Book: Reinforcement Learning and Dynamic Programming Using Function Approximators

Reinforcement learning (RL) can optimally solve decision and control problems involving complex dynamic systems, without requiring a mathematical model of the system. If a model is available, dynamic programming (DP), the model-based counterpart of RL, can be used. RL and DP are applicable in a variety of disciplines, including automatic control, artificial intelligence, economics, and medicine. Recent years have seen a surge of interest RL and DP using compact, approximate representations of the solution, which enable algorithms to scale up to realistic problems.

This book provides an in-depth introduction to RL and DP with function approximators. A concise description of classical RL and DP (Chapter 2) builds the foundation for the remainder of the book. This is followed by an extensive review of the state-of-the-art in RL and DP with approximation, which combines algorithm development with theoretical guarantees, illustrative numerical examples, and insightful comparisons (Chapter 3). Each of the final three chapters (4 to 6) is dedicated to a representative algorithm from the three major classes of methods: value iteration, policy iteration, and policy search. The features and performance of these algorithms are highlighted in extensive experimental studies on a range of control applications.

For graduate students and others new to the field, this book offers a thorough introduction to both the basics and emerging methods. And for those researchers and practitioners working in the fields of optimal and adaptive control, machine learning, artificial intelligence, and operations research, this resource offers a combination of practical algorithms, theoretical analysis, and comprehensive examples that they will be able to adapt and apply to their own work.

1. Introduction

1.1 The dynamic programming and reinforcement learning problem
1.2 Approximation in dynamic programming and reinforcement learning
1.3 About this book

2. An introduction to dynamic programming and reinforcement learning

2.1 Introduction
2.2 Markov decision processes
- 2.2.1 Deterministic setting
- 2.2.2 Stochastic setting
2.3 Value iteration
- 2.3.1 Model-based value iteration
- 2.3.2 Model-free value iteration and the need for exploration
2.4 Policy iteration
- 2.4.1 Model-based policy iteration
- 2.4.2 Model-free policy iteration
2.5 Policy search
2.6 Summary and discussion

3. Dynamic programming and reinforcement learning in large and continuous spaces

3.1 Introduction
3.2 The need for approximation in large and continuous spaces
3.3 Approximation architectures
- 3.3.1 Parametric approximation
- 3.3.2 Nonparametric approximation
- 3.3.3 Comparison of parametric and nonparametric approximation
- 3.3.4 Remarks
3.4 Approximate value iteration
- 3.4.1 Model-based value iteration with parametric approximation
- 3.4.2 Model-free value iteration with parametric approximation
- 3.4.3 Value iteration with nonparametric approximation
- 3.4.4 Convergence and the role of nonexpansive approximation
- 3.4.5 Example: Approximate Q-iteration for a DC motor
3.5 Approximate policy iteration
- 3.5.1 Value iteration-like algorithms for approximate policy
- evaluation
- 3.5.2 Model-free policy evaluation with linearly parameterized approximation
- 3.5.3 Policy evaluation with nonparametric approximation
- 3.5.4 Model-based approximate policy evaluation with rollouts
- 3.5.5 Policy improvement and approximate policy iteration
- 3.5.6 Theoretical guarantees
- 3.5.7 Example: Least-squares policy iteration for a DC motor
3.6 Finding value function approximators automatically
- 3.6.1 Basis function optimization
- 3.6.2 Basis function construction
- 3.6.3 Remarks
3.7 Approximate policy search
- 3.7.1 Policy gradient and actor-critic algorithms
- 3.7.2 Gradient-free policy search
- 3.7.3 Example: Gradient-free policy search for a DC motor
3.8 Comparison of approximate value iteration, policy iteration, and policy search
3.9 Summary and discussion

4. Approximate value iteration with a fuzzy representation

4.1 Introduction
4.2 Fuzzy Q-iteration
- 4.2.1 Approximation and projection mappings of fuzzy Q-iteration
- 4.2.2 Synchronous and asynchronous fuzzy Q-iteration
4.3 Analysis of fuzzy Q-iteration
- 4.3.1 Convergence
- 4.3.2 Consistency
- 4.3.3 Computational complexity
4.4 Optimizing the membership functions
- 4.4.1 A general approach to membership function optimization
- 4.4.2 Cross-entropy optimization
- 4.4.3 Fuzzy Q-iteration with cross-entropy optimization of the membership functions
4.5 Experimental study
- 4.5.1 DC motor: Convergence and consistency study
- 4.5.2 Two-link manipulator: Effects of action interpolation, and comparison with fitted Q-iteration
- 4.5.3 Inverted pendulum: Real-time control
- 4.5.4 Car on the hill: Effects of membership function optimization
4.6 Summary and discussion

5. Approximate policy iteration for online learning and continuous-action control

5.1 Introduction
5.2 A recapitulation of least-squares policy iteration
5.3 Online least-squares policy iteration
5.4 Online LSPI with prior knowledge
- 5.4.1 Online LSPI with policy approximation
- 5.4.2 Online LSPI with monotonic policies
5.5 LSPI with continuous-action, polynomial approximation
5.6 Experimental study
- 5.6.1 Online LSPI for the inverted pendulum
- 5.6.2 Online LSPI for the two-link manipulator
- 5.6.3 Online LSPI with prior knowledge for the DC motor
- 5.6.4 LSPI with continuous-action approximation for the inverted pendulum
5.7 Summary and discussion

6. Approximate policy search with cross-entropy optimization of basis functions

6.1 Introduction
6.2 Cross-entropy optimization
6.3 Cross-entropy policy search
- 6.3.1 General approach
- 6.3.2 Cross-entropy policy search with radial basis functions
6.4 Experimental study
- 6.4.1 Discrete-time double integrator
- 6.4.2 Bicycle balancing
- 6.4.3 Structured treatment interruptions for HIV infection control
6.5 Summary and discussion

Appendix A. Extremely randomized trees

A.1 Structure of the approximator
A.2 Building and using a tree

Appendix B. The cross-entropy method

B.1 Rare-event simulation using the cross-entropy method
B.2 Cross-entropy optimization

Symbols and abbreviations
Bibliography
List of algorithms
Index

About the authors

	Lucian Busoniu is a postdoctoral fellow at the Delft Center for Systems and Control of Delft University of Technology, in the Netherlands. He received his PhD degree (cum laude) in 2009 from the Delft University of Technology, and his MSc degree in 2003 from the Technical University of Cluj-Napoca, Romania. His current research interests include reinforcement learning and dynamic programming with function approximation, intelligent and learning techniques for control problems, and multi-agent learning.
	Robert Babuska is a full professor at the Delft Center for Systems and Control of Delft University of Technology in the Netherlands. He received his PhD degree (cum laude) in Control in 1997 from the Delft University of Technology, and his MSc degree (with honors) in Electrical Engineering in 1990 from Czech Technical University, Prague. His research interests include fuzzy systems modeling and identification, data-driven construction and adaptation of neuro-fuzzy systems, model-based fuzzy control and learning control. He is active in applying these techniques in robotics, mechatronics, and aerospace.
	Bart De Schutter is a full professor at the Delft Center for Systems and Control and at the Marine & Transport Technology department of Delft University of Technology in the Netherlands. He received the PhD degree in Applied Sciences (summa cum laude with congratulations of the examination jury) in 1996 from K.U. Leuven, Belgium. His current research interests include multi-agent systems, hybrid systems control, discrete-event systems, and control of intelligent transportation systems.
	Damien Ernst received the MSc and PhD degrees from the University of Liège in 1998 and 2003, respectively. He is currently a Research Associate of the Belgian FRS-FNRS and he is affiliated with the Systems and Modeling Research Unit of the University of Liège. Damien Ernst spent the period 2003--2006 with the University of Liège as a Postdoctoral Researcher of the FRS-FNRS and held during this period positions as visiting researcher at CMU, MIT and ETH. He spent the academic year 2006--2007 working at Supélec (France) as professor. His main research interests are in the fields of power system dynamics, optimal control, reinforcement learning, and design of dynamic treatment regimes.

Reinforcement Learning and Dynamic Programming Using Function Approximators

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