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E-raamat: First-order and Stochastic Optimization Methods for Machine Learning

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First-order and Stochastic Optimization Methods for Machine LearningSuurem pilt
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This book covers not only foundational materials but also the most recent progresses made during the past few years on the area of machine learning algorithms. In spite of the intensive research and development in this area, there does not exist a systematic treatment to introduce the fundamental concepts and recent progresses on machine learning algorithms, especially on those based on stochastic optimization methods, randomized algorithms, nonconvex optimization, distributed and online learning, and projection free methods. This book will benefit the broad audience in the area of machine learning, artificial intelligence and mathematical programming community by presenting these recent developments in a tutorial style, starting from the basic building blocks to the most carefully designed and complicated algorithms for machine learning.



1 Machine Learning Models
1(20)
1.1 Linear Regression
1(3)
1.2 Logistic Regression
4(3)
1.3 Generalized Linear Models
7(4)
1.3.1 Exponential Family
7(1)
1.3.2 Model Construction
8(3)
1.4 Support Vector Machines
11(4)
1.5 Regularization, Lasso, and Ridge Regression
15(1)
1.6 Population Risk Minimization
16(1)
1.7 Neural Networks
17(3)
1.8 Exercises and Notes
20(1)
2 Convex Optimization Theory
21(32)
2.1 Convex Sets
21(8)
2.1.1 Definition and Examples
21(2)
2.1.2 Projection onto Convex Sets
23(2)
2.1.3 Separation Theorem
25(4)
2.2 Convex Functions
29(8)
2.2.1 Definition and Examples
29(1)
2.2.2 Differentiable Convex Functions
30(1)
2.2.3 Non-differentiable Convex Functions
31(2)
2.2.4 Lipschitz Continuity of Convex Functions
33(2)
2.2.5 Optimality Conditions for Convex Optimization
35(1)
2.2.6 Representer Theorem and Kernel
36(1)
2.3 Lagrange Duality
37(8)
2.3.1 Lagrange Function and Duality
38(1)
2.3.2 Proof of Strong Duality
39(2)
2.3.3 Saddle Points
41(1)
2.3.4 Karush--Kuhn--Tucker Conditions
42(2)
2.3.5 Dual Support Vector Machine
44(1)
2.4 Legendre--Fenchel Conjugate Duality
45(5)
2.4.1 Closure of Convex Functions
45(3)
2.4.2 Conjugate Functions
48(2)
2.5 Exercises and Notes
50(3)
3 Deterministic Convex Optimization
53(60)
3.1 Subgradient Descent
53(7)
3.1.1 General Nonsmooth Convex Problems
54(3)
3.1.2 Nonsmooth Strongly Convex Problems
57(1)
3.1.3 Smooth Convex Problems
58(2)
3.1.4 Smooth and Strongly Convex Problems
60(1)
3.2 Mirror Descent
60(4)
3.3 Accelerated Gradient Descent
64(5)
3.4 Game Interpretation for Accelerated Gradient Descent
69(3)
3.5 Smoothing Scheme for Nonsmooth Problems
72(2)
3.6 Primal-Dual Method for Saddle-Point Optimization
74(10)
3.6.1 General Bilinear Saddle Point Problems
78(1)
3.6.2 Smooth Bilinear Saddle Point Problems
79(1)
3.6.3 Smooth and Strongly Convex Bilinear Saddle Point Problems
80(1)
3.6.4 Linearly Constrained Problems
81(3)
3.7 Alternating Direction Method of Multipliers
84(2)
3.8 Mirror-Prox Method for Variational Inequalities
86(6)
3.8.1 Monotone Variational Inequalities
87(2)
3.8.2 Generalized Monotone Variational Inequalities
89(3)
3.9 Accelerated Level Method
92(16)
3.9.1 Nonsmooth, Smooth, and Weakly Smooth Problems
93(9)
3.9.2 Saddle Point Problems
102(6)
3.10 Exercises and Notes
108(5)
4 Stochastic Convex Optimization
113(108)
4.1 Stochastic Mirror Descent
113(15)
4.1.1 General Nonsmooth Convex Functions
114(4)
4.1.2 Smooth Convex Problems
118(4)
4.1.3 Accuracy Certificates
122(6)
4.2 Stochastic Accelerated Gradient Descent
128(20)
4.2.1 Problems Without Strong Convexity
134(3)
4.2.2 Nonsmooth Strongly Convex Problems
137(2)
4.2.3 Smooth and Strongly Convex Problems
139(5)
4.2.4 Accuracy Certificates
144(4)
4.3 Stochastic Convex-Concave Saddle Point Problems
148(10)
4.3.1 General Algorithmic Framework
149(4)
4.3.2 Minimax Stochastic Problems
153(2)
4.3.3 Bilinear Matrix Games
155(3)
4.4 Stochastic Accelerated Primal-Dual Method
158(23)
4.4.1 Accelerated Primal-Dual Method
160(10)
4.4.2 Stochastic Bilinear Saddle Point Problems
170(11)
4.5 Stochastic Accelerated Mirror-Prox Method
181(18)
4.5.1 Algorithmic Framework
182(2)
4.5.2 Convergence Analysis
184(15)
4.6 Stochastic Block Mirror Descent Method
199(19)
4.6.1 Nonsmooth Convex Optimization
201(10)
4.6.2 Convex Composite Optimization
211(7)
4.7 Exercises and Notes
218(3)
5 Convex Finite-Sum and Distributed Optimization
221(84)
5.1 Random Primal-Dual Gradient Method
221(30)
5.1.1 Multi-Dual-Player Game Reformulation
225(2)
5.1.2 Randomization on Gradient Computation
227(3)
5.1.3 Convergence for Strongly Convex Problems
230(11)
5.1.4 Lower Complexity Bound for Randomized Methods
241(5)
5.1.5 Generalization to Problems Without Strong Convexity
246(5)
5.2 Random Gradient Extrapolation Method
251(27)
5.2.1 Gradient Extrapolation Method
253(7)
5.2.2 Deterministic Finite-Sum Problems
260(11)
5.2.3 Stochastic Finite-Sum Problems
271(5)
5.2.4 Distributed Implementation
276(2)
5.3 Variance-Reduced Mirror Descent
278(9)
5.3.1 Smooth Problems Without Strong Convexity
282(2)
5.3.2 Smooth and Strongly Convex Problems
284(3)
5.4 Variance-Reduced Accelerated Gradient Descent
287(15)
5.4.1 Smooth Problems Without Strong Convexity
290(4)
5.4.2 Smooth and Strongly Convex Problems
294(6)
5.4.3 Problems Satisfying an Error-Bound Condition
300(2)
5.5 Exercises and Notes
302(3)
6 Nonconvex Optimization
305(116)
6.1 Unconstrained Nonconvex Stochastic Optimization
305(23)
6.1.1 Stochastic First-Order Methods
308(10)
6.1.2 Stochastic Zeroth-Order Methods
318(10)
6.2 Nonconvex Stochastic Composite Optimization
328(24)
6.2.1 Some Properties of Prox-Mapping
330(2)
6.2.2 Nonconvex Mirror Descent Methods
332(2)
6.2.3 Nonconvex Stochastic Mirror Descent Methods
334(13)
6.2.4 Stochastic Zeroth-Order Methods for Composite Problems
347(5)
6.3 Nonconvex Stochastic Block Mirror Descent
352(7)
6.4 Nonconvex Stochastic Accelerated Gradient Descent
359(29)
6.4.1 Nonconvex Accelerated Gradient Descent
361(13)
6.4.2 Stochastic Accelerated Gradient Descent Method
374(14)
6.5 Nonconvex Variance-Reduced Mirror Descent
388(7)
6.5.1 Basic Scheme for Deterministic Problems
389(3)
6.5.2 Generalization for Stochastic Optimization Problems
392(3)
6.6 Randomized Accelerated Proximal-Point Methods
395(24)
6.6.1 Nonconvex Finite-Sum Problems
396(12)
6.6.2 Nonconvex Multi-Block Problems
408(11)
6.7 Exercises and Notes
419(2)
7 Projection-Free Methods
421(62)
7.1 Conditional Gradient Method
421(23)
7.1.1 Classic Conditional Gradient
423(10)
7.1.2 New Variants of Conditional Gradient
433(6)
7.1.3 Lower Complexity Bound
439(5)
7.2 Conditional Gradient Sliding Method
444(24)
7.2.1 Deterministic Conditional Gradient Sliding
446(10)
7.2.2 Stochastic Conditional Gradient Sliding Method
456(8)
7.2.3 Generalization to Saddle Point Problems
464(4)
7.3 Nonconvex Conditional Gradient Method
468(1)
7.4 Stochastic Nonconvex Conditional Gradient
469(8)
7.4.1 Basic Scheme for Finite-Sum Problems
470(4)
7.4.2 Generalization for Stochastic Optimization Problems
474(3)
7.5 Stochastic Nonconvex Conditional Gradient Sliding
477(5)
7.5.1 Wolfe Gap vs Projected Gradient
477(1)
7.5.2 Projection-Free Method to Drive Projected Gradient Small
478(4)
7.6 Exercises and Notes
482(1)
8 Operator Sliding and Decentralized Optimization
483(84)
8.1 Gradient Sliding for Composite Optimization
483(26)
8.1.1 Deterministic Gradient Sliding
486(11)
8.1.2 Stochastic Gradient Sliding
497(7)
8.1.3 Strongly Convex and Structured Nonsmooth Problems
504(5)
8.2 Accelerated Gradient Sliding
509(23)
8.2.1 Composite Smooth Optimization
512(15)
8.2.2 Composite Bilinear Saddle Point Problems
527(5)
8.3 Communication Sliding and Decentralized Optimization
532(33)
8.3.1 Problem Formulation
535(3)
8.3.2 Decentralized Communication Sliding
538(11)
8.3.3 Stochastic Decentralized Communication Sliding
549(6)
8.3.4 High Probability Results
555(2)
8.3.5 Convergence Analysis
557(8)
8.4 Exercises and Notes
565(2)
References 567(10)
Index 577
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