強化學(xué)習(xí)的數(shù)學(xué)原理（英文版）

Overview of this Book 1
Chapter 1 Basic Concepts  6
1.1 A grid world example  7
1.2 State and action 8
1.3 State transition  9
1.4 Policy  11
1.5 Reward 13
1.6 Trajectories, returns, and episodes  15
1.7 Markov decision processes 18
1.8 Summary 20
1.9 Q&A 20
Chapter 2 State Values and the Bellman Equation  21
2.1 Motivating example 1: Why are returns important? 23
2.2 Motivating example 2: How to calculate returns?  24
2.3 State values 26
2.4 The Bellman equation  27
2.5 Examples for illustrating the Bellman equation  30
2.6 Matrix-vector form of the Bellman equation 33
2.7 Solving state values from the Bellman equation 35
2.7.1 Closed-form solution  35
2.7.2 Iterative solution 35
2.7.3 Illustrative examples  36
2.8 From state value to action value 38
2.8.1 Illustrative examples  39
2.8.2 The Bellman equation in terms of action values 40
2.9 Summary 41
2.10 Q&A  42
Chapter 3 Optimal State Values and the Bellman Optimality Equation 43
3.1 Motivating example: How to improve policies?  45
3.2 Optimal state values and optimal policies 46
3.3 The Bellman optimality equation 47
3.3.1 Maximization of the right-hand side of the BOE  48
3.3.2 Matrix-vector form of the BOE 49
3.3.3 Contraction mapping theorem  50
3.3.4 Contraction property of the right-hand side of the BOE  53
3.4 Solving an optimal policy from the BOE  55
3.5 Factors that influence optimal policies 58
3.6 Summary 63
3.7 Q&A 63
Chapter 4 Value Iteration and Policy Iteration 66
4.1 Value iteration  68
4.1.1 Elementwise form and implementation  68
4.1.2 Illustrative examples  70
4.2 Policy iteration 72
4.2.1 Algorithm analysis 73
4.2.2 Elementwise form and implementation  76
4.2.3 Illustrative examples  77
4.3 Truncated policy iteration 81
4.3.1 Comparing value iteration and policy iteration  81
4.3.2 Truncated policy iteration algorithm  83
4.4 Summary 85
4.5 Q&A 86
Chapter 5 Monte Carlo Methods 89
5.1 Motivating example: Mean estimation 91
5.2 MC Basic: The simplest MC-based algorithm 93
5.2.1 Converting policy iteration to be model-free 93
5.2.2 The MC Basic algorithm 94
5.2.3 Illustrative examples  96
5.3 MC Exploring Starts  99
5.3.1 Utilizing samples more efficiently  100
5.3.2 Updating policies more efficiently  101
5.3.3 Algorithm description 101
5.4 MC -Greedy: Learning without exploring starts 102
5.4.1 -greedy policies 103
5.4.2 Algorithm description 103
5.4.3 Illustrative examples 105
5.5 Exploration and exploitation of -greedy policies 106
5.6 Summary  111
5.7 Q&A  111
Chapter 6 Stochastic Approximation 114
6.1 Motivating example: Mean estimation 116
6.2 Robbins-Monro algorithm  117
6.2.1 Convergence properties  119
6.2.2 Application to mean estimation  123
6.3 Dvoretzky's convergence theorem  124
6.3.1 Proof of Dvoretzky's theorem  125
6.3.2 Application to mean estimation. 126
6.3.3 Application to the Robbins-Monro theorem  127
6.3.4 An extension of Dvoretzky's theorem  127
6.4 Stochastic gradient descent  128
6.4.1 Application to mean estimation 130
6.4.2 Convergence pattern of SGD 131
6.4.3 A deterministic formulation of SGD 133
6.4.4 BGD, SGD, and mini-batch GD 134
6.4.5 Convergence of SGD 136
6.5 Summary  138
6.6 Q&A  138
Chapter 7 Temporal-Difference Methods 140
7.1 TD learning of state values 142
7.1.1 Algorithm description 142
7.1.2 Property analysis  144
7.1.3 Convergence analysis  146
7.2 TD learning of action values: Sarsa  149
7.2.1 Algorithm description 149
7.2.2 Optimal policy learning via Sarsa  151
7.3 TD learning of action values: n-step Sarsa 154
7.4 TD learning of optimal action values: Q-learning 156
7.4.1 Algorithm description 156
7.4.2 Off-policy vs. on-policy  158
7.4.3 Implementation 160
7.4.4 Illustrative examples 161
7.5 A unified viewpoint  165
7.6 Summary  165
7.7 Q&A  166
Chapter 8 Value Function Approximation 168
8.1 Value representation: From table to function 170
8.2 TD learning of state values with function approximation 174
8.2.1 Objective function 174
8.2.2 Optimization algorithms 180
8.2.3 Selection of function approximators  182
8.2.4 Illustrative examples 183
8.2.5 Theoretical analysis 187
8.3 TD learning of action values with function approximation  198
8.3.1 Sarsa with function approximation 198
8.3.2 Q-learning with function approximation 200
8.4 Deep Q-learning 201
8.4.1 Algorithm description 202
8.4.2 Illustrative examples 204
8.5 Summary  207
8.6 Q&A  207
Chapter 9 Policy Gradient Methods 211
9.1 Policy representation: From table to function  213
9.2 Metrics for defining optimal policies  214
9.3 Gradients of the metrics 219
9.3.1 Derivation of the gradients in the discounted case  221
9.3.2 Derivation of the gradients in the undiscounted case 226
9.4 Monte Carlo policy gradient (REINFORCE) 232
9.5 Summary  235
9.6 Q&A  235
Chapter 10 Actor-Critic Methods  237
10.1 The simplest actor-critic algorithm (QAC)  239
10.2 Advantage actor-critic (A2C) 240
10.2.1 Baseline invariance 240
10.2.2 Algorithm description  243
10.3 Off-policy actor-critic 244
10.3.1 Importance sampling 245
10.3.2 The off-policy policy gradient theorem  247
10.3.3 Algorithm description  249
10.4 Deterministic actor-critic 251
10.4.1 The deterministic policy gradient theorem  251
10.4.2 Algorithm description  258
10.5 Summary 259
10.6 Q&A 260
Appendix A Preliminaries for Probability Theory 262
Appendix B Measure-Theoretic Probability Theory 268
Appendix C Convergence of Sequences  276
C.1 Convergence of deterministic sequences  277
C.2 Convergence of stochastic sequences 280
Appendix D Preliminaries for Gradient Descent 284
Bibliography  290
Symbols 297
Index  299