Accelerated Optimization for Machine Learning
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CHAPTER 1 IntroductionCHAPTER 2 Accelerated Algorithms for Unconstrained Convex Optimization1. Preliminaries2. Accelerated Gradient Method for smooth optimization3. Extension to the Composite Optimization3.1. Nesterovs First Scheme3.2. Nesterovs Second Scheme3.2.1. A Primal Dual Perspective3.3. Nesterovs Third Scheme4. Inexact Proximal and Gradient Computing4.1. Inexact Accelerated Gradient Descent4.2. Inexact Accelerated Proximal Point Method5. Restart6. Smoothing for Nonsmooth Optimization7. Higher Order Accelerated Method8. Explanation: An Variational Perspective8.1. Discretization CHAPTER 3 Accelerated Algorithms for Constrained Convex Optimization1. Preliminaries1.1. Case Study: Linear Equality Constraint2. Accelerated Penalty Method2.1. Non-strongly Convex Objectives2.2. Strong Convex Objectives3. Accelerated Lagrange Multiplier Method3.1. Recovering the Primal Solution3.2. Accelerated Augmented Lagrange Multiplier Method4. Accelerated Alternating Direction Method of Multipliers4.1. Non-strongly Convex and Non-smooth4.2. Strongly Convex and Non-smooth4.3. Non-strongly Convex and Smooth4.4. Strongly Convex and Smooth4.5. Non-ergodic Convergence Rate4.5.1. Original ADMM4.5.2. ADMM with Extrapolation and Increasing Penalty Parameter5. Accelerated Primal Dual Method5.1. Case 15.2. Case 25.3. Case 35.4. Case 4 CHAPTER 4 Accelerated Algorithms for Nonconvex Optimization1. Proximal Gradient with Momentum1.1. Basic Assumptions1.2. Convergence Theorem1.3. Another Method: Monotone APG2. AGD Achieves the Critical Points Quickly2.1. AGD as a Convexity Monitor2.2. Negative Curvature2.3. Accelerating Nonconvex Optimization3. AGD Escapes the Saddle Points Quickly3.1. Almost Convex3.2. Negative Curvature Descent3.3. AGD for Non-Convex Problem3.3.1. Locally Almost Convex! Globally Almost Convex3.3.2. Outer Iterations3.3.3. Inner Iterations CHAPT
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