Lessons from AlphaZero for Optimal, Model Predictive, and Adaptive Control

Dimitri Bertsekas

Mar 2022 · Athena Scientific

About this ebook

The purpose of this book is to propose and develop a new conceptual framework for approximate Dynamic Programming (DP) and Reinforcement Learning (RL). This framework centers around two algorithms, which are designed largely independently of each other and operate in synergy through the powerful mechanism of Newton's method. We call these the off-line training and the on-line play algorithms; the names are borrowed from some of the major successes of RL involving games. Primary examples are the recent (2017) AlphaZero program (which plays chess), and the similarly structured and earlier (1990s) TD-Gammon program (which plays backgammon). In these game contexts, the off-line training algorithm is the method used to teach the program how to evaluate positions and to generate good moves at any given position, while the on-line play algorithm is the method used to play in real time against human or computer opponents.

Both AlphaZero and TD-Gammon were trained off-line extensively using neural networks and an approximate version of the fundamental DP algorithm of policy iteration. Yet the AlphaZero player that was obtained off-line is not used directly during on-line play (it is too inaccurate due to approximation errors that are inherent in off-line neural network training). Instead a separate on-line player is used to select moves, based on multistep lookahead minimization and a terminal position evaluator that was trained using experience with the off-line player. The on-line player performs a form of policy improvement, which is not degraded by neural network approximations. As a result, it greatly improves the performance of the off-line player. 

Similarly, TD-Gammon performs on-line a policy improvement step using one-step or two-step lookahead minimization, which is not degraded by neural network approximations. To this end it uses an off-line neural network-trained terminal position evaluator, and importantly it also extends its on-line lookahead by rollout (simulation with the one-step lookahead player that is based on the position evaluator). 

Significantly, the synergy between off-line training and on-line play also underlies Model Predictive Control (MPC), a major control system design methodology that has been extensively developed since the 1980s. This synergy can be understood in terms of abstract models of infinite horizon DP and simple geometrical constructions, and helps to explain the all-important stability issues within the MPC context. 

An additional benefit of policy improvement by approximation in value space, not observed in the context of games (which have stable rules and environment), is that it works well with changing problem parameters and on-line replanning, similar to indirect adaptive control. Here the Bellman equation is perturbed due to the parameter changes, but approximation in value space still operates as a Newton step. An essential requirement here is that a system model is estimated on-line through some identification method, and is used during the one-step or multistep lookahead minimization process.

In this monograph we aim to provide insights (often based on visualization), which explain the beneficial effects of on-line decision making on top of off-line training. In the process, we will bring out the strong connections between the artificial intelligence view of RL, and the control theory views of MPC and adaptive control. Moreover, we will show that in addition to MPC and adaptive control, our conceptual framework can be effectively integrated with other important methodologies such as multiagent systems and decentralized control, discrete and Bayesian optimization, and heuristic algorithms for discrete optimization.  

One of our principal aims is to show, through the algorithmic ideas of Newton's method and the unifying principles of abstract DP, that the AlphaZero/TD-Gammon methodology of approximation in value space and rollout applies very broadly to deterministic and stochastic optimal control problems. Newton's method here is used for the solution of Bellman's equation, an operator equation that applies universally within DP with both discrete and continuous state and control spaces, as well as finite and infinite horizon.

About the author

Dimitri P. Bertsekas undergraduate studies were in engineering at the National Technical University of Athens, Greece. He obtained his MS in electrical engineering at the George Washington University, Wash. DC in 1969, and his Ph.D. in system science in 1971 at the Massachusetts Institute of Technology.

Dr. Bertsekas has held faculty positions with the Engineering-Economic Systems Dept., Stanford University (1971-1974) and the Electrical Engineering Dept. of the University of Illinois, Urbana (1974-1979). From 1979 to 2019 he was with the Electrical Engineering and Computer Science Department of the Massachusetts Institute of Technology (M.I.T.), where he served as McAfee Professor of Engineering. In 2019, he was appointed Fulton Professor of Computational Decision Making, and a full time faculty member at the department of Computer, Information, and Decision Systems Engineering at Arizona State University (ASU), Tempe, while maintaining a research position at MIT. His research spans several fields, including optimization, control, large-scale computation, and data communication networks, and is closely tied to his teaching and book authoring activities. He has written numerous research papers, and eighteen books and research monographs, several of which are used as textbooks in MIT and ASU classes. Most recently Dr Bertsekas has been focusing on reinforcement learning, and authored a textbook in 2019, and a research monograph on its distributed and multiagent implementation aspects in 2020.

Professor Bertsekas was awarded the INFORMS 1997 Prize for Research Excellence in the Interface Between Operations Research and Computer Science for his book "Neuro-Dynamic Programming", the 2000 Greek National Award for Operations Research, the 2001 ACC John R. Ragazzini Education Award, the 2009 INFORMS Expository Writing Award, the 2014 ACC Richard E. Bellman Control Heritage Award for "contributions to the foundations of deterministic and stochastic optimization-based methods in systems and control," the 2014 Khachiyan Prize for Life-Time Accomplishments in Optimization, the SIAM/MOS 2015 George B. Dantzig Prize, and the 2022 IEEE Control Systems Award. In 2018, he was awarded, jointly with his coauthor John Tsitsiklis, the INFORMS John von Neumann Theory Prize, for the contributions of the research monographs "Parallel and Distributed Computation" and "Neuro-Dynamic Programming". In 2001, he was elected to the United States National Academy of Engineering for "pioneering contributions to fundamental research, practice and education of optimization/control theory, and especially its application to data communication networks."

Dr. Bertsekas' recent books are "Introduction to Probability: 2nd Edition" (2008), "Convex Optimization Theory" (2009), "Dynamic Programming and Optimal Control," Vol. I, (2017), and Vol. II: (2012), "Convex Optimization Algorithms" (2015), "Reinforcement Learning and Optimal Control" (2019), "Rollout, Policy Iteration, and Distributed Reinforcement Learning" (2020), "Abstract Dynamic Programming" (3rd edition, 2022), and "Lessons from AlphaZero for Optimal, Model Predictive, and Adaptive Control" (2022) all published by Athena Scientific.