shen b a Department of Electrical Engineering and Computer Science, Syracuse University b Department
Supplementary Materials: Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 16 Supplementary Materials: Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications Fanhua Shang, Member, IEEE,
Optimization. Benjamin Recht University of California, Berkeley Stephen Wright University of Wisconsin-Madison
Optimization Benjamin Recht University of California, Berkeley Stephen Wright University of Wisconsin-Madison optimization () cost constraints might be too much to cover in 3 hours optimization (for big
Big Data Analytics: Optimization and Randomization
Big Data Analytics: Optimization and Randomization Tianbao Yang Tutorial@ACML 2015 Hong Kong Department of Computer Science, The University of Iowa, IA, USA Nov. 20, 2015 Yang Tutorial for ACML 15 Nov.
Robust PCA. CS5240 Theoretical Foundations in Multimedia. Leow Wee Kheng
Robust PCA CS5240 Theoretical Foundations in Multimedia Leow Wee Kheng Department of Computer Science School of Computing National University of Singapore Leow Wee Kheng (NUS) Robust PCA 1 / 52 Previously...
Agenda. Fast proximal gradient methods. 1 Accelerated first-order methods. 2 Auxiliary sequences. 3 Convergence analysis. 4 Numerical examples
Agenda Fast proximal gradient methods 1 Accelerated first-order methods 2 Auxiliary sequences 3 Convergence analysis 4 Numerical examples 5 Optimality of Nesterov s scheme Last time Proximal gradient method
Tensor-Tensor Product Toolbox
Tensor-Tensor Product Toolbox 1 version 10 Canyi Lu canyilu@gmailcom Carnegie Mellon University https://githubcom/canyilu/tproduct June, 018 1 INTRODUCTION Tensors are higher-order extensions of matrices
Stochastic Optimization: First order method
Stochastic Optimization: First order method Taiji Suzuki Tokyo Institute of Technology Graduate School of Information Science and Engineering Department of Mathematical and Computing Sciences JST, PRESTO
Robust Component Analysis via HQ Minimization
Robust Component Analysis via HQ Minimization Ran He, Wei-shi Zheng and Liang Wang 0-08-6 Outline Overview Half-quadratic minimization principal component analysis Robust principal component analysis Robust
Accelerated Proximal Gradient Methods for Convex Optimization
Accelerated Proximal Gradient Methods for Convex Optimization Paul Tseng Mathematics, University of Washington Seattle MOPTA, University of Guelph August 18, 2008 ACCELERATED PROXIMAL GRADIENT METHODS
Large-scale Stochastic Optimization
Large-scale Stochastic Optimization 11-741/641/441 (Spring 2016) Hanxiao Liu hanxiaol@cs.cmu.edu March 24, 2016 1 / 22 Outline 1. Gradient Descent (GD) 2. Stochastic Gradient Descent (SGD) Formulation
An Alternating Proximal Splitting Method with Global Convergence for Nonconvex Structured Sparsity Optimization
An Alternating Proximal Splitting Method with Global Convergence for Nonconvex Structured Sparsity Optimization Shubao Zhang and Hui Qian and Xiaojin Gong College of Computer Science and Technology Department
Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements
Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements Canyi Lu, Jiashi Feng 2, Zhouchen Li,4 Shuicheng Yan 5,2 Department of Electrical and Computer Engineering, Carnegie Mellon University 2
Selected Topics in Optimization. Some slides borrowed from
Selected Topics in Optimization Some slides borrowed from http://www.stat.cmu.edu/~ryantibs/convexopt/ Overview Optimization problems are almost everywhere in statistics and machine learning. Input Model
Supplementary Material of A Novel Sparsity Measure for Tensor Recovery
Supplementary Material of A Novel Sparsity Measure for Tensor Recovery Qian Zhao 1,2 Deyu Meng 1,2 Xu Kong 3 Qi Xie 1,2 Wenfei Cao 1,2 Yao Wang 1,2 Zongben Xu 1,2 1 School of Mathematics and Statistics,
A Conservation Law Method in Optimization
A Conservation Law Method in Optimization Bin Shi Florida International University Tao Li Florida International University Sundaraja S. Iyengar Florida International University Abstract bshi1@cs.fiu.edu
An ADMM algorithm for optimal sensor and actuator selection
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA An ADMM algorithm for optimal sensor and actuator selection Neil K. Dhingra, Mihailo R. Jovanović, and Zhi-Quan
Sparse Covariance Selection using Semidefinite Programming
Sparse Covariance Selection using Semidefinite Programming A. d Aspremont ORFE, Princeton University Joint work with O. Banerjee, L. El Ghaoui & G. Natsoulis, U.C. Berkeley & Iconix Pharmaceuticals Support
Accelerated primal-dual methods for linearly constrained convex problems
Accelerated primal-dual methods for linearly constrained convex problems Yangyang Xu SIAM Conference on Optimization May 24, 2017 1 / 23 Accelerated proximal gradient For convex composite problem: minimize
Incremental and Stochastic Majorization-Minimization Algorithms for Large-Scale Machine Learning
Incremental and Stochastic Majorization-Minimization Algorithms for Large-Scale Machine Learning Julien Mairal Inria, LEAR Team, Grenoble Journées MAS, Toulouse Julien Mairal Incremental and Stochastic
Generalized Singular Value Thresholding
Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence Generalized Singular Value Thresholding Canyi Lu, Chango Zhu, Chunyan u, Shuicheng Yan, Zhouchen Lin 3, Department of Electrical
Recent Advances in Structured Sparse Models
Recent Advances in Structured Sparse Models Julien Mairal Willow group - INRIA - ENS - Paris 21 September 2010 LEAR seminar At Grenoble, September 21 st, 2010 Julien Mairal Recent Advances in Structured
This can be 2 lectures! still need: Examples: non-convex problems applications for matrix factorization
This can be 2 lectures! still need: Examples: non-convex problems applications for matrix factorization x = prox_f(x)+prox_{f^*}(x) use to get prox of norms! PROXIMAL METHODS WHY PROXIMAL METHODS Smooth
Optimization for Machine Learning
Optimization for Machine Learning (Lecture 3-A - Convex) SUVRIT SRA Massachusetts Institute of Technology Special thanks: Francis Bach (INRIA, ENS) (for sharing this material, and permitting its use) MPI-IS
Improved Optimization of Finite Sums with Minibatch Stochastic Variance Reduced Proximal Iterations
Improved Optimization of Finite Sums with Miniatch Stochastic Variance Reduced Proximal Iterations Jialei Wang University of Chicago Tong Zhang Tencent AI La Astract jialei@uchicago.edu tongzhang@tongzhang-ml.org
Block stochastic gradient update method
Block stochastic gradient update method Yangyang Xu and Wotao Yin IMA, University of Minnesota Department of Mathematics, UCLA November 1, 2015 This work was done while in Rice University 1 / 26 Stochastic
A direct formulation for sparse PCA using semidefinite programming
A direct formulation for sparse PCA using semidefinite programming A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley Available online at www.princeton.edu/~aspremon
Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization
Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization Canyi Lu 1, Jiashi Feng 1, Yudong Chen, Wei Liu 3, Zhouchen Lin 4,5,, Shuicheng Yan 6,1
A Multilevel Proximal Algorithm for Large Scale Composite Convex Optimization
A Multilevel Proximal Algorithm for Large Scale Composite Convex Optimization Panos Parpas Department of Computing Imperial College London www.doc.ic.ac.uk/ pp500 p.parpas@imperial.ac.uk jointly with D.V.
arxiv: v2 [cs.na] 10 Feb 2018
![arxiv: v2 [cs.na] 10 Feb 2018](/thumbs/78/78591070.jpg "arxiv: v2 [cs.na] 10 Feb 2018")
Efficient Robust Matrix Factorization with Nonconvex Penalties Quanming Yao, James T. Kwok Department of Computer Science and Engineering Hong Kong University of Science and Technology Hong Kong { qyaoaa,
Proximal Gradient Descent and Acceleration. Ryan Tibshirani Convex Optimization /36-725
Proximal Gradient Descent and Acceleration Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: subgradient method Consider the problem min f(x) with f convex, and dom(f) = R n. Subgradient method:
Stochastic Composition Optimization
Stochastic Composition Optimization Algorithms and Sample Complexities Mengdi Wang Joint works with Ethan X. Fang, Han Liu, and Ji Liu ORFE@Princeton ICCOPT, Tokyo, August 8-11, 2016 1 / 24 Collaborators
PART II: Basic Theory of Half-quadratic Minimization
PART II: Basic Theory of Half-quadratic Minimization Ran He, Wei-Shi Zheng and Liang Wang 013-1-01 Outline History of half-quadratic (HQ) minimization Half-quadratic minimization HQ loss functions HQ in
Optimization methods
Lecture notes 3 February 8, 016 1 Introduction Optimization methods In these notes we provide an overview of a selection of optimization methods. We focus on methods which rely on first-order information,
Accelerating SVRG via second-order information
Accelerating via second-order information Ritesh Kolte Department of Electrical Engineering rkolte@stanford.edu Murat Erdogdu Department of Statistics erdogdu@stanford.edu Ayfer Özgür Department of Electrical
Ranking from Crowdsourced Pairwise Comparisons via Matrix Manifold Optimization
Ranking from Crowdsourced Pairwise Comparisons via Matrix Manifold Optimization Jialin Dong ShanghaiTech University 1 Outline Introduction FourVignettes: System Model and Problem Formulation Problem Analysis
Coordinate Descent and Ascent Methods
Coordinate Descent and Ascent Methods Julie Nutini Machine Learning Reading Group November 3 rd, 2015 1 / 22 Projected-Gradient Methods Motivation Rewrite non-smooth problem as smooth constrained problem:
Fast Nonnegative Matrix Factorization with Rank-one ADMM
Fast Nonnegative Matrix Factorization with Rank-one Dongjin Song, David A. Meyer, Martin Renqiang Min, Department of ECE, UCSD, La Jolla, CA, 9093-0409 dosong@ucsd.edu Department of Mathematics, UCSD,
Contents. Preface. 1 Introduction Optimization view on mathematical models NLP models, black-box versus explicit expression 3
Contents Preface ix 1 Introduction 1 1.1 Optimization view on mathematical models 1 1.2 NLP models, black-box versus explicit expression 3 2 Mathematical modeling, cases 7 2.1 Introduction 7 2.2 Enclosing
Parallel Successive Convex Approximation for Nonsmooth Nonconvex Optimization
Parallel Successive Convex Approximation for Nonsmooth Nonconvex Optimization Meisam Razaviyayn meisamr@stanford.edu Mingyi Hong mingyi@iastate.edu Zhi-Quan Luo luozq@umn.edu Jong-Shi Pang jongship@usc.edu
2 Regularized Image Reconstruction for Compressive Imaging and Beyond
EE 367 / CS 448I Computational Imaging and Display Notes: Compressive Imaging and Regularized Image Reconstruction (lecture ) Gordon Wetzstein gordon.wetzstein@stanford.edu This document serves as a supplement
Optimization for Learning and Big Data
Optimization for Learning and Big Data Donald Goldfarb Department of IEOR Columbia University Department of Mathematics Distinguished Lecture Series May 17-19, 2016. Lecture 1. First-Order Methods for
Sparse Optimization Lecture: Dual Methods, Part I
Sparse Optimization Lecture: Dual Methods, Part I Instructor: Wotao Yin July 2013 online discussions on piazza.com Those who complete this lecture will know dual (sub)gradient iteration augmented l 1 iteration
Topographic Dictionary Learning with Structured Sparsity
Topographic Dictionary Learning with Structured Sparsity Julien Mairal 1 Rodolphe Jenatton 2 Guillaume Obozinski 2 Francis Bach 2 1 UC Berkeley 2 INRIA - SIERRA Project-Team San Diego, Wavelets and Sparsity
SPARSE SIGNAL RESTORATION. 1. Introduction
SPARSE SIGNAL RESTORATION IVAN W. SELESNICK 1. Introduction These notes describe an approach for the restoration of degraded signals using sparsity. This approach, which has become quite popular, is useful
Master 2 MathBigData. 3 novembre CMAP - Ecole Polytechnique
Master 2 MathBigData S. Gaïffas 1 3 novembre 2014 1 CMAP - Ecole Polytechnique 1 Supervised learning recap Introduction Loss functions, linearity 2 Penalization Introduction Ridge Sparsity Lasso 3 Some
Numerisches Rechnen. (für Informatiker) M. Grepl P. Esser & G. Welper & L. Zhang. Institut für Geometrie und Praktische Mathematik RWTH Aachen
Numerisches Rechnen (für Informatiker) M. Grepl P. Esser & G. Welper & L. Zhang Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2011/12 IGPM, RWTH Aachen Numerisches Rechnen
Sparse and Regularized Optimization
Sparse and Regularized Optimization In many applications, we seek not an exact minimizer of the underlying objective, but rather an approximate minimizer that satisfies certain desirable properties: sparsity
A direct formulation for sparse PCA using semidefinite programming
A direct formulation for sparse PCA using semidefinite programming A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley A. d Aspremont, INFORMS, Denver,
arxiv: v3 [math.oc] 20 Mar 2013
![arxiv: v3 [math.oc] 20 Mar 2013](/thumbs/89/99208921.jpg "arxiv: v3 [math.oc] 20 Mar 2013")
1 Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers Fu Lin, Makan Fardad, and Mihailo R. Jovanović Abstract arxiv:1111.6188v3 [math.oc] 20 Mar 2013 We design sparse
Proximal Methods for Optimization with Spasity-inducing Norms
Proximal Methods for Optimization with Spasity-inducing Norms Group Learning Presentation Xiaowei Zhou Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology
IPAM Summer School Optimization methods for machine learning. Jorge Nocedal
IPAM Summer School 2012 Tutorial on Optimization methods for machine learning Jorge Nocedal Northwestern University Overview 1. We discuss some characteristics of optimization problems arising in deep
Lecture 2 February 25th
Statistical machine learning and convex optimization 06 Lecture February 5th Lecturer: Francis Bach Scribe: Guillaume Maillard, Nicolas Brosse This lecture deals with classical methods for convex optimization.
An ADMM algorithm for optimal sensor and actuator selection
An ADMM algorithm for optimal sensor and actuator selection Neil K. Dhingra, Mihailo R. Jovanović, and Zhi-Quan Luo 53rd Conference on Decision and Control, Los Angeles, California, 2014 1 / 25 2 / 25
Characterization of Gradient Dominance and Regularity Conditions for Neural Networks
Characterization of Gradient Dominance and Regularity Conditions for Neural Networks Yi Zhou Ohio State University Yingbin Liang Ohio State University Abstract zhou.1172@osu.edu liang.889@osu.edu The past
Distributed Consensus Optimization
Distributed Consensus Optimization Ming Yan Michigan State University, CMSE/Mathematics September 14, 2018 Decentralized-1 Backgroundwhy andwe motivation need decentralized optimization? I Decentralized
Optimized first-order minimization methods
Optimized first-order minimization methods Donghwan Kim & Jeffrey A. Fessler EECS Dept., BME Dept., Dept. of Radiology University of Michigan web.eecs.umich.edu/~fessler UM AIM Seminar 2014-10-03 1 Disclosure
Optimizing Nonconvex Finite Sums by a Proximal Primal-Dual Method
Optimizing Nonconvex Finite Sums by a Proximal Primal-Dual Method Davood Hajinezhad Iowa State University Davood Hajinezhad Optimizing Nonconvex Finite Sums by a Proximal Primal-Dual Method 1 / 35 Co-Authors
CSC 576: Variants of Sparse Learning
CSC 576: Variants of Sparse Learning Ji Liu Department of Computer Science, University of Rochester October 27, 205 Introduction Our previous note basically suggests using l norm to enforce sparsity in
ECS289: Scalable Machine Learning
ECS289: Scalable Machine Learning Cho-Jui Hsieh UC Davis Sept 29, 2016 Outline Convex vs Nonconvex Functions Coordinate Descent Gradient Descent Newton s method Stochastic Gradient Descent Numerical Optimization
Lecture 8: February 9
0-725/36-725: Convex Optimiation Spring 205 Lecturer: Ryan Tibshirani Lecture 8: February 9 Scribes: Kartikeya Bhardwaj, Sangwon Hyun, Irina Caan 8 Proximal Gradient Descent In the previous lecture, we
The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices
Noname manuscript No. (will be inserted by the editor) The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices Zhouchen Lin Minming Chen Yi Ma Received: date / Accepted:
Stochastic Gradient Descent with Variance Reduction
Stochastic Gradient Descent with Variance Reduction Rie Johnson, Tong Zhang Presenter: Jiawen Yao March 17, 2015 Rie Johnson, Tong Zhang Presenter: JiawenStochastic Yao Gradient Descent with Variance Reduction
Accelerated Dual Gradient-Based Methods for Total Variation Image Denoising/Deblurring Problems (and other Inverse Problems)
Accelerated Dual Gradient-Based Methods for Total Variation Image Denoising/Deblurring Problems (and other Inverse Problems) Donghwan Kim and Jeffrey A. Fessler EECS Department, University of Michigan
CPSC 540: Machine Learning
CPSC 540: Machine Learning First-Order Methods, L1-Regularization, Coordinate Descent Winter 2016 Some images from this lecture are taken from Google Image Search. Admin Room: We ll count final numbers
A Bregman alternating direction method of multipliers for sparse probabilistic Boolean network problem
A Bregman alternating direction method of multipliers for sparse probabilistic Boolean network problem Kangkang Deng, Zheng Peng Abstract: The main task of genetic regulatory networks is to construct a
Numerical Optimization Professor Horst Cerjak, Horst Bischof, Thomas Pock Mat Vis-Gra SS09
Numerical Optimization 1 Working Horse in Computer Vision Variational Methods Shape Analysis Machine Learning Markov Random Fields Geometry Common denominator: optimization problems 2 Overview of Methods
6. Proximal gradient method
L. Vandenberghe EE236C (Spring 2016) 6. Proximal gradient method motivation proximal mapping proximal gradient method with fixed step size proximal gradient method with line search 6-1 Proximal mapping
BENEFITING from the success of Compressive Sensing
Nonconvex Nonsmooth Low Rank Minimization via Iteratively Reweighted Nuclear Norm Canyi Lu, Student Member, IEEE, Jinhui Tang, Senior Member, IEEE, Shuicheng Yan, Senior Member, IEEE, and Zhouchen Lin,
A DELAYED PROXIMAL GRADIENT METHOD WITH LINEAR CONVERGENCE RATE. Hamid Reza Feyzmahdavian, Arda Aytekin, and Mikael Johansson
204 IEEE INTERNATIONAL WORKSHOP ON ACHINE LEARNING FOR SIGNAL PROCESSING, SEPT. 2 24, 204, REIS, FRANCE A DELAYED PROXIAL GRADIENT ETHOD WITH LINEAR CONVERGENCE RATE Hamid Reza Feyzmahdavian, Arda Aytekin,
Frist order optimization methods for sparse inverse covariance selection
Frist order optimization methods for sparse inverse covariance selection Katya Scheinberg Lehigh University ISE Department (joint work with D. Goldfarb, Sh. Ma, I. Rish) Introduction l l l l l l The field
SVRG++ with Non-uniform Sampling
SVRG++ with Non-uniform Sampling Tamás Kern András György Department of Electrical and Electronic Engineering Imperial College London, London, UK, SW7 2BT {tamas.kern15,a.gyorgy}@imperial.ac.uk Abstract
STA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 8: Optimization Cho-Jui Hsieh UC Davis May 9, 2017 Optimization Numerical Optimization Numerical Optimization: min X f (X ) Can be applied
Matrix-Tensor and Deep Learning in High Dimensional Data Analysis
Matrix-Tensor and Deep Learning in High Dimensional Data Analysis Tien D. Bui Department of Computer Science and Software Engineering Concordia University 14 th ICIAR Montréal July 5-7, 2017 Introduction
Convergence of Cubic Regularization for Nonconvex Optimization under KŁ Property
Convergence of Cubic Regularization for Nonconvex Optimization under KŁ Property Yi Zhou Department of ECE The Ohio State University zhou.1172@osu.edu Zhe Wang Department of ECE The Ohio State University
arxiv: v1 [math.oc] 1 Jul 2016
![arxiv: v1 [math.oc] 1 Jul 2016](/thumbs/78/78221552.jpg "arxiv: v1 [math.oc] 1 Jul 2016")
Convergence Rate of Frank-Wolfe for Non-Convex Objectives Simon Lacoste-Julien INRIA - SIERRA team ENS, Paris June 8, 016 Abstract arxiv:1607.00345v1 [math.oc] 1 Jul 016 We give a simple proof that the
Conditional Gradient (Frank-Wolfe) Method
Conditional Gradient (Frank-Wolfe) Method Lecturer: Aarti Singh Co-instructor: Pradeep Ravikumar Convex Optimization 10-725/36-725 1 Outline Today: Conditional gradient method Convergence analysis Properties
One Mirror Descent Algorithm for Convex Constrained Optimization Problems with Non-Standard Growth Properties
One Mirror Descent Algorithm for Convex Constrained Optimization Problems with Non-Standard Growth Properties Fedor S. Stonyakin 1 and Alexander A. Titov 1 V. I. Vernadsky Crimean Federal University, Simferopol,
Finite-sum Composition Optimization via Variance Reduced Gradient Descent
Finite-sum Composition Optimization via Variance Reduced Gradient Descent Xiangru Lian Mengdi Wang Ji Liu University of Rochester Princeton University University of Rochester xiangru@yandex.com mengdiw@princeton.edu
PRINCIPAL Component Analysis (PCA) is a fundamental approach
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm Canyi Lu, Jiashi Feng, Yudong Chen, Wei Liu, Member, IEEE, Zhouchen
Support Vector Machines: Training with Stochastic Gradient Descent. Machine Learning Fall 2017
Support Vector Machines: Training with Stochastic Gradient Descent Machine Learning Fall 2017 1 Support vector machines Training by maximizing margin The SVM objective Solving the SVM optimization problem
ONLINE VARIANCE-REDUCING OPTIMIZATION
ONLINE VARIANCE-REDUCING OPTIMIZATION Nicolas Le Roux Google Brain nlr@google.com Reza Babanezhad University of British Columbia rezababa@cs.ubc.ca Pierre-Antoine Manzagol Google Brain manzagop@google.com
1 Sparsity and l 1 relaxation
6.883 Learning with Combinatorial Structure Note for Lecture 2 Author: Chiyuan Zhang Sparsity and l relaxation Last time we talked about sparsity and characterized when an l relaxation could recover the
Journal Club. A Universal Catalyst for First-Order Optimization (H. Lin, J. Mairal and Z. Harchaoui) March 8th, CMAP, Ecole Polytechnique 1/19
Journal Club A Universal Catalyst for First-Order Optimization (H. Lin, J. Mairal and Z. Harchaoui) CMAP, Ecole Polytechnique March 8th, 2018 1/19 Plan 1 Motivations 2 Existing Acceleration Methods 3 Universal
Proximal methods. S. Villa. October 7, 2014
Proximal methods S. Villa October 7, 2014 1 Review of the basics Often machine learning problems require the solution of minimization problems. For instance, the ERM algorithm requires to solve a problem
Newton-Like Methods for Sparse Inverse Covariance Estimation
Newton-Like Methods for Sparse Inverse Covariance Estimation Peder A. Olsen Figen Oztoprak Jorge Nocedal Stephen J. Rennie June 7, 2012 Abstract We propose two classes of second-order optimization methods
Stochastic gradient descent and robustness to ill-conditioning
Stochastic gradient descent and robustness to ill-conditioning Francis Bach INRIA - Ecole Normale Supérieure, Paris, France ÉCOLE NORMALE SUPÉRIEURE Joint work with Aymeric Dieuleveut, Nicolas Flammarion,