Dimitri P. Bertsekas


Lecture slides for a course in Reinforcement Learning and Optimal Control (January 8-February 21, 2019), at Arizona State University:

Slides-Lecture 1, Slides-Lecture 2, Slides-Lecture 3, Slides-Lecture 4, Slides-Lecture 5, Slides-Lecture 6, Slides-Lecture 7, Slides-Lecture 8, Slides-Lecture 9, Slides-Lecture 10, Slides-Lecture 11, Slides-Lecture 12, Slides-Lecture 13.

Videos of lectures from Reinforcement Learning and Optimal Control course at Arizona State University: (Click around the screen to see just the video, or just the slides, or both simultaneously).

Video-Lecture 1, Video-Lecture 2, Video-Lecture 3,Video-Lecture 4, Video-Lecture 5, Video-Lecture 6, Video-Lecture 7, Video-Lecture 8, Video-Lecture 9, Video-Lecture 10, Video-Lecture 11, Video-Lecture 12, Video-Lecture 13.

Lecture 13 is an overview of the entire course.



The book is available from the publishing company Athena Scientific, or from

Click here for an extended lecture/summary of the book: Ten Key Ideas for Reinforcement Learning and Optimal Control.

The purpose of the book is to consider large and challenging multistage decision problems, which can be solved in principle by dynamic programming and optimal control, but their exact solution is computationally intractable. We discuss solution methods that rely on approximations to produce suboptimal policies with adequate performance. These methods are collectively referred to as reinforcement learning, and also by alternative names such as approximate dynamic programming, and neuro-dynamic programming.

Our subject has benefited enormously from the interplay of ideas from optimal control and from artificial intelligence. One of the aims of this monograph is to explore the common boundary between these two fields and to form a bridge that is accessible by workers with background in either field.

The mathematical style of the book is somewhat different from the author's dynamic programming books, and the neuro-dynamic programming monograph, written jointly with John Tsitsiklis. We rely more on intuitive explanations and less on proof-based insights. Still we provide a rigorous short account of the theory of finite and infinite horizon dynamic programming, and some basic approximation methods, in an appendix. For this we require a modest mathematical background: calculus, elementary probability, and a minimal use of matrix-vector algebra.

The methods of this book have been successful in practice, and often spectacularly so, as evidenced by recent amazing accomplishments in the games of chess and Go. However, across a wide range of problems, their performance properties may be less than solid. This is a reflection of the state of the art in the field: there are no methods that are guaranteed to work for all or even most problems, but there are enough methods to try on a given challenging problem with a reasonable chance that one or more of them will be successful in the end. Accordingly, we have aimed to present a broad range of methods that are based on sound principles, and to provide intuition into their properties, even when these properties do not include a solid performance guarantee. Hopefully, with enough exploration with some of these methods and their variations, the reader will be able to address adequately his/her own problem.


Click here for preface and table of contents.

Selected sections from book chapters:

Chapter 1: Exact Dynamic Programming,

Chapter 2: Approximation in Value Space,

The following papers and reports have a strong connection to material in the book, and amplify on its analysis and its range of applications.

  • D. P. Bertsekas, "Multiagent Rollout Algorithms and Reinforcement Learning," arXiv preprint arXiv:1910.00120, September 2019.

  • Bhattacharya, S., Badyal, S., Wheeler, W., Gil, S., Bertsekas, D.,"Reinforcement Learning for POMDP: Partitioned Rollout and Policy Iteration with Application to Autonomous Sequential Repair Problems," IEEE Robotics and Automation Letters, to appear, 2020.

  • Bertsekas, D., "Constrained Multiagent Rollout and Multidimensional Assignment with the Auction Algorithm," arXiv preprint, arXiv:2002.07407 February 2020.

  • D. P. Bertsekas, "Biased Aggregation, Rollout, and Enhanced Policy Improvement for Reinforcement Learning," Lab. for Information and Decision Systems Report, MIT, October 2018; a shorter version appears as arXiv preprint arXiv:1910.02426, Oct. 2019.

  • D. P. Bertsekas, "Feature-Based Aggregation and Deep Reinforcement Learning: A Survey and Some New Implementations," Lab. for Information and Decision Systems Report, MIT, April 2018 (revised August 2018); arXiv preprint arXiv:1804.04577; a version published in IEEE/CAA Journal of Automatica Sinica. (Lecture Slides). (Related Video Lecture).

    Dynamic Programming and Optimal Control, Vol. 1, 4th Edition

    Dimitri P. Bertsekas

    Published February 2017Volume 1 cover image

    The fourth edition (February 2017) contains a substantial amount of new material, particularly on approximate DP in Chapter 6. This chapter was thoroughly reorganized and rewritten, to bring it in line, both with the contents of Vol. II, whose latest edition appeared in 2012, and with recent developments, which have propelled approximate DP to the forefront of attention.

    Some of the highlights of the revision of Chapter 6 are an increased emphasis on one-step and multistep lookahead methods, parametric approximation architectures, neural networks, rollout, and Monte Carlo tree search. Among other applications, these methods have been instrumental in the recent spectacular success of computer Go programs. The material on approximate DP also provides an introduction and some perspective for the more analytically oriented treatment of Vol. II.

    Click here for direct ordering from the publisher and preface, table of contents, supplementary educational material, lecture slides, videos, etc

    Dynamic Programming and Optimal Control, Vol. I, ISBN-13: 978-1-886529-43-4, 576 pp., hardcover, 2017

    Dynamic Programming and Optimal Control, Vol. II, 4th Edition: Approximate Dynamic Programming

    Dimitri P. Bertsekas

    Published June 2012Volume 2 cover image

    The fourth edition of Vol. II of the two-volume DP textbook was published in June 2012. This is a major revision of Vol. II and contains a substantial amount of new material, as well as a reorganization of old material. The length has increased by more than 60% from the third edition, and most of the old material has been restructured and/or revised. Volume II now numbers more than 700 pages and is larger in size than Vol. I. It can arguably be viewed as a new book!

    Approximate DP has become the central focal point of this volume, and occupies more than half of the book (the last two chapters, and large parts of Chapters 1-3). Thus one may also view this new edition as a followup of the author's 1996 book "Neuro-Dynamic Programming" (coauthored with John Tsitsiklis). A lot of new material, the outgrowth of research conducted in the six years since the previous edition, has been included.

    A new printing of the fourth edition (January 2018) contains some updated material, particularly on undiscounted problems in Chapter 4, and approximate DP in Chapter 6. References were also made to the contents of the 2017 edition of Vol. I, and to high profile developments in deep reinforcement learning, which have brought approximate DP to the forefront of attention.

    Dynamic Programming and Optimal Control, Vol. II: Approximate Dynamic Programming, ISBN-13: 978-1-886529-44-1, 712 pp., hardcover, 2012


    Click here for an updated version of Chapter 4, which incorporates recent research on a variety of undiscounted problem topics, including

  • Deterministic optimal control and adaptive DP (Sections 4.2 and 4.3).

  • Stochastic shortest path problems under weak conditions and their relation to positive cost problems (Sections 4.1.4 and 4.4).

  • Affine monotonic and multiplicative cost models (Section 4.5).


    Click here for preface and detailed information.

    Click here to order at

    Lectures on Exact and Approximate Finite Horizon DP: Videos from a 4-lecture, 4-hour short course at the University of Cyprus on finite horizon DP, Nicosia, 2017. Videos from Youtube. (Lecture Slides: Lecture 1, Lecture 2, Lecture 3, Lecture 4.)

    Videos from a 6-lecture, 12-hour short course at Tsinghua Univ., Beijing, China, 2014. From the Tsinghua course site, and from Youtube. Click here to download Approximate Dynamic Programming Lecture slides, for this 12-hour video course.

    Click here to download lecture slides for a 7-lecture short course on Approximate Dynamic Programming, Caradache, France, 2012.

    Click here to download lecture slides for the MIT course "Dynamic Programming and Stochastic Control (6.231), Dec. 2015. The last six lectures cover a lot of the approximate dynamic programming material.

    Click here to download research papers and other material on Dynamic Programming and Approximate Dynamic Programming.

    Abstract Dynamic Programming, 2nd Edition, 2018

    by Dimitri P. Bertsekas


    The 2nd edition of the research monograph "Abstract Dynamic Programming," is available in hardcover from the publishing company, Athena Scientific, or from

    The 2nd edition aims primarily to amplify the presentation of the semicontractive models of Chapter 3 and Chapter 4 of the first (2013) edition, and to supplement it with a broad spectrum of research results that I obtained and published in journals and reports since the first edition was written (see below). As a result, the size of this material more than doubled, and the size of the book increased by nearly 40%.

    Contents and Preface, 2ND EDITION

    Chapter 1, 2ND EDITION, Introduction

    Chapter 2, 2ND EDITION, Contractive Models

    Chapter 3, 2ND EDITION, Semicontractive Models

    Chapter 4, 2ND EDITION, Noncontractive Models

    Appendixes and References, 2ND EDITION

    In addition to the changes in Chapters 3, and 4, I have also eliminated from the second edition the material of the first edition that deals with restricted policies and Borel space models (Chapter 5 and Appendix C). These models are motivated in part by the complex measurability questions that arise in mathematically rigorous theories of stochastic optimal control involving continuous probability spaces. The restricted policies framework aims primarily to extend abstract DP ideas to Borel space models. Since this material is fully covered in Chapter 6 of the 1978 monograph by Bertsekas and Shreve, and followup research on the subject has been limited, I decided to omit Chapter 5 and Appendix C of the first edition from the second edition and just post them below.

    Chapter 5 of 1st Edition

    Appendix C of 1st Edition

    The following papers and reports have a strong connection to the book, and amplify on the analysis and the range of applications of the semicontractive models of Chapters 3 and 4:

  • D. P. Bertsekas, "Regular Policies in Abstract Dynamic Programming", Lab. for Information and Decision Systems Report LIDS-P-3173, MIT, May 2015; SIAM J. on Optimization, Vol. 27, No. 3, pp. 1694-1727. (Related Lecture Slides); (Related Video Lectures).

  • D. P. Bertsekas, "Value and Policy Iteration in Deterministic Optimal Control and Adaptive Dynamic Programming", Lab. for Information and Decision Systems Report LIDS-P-3174, MIT, May 2015 (revised Sept. 2015); IEEE Transactions on Neural Networks and Learning Systems, Vol. 28, 2017, pp. 500-509.

  • D. P. Bertsekas and H. Yu, "Stochastic Shortest Path Problems Under Weak Conditions", Lab. for Information and Decision Systems Report LIDS-P-2909, MIT, January 2016.

  • D. P. Bertsekas, "Robust Shortest Path Planning and Semicontractive Dynamic Programming," Lab. for Information and Decision Systems Report LIDS-P-2915, MIT, Feb. 2014 (revised Jan. 2015 and June 2016); arXiv preprint arXiv:1608.01670; Naval Research Logistics (NRL), 66(1), pp.15-37.

  • D. P. Bertsekas, "Affine Monotonic and Risk-Sensitive Models in Dynamic Programming", Lab. for Information and Decision Systems Report LIDS-3204, MIT, June 2016; to appear in IEEE Transactions on Aut. Control.

  • D. P. Bertsekas, "Stable Optimal Control and Semicontractive Dynamic Programming," SIAM J. on Control and Optimization, Vol. 56, 2018, pp. 231-252, (Related Lecture Slides), (Related Video Lecture from MIT, May 2017). (Related Lecture Slides from UConn, Oct. 2017). (Related Video Lecture from UConn, Oct. 2017).

  • D. P. Bertsekas, "Proper Policies in Infinite-State Stochastic Shortest Path Problems," IEEE Transactions on Automatic Control, Vol. 63, 2018, pp. 3787-3792. (Related Lecture Slides).

  • An updated version of Chapter 4 of the author's Dynamic Programming book, Vol. II, which incorporates recent research on a variety of undiscounted problems and relates to abstract DP topics; (Related Lecture Slides).


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