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Distributed Optimization and Statistical Learning via the Altern...

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电子书Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers，包含了拉格朗日乘子法等多种最优化算法，描述清晰，是国外一些论文的参考文献之一

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Foundations and Trends

Machine Learning

Vol. 3, No. 1 (2010) 1–122

! 2011 S. Boyd, N. Parikh, E. Chu, B. Peleato

and J. Eckstein

DOI: 10.1561/2200000016

Distributed Optimization and Statistical

Learning via the Alternating Direction

Method of Multipliers

Stephen Boyd

, Neal Parikh

, Eric Chu

Borja Peleato

and Jonathan Eckstein

Electrical Engineering Department, Stanford University, Stanford, CA

94305, USA, boyd@stanford.edu

Computer Science Department, Stanford University, Stanford, CA 94305,

USA, npparikh@cs.stanford.edu

Electrical Engineering Department, Stanford University, Stanford, CA

94305, USA, echu508@stanford.edu

Electrical Engineering Department, Stanford University, Stanford, CA

94305, USA, peleato@stanford.edu

Management Science and Information Systems Department and

RUTCOR, Rutgers University, Piscataway, NJ 08854, USA,

jeckstei@rci.rutgers.edu

Contents

1 Introduction 3

2 Precursors 7

2.1 Dual Ascent 7

2.2 Dual Decomposition 9

2.3 Augmented Lagrangians and the Method of Multipliers 10

3 Alternating Direction Method of Multipliers 13

3.1 Algorithm 13

3.2 Convergence 15

3.3 Optimality Conditions and Stopping Criterion 18

3.4 Extensions and Variations 20

3.5 Notes and References 23

4 General Patterns 25

4.1 Proximity Operator 25

4.2 Quadratic Objective Terms 26

4.3 Smooth Objective Terms 30

4.4 Decomposition 31

5 Constrained Convex Optimization 33

5.1 Convex Feasibility 34

5.2 Linear and Quadratic Programming 36

6 !

-Norm Problems 38

6.1 Least Absolute Deviations 39

6.2 Basis Pursuit 41

6.3 General !

Regularized Loss Minimization 42

6.4 Lasso 43

6.5 Sparse Inverse Covariance Selection 45

7 Consensus and Sharing 48

7.1 Global Variable Consensus Optimization 48

7.2 General Form Consensus Optimization 53

7.3 Sharing 56

8 Distributed Model Fitting 61

8.1 Examples 62

8.2 Splitting across Examples 64

8.3 Splitting across Features 66

9 Nonconvex Problems 73

9.1 Nonconvex Constraints 73

9.2 Bi-convex Problems 76

10 Implementation 78

10.1 Abstract Implementation 78

10.2 MPI 80

10.3 Graph Computing Frameworks 81

10.4 MapReduce 82

11 Numerical Examples 87

11.1 Small Dense Lasso 88

11.2 Distributed !

Regularized Logistic Regression 92

11.3 Group Lasso with Feature Splitting 95

11.4 Distributed Large-Scale Lasso with MPI 97

11.5 Regressor Selection 100

12 Conclusions 103

Acknowledgments 105

A Convergence Proof 106

References 111

Abstract

Many problems of recent interest in statistics and machine learning

can be posed in the framework of convex optimization. Due to the

explosion in size and complexity of modern datasets, it is increasingly

important to be able to solve problems with a very large number of fea-

tures or training examples. As a result, both the decentralized collection

or storage of these datasets as well as accompanying distributed solu-

tion methods are either necessary or at least highly desirable. In this

review, we argue that the alternating direction method of multipliers

is well suited to distributed convex optimization, and in particular to

large-scale problems arising in statistics, machine learning, and related

areas. The method was developed in the 1970s, with roots in the 1950s,

and is equivalent or closely related to many other algorithms, such

as dual decomposition, the method of multipliers, Douglas–Rachford

splitting, Spingarn’s method of partial inverses, Dykstra’s alternating

projections, Bregman iterative algorithms for !

problems, proximal

methods, and others. After brieﬂy surveying the theory and history of

the algorithm, we discuss applications to a wide variety of statistical

and machine learning problems of recent interest, including the lasso,

sparse logistic regression, basis pursuit, covariance selection, support

vector machines, and many others. We also discuss general distributed

optimization, extensions to the nonconvex setting, and eﬃcient imple-

mentation, including some details on distributed MPI and Hadoop

MapReduce implementations.

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