Stochastic Optimization for Large-scale Machine Learning

List of Figures
List of Tables
Preface

Section I BACKGROUND

Introduction
1.1 LARGE-SCALE MACHINE LEARNING
1.2 OPTIMIZATION PROBLEMS
1.3 LINEAR CLASSIFICATION
1.3.1 Support Vector Machine (SVM)
1.3.2 Logistic Regression
1.3.3 First and Second Order Methods
1.3.3.1 First Order Methods
1.3.3.2 Second Order Methods
1.4 STOCHASTIC APPROXIMATION APPROACH
1.5 COORDINATE DESCENT APPROACH
1.6 DATASETS
1.7 ORGANIZATION OF BOOK

Optimisation Problem, Solvers, Challenges and Research Directions
2.1 INTRODUCTION
2.1.1 Contributions
2.2 LITERATURE
2.3 PROBLEM FORMULATIONS
2.3.1 Hard Margin SVM (1992)
2.3.2 Soft Margin SVM (1995)
2.3.3 One-versus-Rest (1998)
2.3.4 One-versus-One (1999)
2.3.5 Least Squares SVM (1999)
2.3.6 v-SVM (2000)
2.3.7 Smooth SVM (2001)
2.3.8 Proximal SVM (2001)
2.3.9 Crammer Singer SVM (2002)
2.3.10 Ev-SVM (2003)
2.3.11 Twin SVM (2007)
2.3.12 Capped lp-norm SVM (2017)
2.4 PROBLEM SOLVERS
2.4.1 Exact Line Search Method
2.4.2 Backtracking Line Search
2.4.3 Constant Step Size
2.4.4 Lipschitz & Strong Convexity Constants
2.4.5 Trust Region Method
2.4.6 Gradient Descent Method
2.4.7 Newton Method
2.4.8 Gauss-Newton Method
2.4.9 Levenberg-Marquardt Method
2.4.10 Quasi-Newton Method
2.4.11 Subgradient Method
2.4.12 Conjugate Gradient Method
2.4.13 Truncated Newton Method
2.4.14 Proximal Gradient Method
2.4.15 Recent Algorithms
2.5 COMPARATIVE STUDY
2.5.1 Results from Literature
2.5.2 Results from Experimental Study
2.5.2.1 Experimental Setup and Implementation Details
2.5.2.2 Results and Discussions
2.6 CURRENT CHALLENGES AND RESEARCH DIRECTIONS
2.6.1 Big Data Challenge
2.6.2 Areas of Improvement
2.6.2.1 Problem Formulations
2.6.2.2 Problem Solvers
2.6.2.3 Problem Solving Strategies/Approaches
2.6.2.4 Platforms/Frameworks
2.6.3 Research Directions
2.6.3.1 Stochastic Approximation Algorithms
2.6.3.2 Coordinate Descent Algorithms
2.6.3.3 Proximal Algorithms
2.6.3.4 Parallel/Distributed Algorithms
2.6.3.5 Hybrid Algorithms
2.7 CONCLUSION

Section II FIRST ORDER METHODS
Mini-batch and Block-coordinate Approach
3.1 INTRODUCTION
3.1.1 Motivation
3.1.2 Batch Block Optimization Framework (BBOF)
3.1.3 Brief Literature Review
3.1.4 Contributions
3.2 STOCHASTIC AVERAGE ADJUSTED GRADIENT (SAAG) METHODS
3.3 ANALYSIS
3.4 NUMERICAL EXPERIMENTS
3.4.1 Experimental setup
3.4.2 Convergence against epochs
3.4.3 Convergence against Time
3.5 CONCLUSION AND FUTURE SCOPE

Variance Reduction Methods
4.1 INTRODUCTION
4.1.1 Optimization Problem
4.1.2 Solution Techniques for Optimization Problem
4.1.3 Contributions
4.2 NOTATIONS AND RELATED WORK
4.2.1 Notations
4.2.2 Related Work
4.3 SAAG-I, II AND PROXIMAL EXTENSIONS
4.4 SAAG-III AND IV ALGORITHMS
4.5 ANALYSIS
4.6 EXPERIMENTAL RESULTS
4.6.1 Experimental Setup
4.6.2 Results with Smooth Problem
4.6.3 Results with non-smooth Problem
4.6.4 Mini-batch Block-coordinate versus mini-batch setting
4.6.5 Results with SVM
4.7 CONCLUSION

Learning and Data Access
5.1 INTRODUCTION
5.1.1 Optimization Problem
5.1.2 Literature Review
5.1.3 Contributions
5.2 SYSTEMATIC SAMPLING
5.2.1 Definitions
5.2.2 Learning using Systematic Sampling
5.3 ANALYSIS
5.4 EXPERIMENTS
5.4.1 Experimental Setup
5.4.2 Implementation Details
5.4.3 Results
5.5 CONCLUSION

Section III SECOND ORDER METHODS

Mini-batch Block-coordinate Newton Method
6.1 INTRODUCTION
6.1.1 Contributions
6.2 MBN
6.3 EXPERIMENTS
6.3.1 Experimental Setup
6.3.2 Comparative Study
6.4 CONCLUSION

Stochastic Trust Region Inexact Newton Method
7.1 INTRODUCTION
7.1.1 Optimization Problem
7.1.2 Solution Techniques
7.1.3 Contributions
7.2 LITERATURE REVIEW
7.3 TRUST REGION INEXACT NEWTON METHOD
7.3.1 Inexact Newton Method
7.3.2 Trust Region Inexact Newton Method
7.4 STRON
7.4.1 Complexity
7.4.2 Analysis
7.5 EXPERIMENTAL RESULTS
7.5.1 Experimental Setup
7.5.2 Comparative Study
7.5.3 Results with SVM
7.6 EXTENSIONS
7.6.1 PCG Subproblem Solver 1
7.6.2 Stochastic Variance Reduced Trust Region Inexact Newton Method
7.7 CONCLUSION

Section IV CONCLUSION
Conclusion and Future Scope
8.1 FUTURE SCOPE 142

Bibliography

Index