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Reinforcement Learning and Optimal Control
Reinforcement Learning and Optimal Control by Dimitri P. Bertsekas Massachusetts Institute of Technology WWW site for book information and orders http://www.athenasc.com Athena Scientific, Belmont, Massachusetts Athena Scientific Post Office Box 805 Nashua, NH 03060 U.S.A. Email: [email protected] WWW: http://www.athenasc.com Cover photography: Dimitri Bertsekas c 2019 Dimitri P. Bertsekas All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. Publisher’s Cataloging-in-Publication Data Bertsekas, Dimitri P. Reinforcement Learning and Optimal Control Includes Bibliography and Index 1. Mathematical Optimization. 2. Dynamic Programming. I. Title. QA402.5.B4652019 519.703 00-91281 ISBN-10: 1-886529-39-6, ISBN-13: 978-1-886529-39-7 ABOUT THE AUTHOR Dimitri Bertsekas studied Mechanical and Electrical Engineering at the National Technical University of Athens, Greece, and obtained his Ph.D. in system science from the Massachusetts Institute of Technology. He has held faculty positions with the Engineering-Economic Systems Department, Stanford University, and the Electrical Engineering Department of the Uni- versity of Illinois, Urbana. Since 1979 he has been teaching at the Electrical Engineering and Computer Science Department of the Massachusetts In- stitute of Technology (M.I.T.), where he is currently McAfee Professor of Engineering. Starting in August 2019, he will also be Fulton Professor of Computational Decision Making at the Arizona State University, Tempe, AZ. Professor Bertsekas’ teaching and research have spanned several fields, including deterministic optimization, dynamic programming and stochas- tic control, large-scale and distributed computation, and data communi- cation networks. -
The Bibliography
Referenced Books [Ach92] N. I. Achieser. Theory of Approximation. Dover Publications Inc., New York, 1992. Reprint of the 1956 English translation of the 1st Rus- sian edition; the 2nd augmented Russian edition is available, Moscow, Nauka, 1965. [AH05] Kendall Atkinson and Weimin Han. Theoretical Numerical Analysis: A Functional Analysis Framework, volume 39 of Texts in Applied Mathe- matics. Springer, New York, second edition, 2005. [Atk89] Kendall E. Atkinson. An Introduction to Numerical Analysis. John Wiley & Sons Inc., New York, second edition, 1989. [Axe94] Owe Axelsson. Iterative Solution Methods. Cambridge University Press, Cambridge, 1994. [Bab86] K. I. Babenko. Foundations of Numerical Analysis [Osnovy chislennogo analiza]. Nauka, Moscow, 1986. [Russian]. [BD92] C. A. Brebbia and J. Dominguez. Boundary Elements: An Introductory Course. Computational Mechanics Publications, Southampton, second edition, 1992. [Ber52] S. N. Bernstein. Collected Works. Vol. I. The Constructive Theory of Functions [1905–1930]. Izdat. Akad. Nauk SSSR, Moscow, 1952. [Russian]. [Ber54] S. N. Bernstein. Collected Works. Vol. II. The Constructive Theory of Functions [1931–1953]. Izdat. Akad. Nauk SSSR, Moscow, 1954. [Russian]. [BH02] K. Binder and D. W. Heermann. Monte Carlo Simulation in Statistical Physics: An Introduction, volume 80 of Springer Series in Solid-State Sciences. Springer-Verlag, Berlin, fourth edition, 2002. [BHM00] William L. Briggs, Van Emden Henson, and Steve F. McCormick. A Multigrid Tutorial. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2000. [Boy01] John P. Boyd. Chebyshev and Fourier Spectral Methods. Dover Publi- cations Inc., Mineola, NY, second edition, 2001. [Bra84] Achi Brandt. Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, volume 85 of GMD-Studien [GMD Studies]. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Algorithms for Optimal Paths of One, Many, and an Infinite Number of Agents Permalink https://escholarship.org/uc/item/3qj5d7dj Author Lin, Alex Tong Publication Date 2020 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Algorithms for Optimal Paths of One, Many, and an Infinite Number of Agents A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Alex Tong Lin 2020 c Copyright by Alex Tong Lin 2020 ABSTRACT OF THE DISSERTATION Algorithms for Optimal Paths of One, Many, and an Infinite Number of Agents by Alex Tong Lin Doctor of Philosophy in Mathematics University of California, Los Angeles, 2020 Professor Stanley J. Osher, Chair In this dissertation, we provide efficient algorithms for modeling the behavior of a single agent, multiple agents, and a continuum of agents. For a single agent, we combine the modeling framework of optimal control with advances in optimization splitting in order to efficiently find optimal paths for problems in very high-dimensions, thus providing allevia- tion from the curse of dimensionality. For a multiple, but finite, number of agents, we take the framework of multi-agent reinforcement learning and utilize imitation learning in order to decentralize a centralized expert, thus obtaining optimal multi-agents that act in a de- centralized fashion. For a continuum of agents, we take the framework of mean-field games and use two neural networks, which we train in an alternating scheme, in order to efficiently find optimal paths for high-dimensional and stochastic problems. -
Boundary Value Problems for Systems That Are Not Strictly
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 24 (2011) 757–761 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On mixed initial–boundary value problems for systems that are not strictly hyperbolic Corentin Audiard ∗ Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, Rhone, France article info a b s t r a c t Article history: The classical theory of strictly hyperbolic boundary value problems has received several Received 25 June 2010 extensions since the 70s. One of the most noticeable is the result of Metivier establishing Received in revised form 23 December 2010 Majda's ``block structure condition'' for constantly hyperbolic operators, which implies Accepted 28 December 2010 well-posedness for the initial–boundary value problem (IBVP) with zero initial data. The well-posedness of the IBVP with non-zero initial data requires that ``L2 is a continuable Keywords: initial condition''. For strictly hyperbolic systems, this result was proven by Rauch. We Boundary value problem prove here, by using classical matrix theory, that his fundamental a priori estimates are Hyperbolicity Multiple characteristics valid for constantly hyperbolic IBVPs. ' 2011 Elsevier Ltd. All rights reserved. 1. Introduction In his seminal paper [1] on hyperbolic initial–boundary value problems, H.O. Kreiss performed the algebraic construction of a tool, now called the Kreiss symmetrizer, that leads to a priori estimates. Namely, if u is a solution of 8 d X C >@ u C A .x; t/@ u D f ;.t; x/ 2 × Ω; <> t j xj R jD1 (1) C >Bu D g;.t; x/ 2 @ × @Ω; :> R ujtD0 D 0; C Pd where the operator @t jD1 Aj@xj is assumed to be strictly hyperbolic and B satisfies the uniform Lopatinski˘ı condition, there is some γ0 > 0 such that u satisfies the a priori estimate p γ kuk 2 C C kuk 2 C ≤ C kf k 2 C C kgk 2 C ; (2) Lγ .R ×Ω/ Lγ .R ×@Ω/ Lγ .R ×Ω/ Lγ .R ×@Ω/ 2 2 −γ t for γ ≥ γ0. -
Sampling and Inference in Complex Networks Jithin Kazhuthuveettil Sreedharan
Sampling and inference in complex networks Jithin Kazhuthuveettil Sreedharan To cite this version: Jithin Kazhuthuveettil Sreedharan. Sampling and inference in complex networks. Other [cs.OH]. Université Côte d’Azur, 2016. English. NNT : 2016AZUR4121. tel-01485852 HAL Id: tel-01485852 https://tel.archives-ouvertes.fr/tel-01485852 Submitted on 9 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. École doctorale STIC Sciences et Technologies de l’Information et de la Communication Unité de recherche: INRIA (équipe Maestro) Thèse de doctorat Présentée en vue de l’obtention du grade de Docteur en Sciences de l’UNIVERSITE COTE D’AZUR Mention : Informatique par Jithin Kazhuthuveettil Sreedharan Sampling and Inference in Complex Networks (Échantillonnage et Inférence dans Réseaux Complexes) Dirigé par Konstantin Avrachenkov Soutenue le 2 décembre 2016 Devant le jury composé de: Konstantin Avrachenkov - Inria, France Directeur Nelly Litvak - University of Twente, The Netherlands Rapporteur Don Towsley - University of Massachusetts, USA Rapporteur Philippe Jacquet - Nokia Bell Labs, France Examinateur Alain Jean-Marie - Inria, France Président Abstract The recent emergence of large evolving networks, mainly due to the rise of Online Social Networks (OSNs), brought out the difficulty to gather a complete picture of a network and it opened up the development of new distributed techniques. -
Society Reports USNC/TAM
Appendix J 2008 Society Reports USNC/TAM Table of Contents J.1 AAM: Ravi-Chandar.............................................................................................. 1 J.2 AIAA: Chen............................................................................................................. 2 J.3 AIChE: Higdon ....................................................................................................... 3 J.4 AMS: Kinderlehrer................................................................................................. 5 J.5 APS: Foss................................................................................................................. 5 J.6 ASA: Norris............................................................................................................. 6 J.7 ASCE: Iwan............................................................................................................. 7 J.8 ASME: Kyriakides.................................................................................................. 8 J.9 ASTM: Chona ......................................................................................................... 9 J.10 SEM: Shukla ....................................................................................................... 11 J.11 SES: Jasiuk.......................................................................................................... 13 J.12 SIAM: Healey...................................................................................................... 14 J.13 SNAME: Karr.................................................................................................... -
Madeleine Udell's Thesis
GENERALIZED LOW RANK MODELS A DISSERTATION SUBMITTED TO THE INSTITUTE FOR COMPUTATIONAL AND MATHEMATICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Madeleine Udell May 2015 c Copyright by Madeleine Udell 2015 All Rights Reserved ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Professor Stephen Boyd) Principal Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Professor Ben Van Roy) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Professor Lester Mackey) Approved for the Stanford University Committee on Graduate Studies iii Abstract Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. This dissertation extends the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, k-means, k-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compress- ing, denoising, and imputing missing entries across all data types simultaneously. -
Nash Q-Learning for General-Sum Stochastic Games
Journal of Machine Learning Research 4 (2003) 1039-1069 Submitted 11/01; Revised 10/02; Published 11/03 Nash Q-Learning for General-Sum Stochastic Games Junling Hu [email protected] Talkai Research 843 Roble Ave., 2 Menlo Park, CA 94025, USA Michael P. Wellman [email protected] Artificial Intelligence Laboratory University of Michigan Ann Arbor, MI 48109-2110, USA Editor: Craig Boutilier Abstract We extend Q-learning to a noncooperative multiagent context, using the framework of general- sum stochastic games. A learning agent maintains Q-functions over joint actions, and performs updates based on assuming Nash equilibrium behavior over the current Q-values. This learning protocol provably converges given certain restrictions on the stage games (defined by Q-values) that arise during learning. Experiments with a pair of two-player grid games suggest that such restric- tions on the game structure are not necessarily required. Stage games encountered during learning in both grid environments violate the conditions. However, learning consistently converges in the first grid game, which has a unique equilibrium Q-function, but sometimes fails to converge in the second, which has three different equilibrium Q-functions. In a comparison of offline learn- ing performance in both games, we find agents are more likely to reach a joint optimal path with Nash Q-learning than with a single-agent Q-learning method. When at least one agent adopts Nash Q-learning, the performance of both agents is better than using single-agent Q-learning. We have also implemented an online version of Nash Q-learning that balances exploration with exploitation, yielding improved performance. -
LIDS – a BRIEF HISTORY Alan S
LIDS – A BRIEF HISTORY Alan S. Willsky The Laboratory for Information and Decision Systems (LIDS) is MIT’s oldest laboratory and has evolved in scope and changed its name several times during its long and rich history. For a more detailed account of its early years, see From Servo Loops to Fiber Nets, a document produced in 1990 by then-co-directors Sanjoy K. Mitter and Robert G. Gallager as part LIDS’ 50th anniversary celebration. The 1940s into the 1950s The Servomechanism Laboratory was established in 1940 by Professor Gordon Brown, one year before the United States entered World War II. At that time, the concepts of feedback control and servomechanisms were being codified and recognized as important disciplines, and the need for advances – initially driven by military challenges and later by industrial automation – were substantial. Indeed, the first electrical engineering department courses in servomechanisms were created only in 1939 and were codified in the classic 1948 textbook Principles of Servomechanisms by Gordon Brown and Donald Campbell. Wartime applications such as servo controls for the pointing of guns pushed the application and the foundational methodology and theory of servomechanisms and control. On the theoretical/methodological side, this push led to the 1957 text Analytical Design of Linear Feedback Controls, by George Newton, Leonard Gould, and James Kaiser, which combined Norbert Wiener’s theory of estimation of time series and the principles of servomechanisms to create new approaches to control design and the understanding of fundamental limits of performance. On the applications side, military and industrial post-war challenges led to major programs and contributions by researchers in the Lab. -
Mathematics People
NEWS Mathematics People or up to ten years post-PhD, are eligible. Awardees receive Braverman Receives US$1 million distributed over five years. NSF Waterman Award —From an NSF announcement Mark Braverman of Princeton University has been selected as a Prizes of the Association cowinner of the 2019 Alan T. Wa- terman Award of the National Sci- for Women in Mathematics ence Foundation (NSF) for his work in complexity theory, algorithms, The Association for Women in Mathematics (AWM) has and the limits of what is possible awarded a number of prizes in 2019. computationally. According to the Catherine Sulem of the Univer- prize citation, his work “focuses on sity of Toronto has been named the Mark Braverman complexity, including looking at Sonia Kovalevsky Lecturer for 2019 by algorithms for optimization, which, the Association for Women in Math- when applied, might mean planning a route—how to get ematics (AWM) and the Society for from point A to point B in the most efficient way possible. Industrial and Applied Mathematics “Algorithms are everywhere. Most people know that (SIAM). The citation states: “Sulem every time someone uses a computer, algorithms are at is a prominent applied mathemati- work. But they also occur in nature. Braverman examines cian working in the area of nonlin- randomness in the motion of objects, down to the erratic Catherine Sulem ear analysis and partial differential movement of particles in a fluid. equations. She has specialized on “His work is also tied to algorithms required for learning, the topic of singularity development in solutions of the which serve as building blocks to artificial intelligence, and nonlinear Schrödinger equation (NLS), on the problem of has even had implications for the foundations of quantum free surface water waves, and on Hamiltonian partial differ- computing. -
Value and Policy Iteration in Optimal Control and Adaptive Dynamic Programming
May 2015 Report LIDS-P-3174 Value and Policy Iteration in Optimal Control and Adaptive Dynamic Programming Dimitri P. Bertsekasy Abstract In this paper, we consider discrete-time infinite horizon problems of optimal control to a terminal set of states. These are the problems that are often taken as the starting point for adaptive dynamic programming. Under very general assumptions, we establish the uniqueness of solution of Bellman's equation, and we provide convergence results for value and policy iteration. 1. INTRODUCTION In this paper we consider a deterministic discrete-time optimal control problem involving the system xk+1 = f(xk; uk); k = 0; 1;:::; (1.1) where xk and uk are the state and control at stage k, lying in sets X and U, respectively. The control uk must be chosen from a constraint set U(xk) ⊂ U that may depend on the current state xk. The cost for the kth stage is g(xk; uk), and is assumed nonnnegative and real-valued: 0 ≤ g(xk; uk) < 1; xk 2 X; uk 2 U(xk): (1.2) We are interested in feedback policies of the form π = fµ0; µ1;:::g, where each µk is a function mapping every x 2 X into the control µk(x) 2 U(x). The set of all policies is denoted by Π. Policies of the form π = fµ, µ, : : :g are called stationary, and for convenience, when confusion cannot arise, will be denoted by µ. No restrictions are placed on X and U: for example, they may be finite sets as in classical shortest path problems involving a graph, or they may be continuous spaces as in classical problems of control to the origin or some other terminal set. -
19880014833.Pdf
;V,45Il t'£- /?~ ~f5-6 NASA Contractor Report 181656 lCASE REPORT NO. 88-24 NASA-CR-181656 19880014833 I I .t. ICASE EFFICIENT IMPLEMENTATION OF ESSENTIALLY NON-DSCILLATORY SHOCK CAPTURING SCHEMES, II Chi-nang Shu Stanley Osher Contract No. NASI-18ID7 April 1988 INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING NASA Langley Research Center, Hampton, Virginia 23665 Operated by the Universities Space Research Association ~r:l·l~~~ H"~) r~R·'-:.v _ ~,1\Uf'i::f},~ . ..... L .... ~ b,i' ~ L 'u~l 1 .,' \c.~:~ NI\SI\ J , "l/\..~' National Aeronautics and L\~!~L::Y nEsr,",I'.:-'~ c~~~~~:" Space Administration I !t ~~~ e," ,I' i \ ~ Langley Research Center Hampton. Virginia 23665 llllmnllllllmrllllllillmlllllllilif NF00885 4B~ EM3287.PRT DISPLAY 22/2/2 88N24217*t ISSUE 17 PAGE 2392 CATEGORY 64 RPT#: NASA-CR-181656 ICASE-88-24 NAS 1.26:181656 CNTt: NAS1-18107 NAGl-270 88/04/00 64 PAGES UNCLASSIFIED DOCUMENT UTTL: Efficient implementation of essentially non-oscillatory shock capturing schemes, 2 TLSP: Final Report AUTH: A/SHU, CHI-WANG; B/OSHER, STANLEY PAA: B/(California Univ., Los Angeles.) CORP: National Aeronautics and Space Administration. Langley Research Center, Hampton, Va. SAP: Avail: NTIS HC A04/MF A01 cro: UNITED STATES Submitted for publication Sponsored in part by Defense Advanced Research Projects Agency, Arlington, Va. MAJS: /*NUMERICAL FLOW VISUALIZATION/*OSCILLATIONS/*SHOCK WAVES MINS: I CONSERVATION LAWS/ EULER EQUATIONS OF MOTION/ RUNGE-KUTTA METHOD ABA: Author ABS: Earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock capturing schemes is continued. Anew simplified expression is provided for the ENa construction procedure based again on numerical fluxes rather than cell averages.