Accelerated Optimization for Machine Learning: First-Order Algorithms

ISBN-13: 9789811529092 / Angielski / Twarda / 2020 / 275 str.

Zhouchen Lin; Huan Li; Cong Fang

Spis treści

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 Perspective

8.1. 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 Method

3.1. 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 Optimization

3. 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

O autorze

Zhouchen Lin is a leading expert in the fields of machine learning and computer vision. He is currently a Professor at the Key Laboratory of Machine Perception (Ministry of Education), School of EECS, Peking University. He served as an area chair for several prestigious conferences, including CVPR, ICCV, ICML, NIPS, AAAI and IJCAI. He is an associate editor of the IEEE Transactions on Pattern Analysis and Machine Intelligence and the International Journal of Computer Vision. He is a Fellow of IAPR and IEEE.

Huan Li received his Ph.D. degree in machine learning from Peking University in 2019. He is currently an Assistant Professor at the College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics. His current research interests include optimization and machine learning.

Cong Fang received his Ph.D. degree from Peking University in 2019. He is currently a Postdoctoral Researcher at Princeton University. His research interests include machine learning and optimization.

Blurb

This book on optimization includes forewords by Michael I. Jordan, Zongben Xu and Zhi-Quan Luo. Machine learning relies heavily on optimization to solve problems with its learning models, and first-order optimization algorithms are the mainstream approaches. The acceleration of first-order optimization algorithms is crucial for the efficiency of machine learning.

Written by leading experts in the field, this book provides a comprehensive introduction to, and state-of-the-art review of accelerated first-order optimization algorithms for machine learning. It discusses a variety of methods, including deterministic and stochastic algorithms, where the algorithms can be synchronous or asynchronous, for unconstrained and constrained problems, which can be convex or non-convex. Offering a rich blend of ideas, theories and proofs, the book is up-to-date and self-contained. It is an excellent reference resource for users who are seeking faster optimization algorithms, as well as for graduate students and researchers wanting to grasp the frontiers of optimization in machine learning in a short time.