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CHAPTER 1 Introduction
CHAPTER 2 Accelerated Algorithms for Unconstrained Convex Optimization
1. Preliminaries
2. Accelerated Gradient Method for smooth optimization
3. Extension to the Composite Optimization
3.1. Nesterov's First Scheme
3.2. Nesterov's Second Scheme
3.2.1. A Primal Dual Perspective
3.3. Nesterov's Third Scheme
4. Inexact Proximal and Gradient Computing
4.1. Inexact Accelerated Gradient Descent
4.2. Inexact Accelerated Proximal Point Method
5. Restart
6. Smoothing for Nonsmooth Optimization
7. Higher Order Accelerated Method
8. Explanation: An Variational Perspective8.1. Discretization
CHAPTER 3 Accelerated Algorithms for Constrained Convex Optimization
1. Preliminaries
1.1. Case Study: Linear Equality Constraint
2. Accelerated Penalty Method
2.1. Non-strongly Convex Objectives
2.2. Strong Convex Objectives
3. Accelerated Lagrange Multiplier Method3.1. Recovering the Primal Solution
3.2. Accelerated Augmented Lagrange Multiplier Method
4. Accelerated Alternating Direction Method of Multipliers
4.1. Non-strongly Convex and Non-smooth
4.2. Strongly Convex and Non-smooth
4.3. Non-strongly Convex and Smooth
4.4. Strongly Convex and Smooth
4.5. Non-ergodic Convergence Rate
4.5.1. Original ADMM
4.5.2. ADMM with Extrapolation and Increasing Penalty Parameter
5. Accelerated Primal Dual Method
5.1. Case 1
5.2. Case 2
5.3. Case 3
5.4. Case 4
CHAPTER 4 Accelerated Algorithms for Nonconvex Optimization
1. Proximal Gradient with Momentum
1.1. Basic Assumptions
1.2. Convergence Theorem
1.3. Another Method: Monotone APG
2. AGD Achieves the Critical Points Quickly
2.1. AGD as a Convexity Monitor
2.2. Negative Curvature
2.3. Accelerating Nonconvex Optimization3. AGD Escapes the Saddle Points Quickly
3.1. Almost Convex
3.2. Negative Curvature Descent
3.3. AGD for Non-Convex Problem
3.3.1. Locally Almost Convex! Globally Almost Convex
3.3.2. Outer Iterations
3.3.3. Inner Iterations
CHAPTER 5 Accelerated Stochastic Algorithms
1. The Individual Convexity Case
1.1. Accelerated Stochastic Coordinate Descent
1.2. Background for Variance Reduction Methods
1.3. Accelerated Stochastic Variance Reduction Method
1.4. Black-Box Acceleration
2. The Individual Non-convexity Case
2.1. Individual Non-convex but Integrally Convex
3. The Non-Convexity Case
3.1. SPIDER
3.2. Momentum Acceleration
4. Constrained Problem
5. Infinity Case
CHAPTER 6 Paralleling Algorithms
1. Accelerated Asynchronous Algorithms
1.1. Asynchronous Accelerated Gradient Descent
1.2. Asynchronous Accelerated Stochastic Coordinate Descent
2. Accelerated Distributed Algorithms
2.1. Centralized Topology
2.1.1. Large Mini-batch Algorithms
2.1.2. Dual Communication-Efficient Methods
2.2. Decentralized Topology
CHAPTER 7 Conclusions
APPENDIX Mathematical Preliminaries
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