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E-raamat: Introduction to Online Convex Optimization

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New edition of a graduate-level textbook on that focuses on online convex optimization, a machine learning framework that views optimization as a process.

In many practical applications, the environment is so complex that it is not feasible to lay out a comprehensive theoretical model and use classical algorithmic theory and/or mathematical optimization. Introduction to Online Convex Optimization presents a robust machine learning approach that contains elements of mathematical optimization, game theory, and learning theory: an optimization method that learns from experience as more aspects of the problem are observed. This view of optimization as a process has led to some spectacular successes in modeling and systems that have become part of our daily lives.

Based on the “Theoretical Machine Learning” course taught by the author at Princeton University, the second edition of this widely used graduate level text features:
  • Thoroughly updated material throughout
  • New chapters on boosting, adaptive regret, and approachability and expanded exposition on optimization
  • Examples of applications, including prediction from expert advice, portfolio selection, matrix completion and recommendation systems, SVM training, offered throughout
  • Exercises that guide students in completing parts of proofs
    • Preface xi
    Acknowledgments xv
    List of Figures
    		xvii
    List of Symbols
    		xix
    1 Introduction
    		1(14)
    1.1 The Online Convex Optimization Setting
    		1(2)
    1.2 Examples of Problems That Can Be Modeled via Online Convex Optimization
    		3(4)
    1.3 A Gentle Start: Learning from Expert Advice
    		7(5)
    1.4 Bibliographic Remarks
    		12(1)
    1.5 Exercises
    		13(2)
    2 Basic Concepts in Convex Optimization
    		15(22)
    2.1 Basic Definitions and Setup
    		15(4)
    2.2 Gradient Descent
    		19(6)
    2.3 Constrained Gradient/Subgradient Descent
    		25(2)
    2.4 Reductions to Nonsmooth and Nonstrongly Convex Functions
    		27(4)
    2.5 Example: Support Vector Machine Training
    		31(2)
    2.6 Bibliographic Remarks
    		33(1)
    2.7 Exercises
    		34(3)
    3 First-Order Algorithms for Online Convex Optimization
    		37(12)
    3.1 Online Gradient Descent
    		38(2)
    3.2 Lower Bounds
    		40(1)
    3.3 Logarithmic Regret
    		41(2)
    3.4 Application: Stochastic Gradient Descent
    		43(3)
    3.5 Bibliographic Remarks
    		46(1)
    3.6 Exercises
    		46(3)
    4 Second-Order Methods
    		49(14)
    4.1 Motivation: Universal Portfolio Selection
    		49(3)
    4.2 Exp-Concave Functions
    		52(1)
    4.3 Exponentially Weighted Online Convex Optimization
    		53(2)
    4.4 The Online Newton Step Algorithm
    		55(5)
    4.5 Bibliographic Remarks
    		60(1)
    4.6 Exercises
    		61(2)
    5 Regularization
    		63(26)
    5.1 Regularization Functions
    		64(1)
    5.2 The RFTL Algorithm and Its Analysis
    		65(4)
    5.3 Online Mirror Descent
    		69(3)
    5.4 Application and Special Cases
    		72(2)
    5.5 Randomized Regularization
    		74(7)
    5.6 Adaptive Gradient Descent
    		81(5)
    5.7 Bibliographic Remarks
    		86(1)
    5.8 Exercises
    		87(2)
    6 Bandit Convex Optimization
    		89(18)
    6.1 The Bandit Convex Optimization Setting
    		89(1)
    6.2 The Multiarmed Bandit (MAB) Problem
    		90(4)
    6.3 A Reduction from Limited Information to Full Information
    		94(5)
    6.4 Online Gradient Descent without a Gradient
    		99(2)
    6.5 Optimal Regret Algorithms for Bandit Linear Optimization
    		101(4)
    6.6 Bibliographic Remarks
    		105(1)
    6.7 Exercises
    		106(1)
    7 Projection-Free Algorithms
    		107(16)
    7.1 Review: Relevant Concepts from Linear Algebra
    		107(1)
    7.2 Motivation: Recommender Systems
    		108(2)
    7.3 The Conditional Gradient Method
    		110(4)
    7.4 Projections versus Linear Optimization
    		114(2)
    7.5 The Online Conditional Gradient Algorithm
    		116(3)
    7.6 Bibliographic Remarks
    		119(1)
    7.7 Exercises
    		120(3)
    8 Games, Duality, and Regret
    		123(10)
    8.1 Linear Programming and Duality
    		124(1)
    8.2 Zero-Sum Games and Equilibria
    		125(3)
    8.3 Proof of von Neumann Theorem
    		128(2)
    8.4 Approximating Linear Programs
    		130(1)
    8.5 Bibliographic Remarks
    		131(1)
    8.6 Exercises
    		131(2)
    9 Learning Theory, Generalization, and Online Convex Optimization
    		133(14)
    9.1 Statistical Learning Theory
    		133(5)
    9.2 Agnostic Learning Using Online Convex Optimization
    		138(5)
    9.3 Learning and Compression
    		143(2)
    9.4 Bibliographic Remarks
    		145(1)
    9.5 Exercises
    		145(2)
    10 Learning in Changing Environments
    		147(16)
    10.1 A Simple Start: Dynamic Regret
    		148(1)
    10.2 The Notion of Adaptive Regret
    		149(2)
    10.3 Tracking the Best Expert
    		151(2)
    10.4 Efficient Adaptive Regret for Online Convex Optimization
    		153(2)
    10.5 Computationally Efficient Methods
    		155(4)
    10.6 Bibliographic Remarks
    		159(1)
    10.7 Exercises
    		160(3)
    11 Boosting and Regret
    		163(8)
    11.1 The Problem of Boosting
    		164(1)
    11.2 Boosting by Online Convex Optimization
    		165(4)
    11.3 Bibliographic Remarks
    		169(1)
    11.4 Exercises
    		170(1)
    12 Online Boosting
    		171(10)
    12.1 Motivation: Learning from a Huge Set of Experts
    		171(2)
    12.2 The Contextual Learning Model
    		173(1)
    12.3 The Extension Operator
    		174(1)
    12.4 The Online Boosting Method
    		175(4)
    12.5 Bibliographic Remarks
    		179(1)
    12.6 Exercises
    		180(1)
    13 Blackwell Approachability and Online Convex Optimization
    		181(10)
    13.1 Vector-Valued Games and Approachability
    		181(2)
    13.2 From OCO to Approachability
    		183(3)
    13.3 From Approachability to OCO
    		186(3)
    13.4 Bibliographic Remarks
    		189(1)
    13.5 Exercises
    		190(1)
    Notes 191(2)
    References 193(14)
    Index 207
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