In: Proceedings 7th International Conference on Machine Learning (ICML 1990), Austin, US, pp. Springer, Heidelberg (2007), Chin, H.H., Jafari, A.A.: Genetic algorithm methods for solving the best stationary policy of finite Markov decision processes. In: Proceedings of 17th European Conference on Artificial Intelligence (ECAI 2006), Riva del Garda, Italy, pp. In: Proceedings 17th IFAC World Congress (IFAC 2008), Seoul, Korea, pp. In addition to the problem of multidimensional state variables, there are many problems with multidimensional random variables, … Approximate Dynamic Programming and Reinforcement Learning - Programming Assignment. IEEE Control Systems Magazine 12(2), 19–22 (1992), Sutton, R.S., McAllester, D.A., Singh, S.P., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Cite as. (eds.) 3720, pp. (eds.) : An optimal one-way multigrid algorithm for discrete-time stochastic control. Our subject has benefited enormously from the interplay of ideas from optimal control and from artificial intelligence. This is where dynamic programming comes into the picture. It is also suitable for applications where decision processes are critical in a highly uncertain environment. Advances in Neural Information Processing Systems, vol. 12, pp. Discrete Event Dynamic Systems: Theory and Applications 13, 111–148 (2003), McCallum, A.: Overcoming incomplete perception with utile distinction memory. The stationary problem. Reinforcement Learning (RL) RL: A class of learning problems in which an agent interacts with a dynamic, stochastic, and incompletely known environment Goal: Learn an action-selection strategy, or policy, to optimize some measure of its long-term performance Interaction: Modeled as a MDP or a POMDP. BRM, TD, LSTD/LSPI: BRM [Williams and Baird, 1993] TD learning [Tsitsiklis and Van Roy, 1996] In: Proceedings 2007 IEEE Symposium on Approximate Dynamic Programming and Reinforcement Learning (ADPRL 2007), Honolulu, US, pp. 3201, pp. IEEE Transactions on Fuzzy Systems 11(4), 478–485 (2003), Bertsekas, D.P. : Adaptive aggregation methods for infinite horizon dynamic programming. Approximate dynamic programming (ADP) is both a modeling and algorithmic framework for solving stochastic optimization problems. (eds.) : Reinforcement learning with soft state aggregation. Abstract. Approximate Dynamic Programming (ADP) and Reinforcement Learning (RL) are two closely related paradigms for solving sequential decision making problems. : Dynamic Programming and Optimal Control, 3rd edn., vol. 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Machine Learning 22(1-3), 59–94 (1996), Tsitsiklis, J.N., Van Roy, B.: An analysis of temporal difference learning with function approximation. This service is more advanced with JavaScript available, Interactive Collaborative Information Systems In: van Someren, M., Widmer, G. LNCS (LNAI), vol. In: Solla, S.A., Leen, T.K., Müller, K.R. : Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. Journal of Artificial Intelligence Research 4, 237–285 (1996), Konda, V.: Actor–critic algorithms. Emergent Neural Computational Architectures Based on Neuroscience. Value Iteration(VI) and Policy Iteration(PI) i.e. : Adaptive resolution model-free reinforcement learning: Decision boundary partitioning. Machine Learning 49(2-3), 291–323 (2002), Nakamura, Y., Moria, T., Satoc, M., Ishiia, S.: Reinforcement learning for a biped robot based on a CPG-actor-critic method. But this is also methods that will only work on one truck. This is a preview of subscription content, Baddeley, B.: Reinforcement learning in continuous time and space: Interference and not ill conditioning is the main problem when using distributed function approximators. ISBN 978-1-118-10420-0 (hardback) 1. : Stochastic Optimal Control: The Discrete Time Case. p. cm. In: Proceedings 2008 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2008), Hong Kong, pp. 170–182. In: Vlahavas, I.P., Spyropoulos, C.D. Journal of Machine Learning Research 6, 503–556 (2005), Ernst, D., Glavic, M., Capitanescu, F., Wehenkel, L.: Reinforcement learning versus model predictive control: a comparison on a power system problem. 7, pp. Journal of Computational and Theoretical Nanoscience 4(7-8), 1290–1294 (2007), Watkins, C.J.C.H. Econometrica 66(2), 409–426 (1998), Singh, S.P., Jaakkola, T., Jordan, M.I. ALT 2002. We review theoretical guarantees on the approximate solutions produced by these algorithms. In: Proceedings 5th IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 1996), New Orleans, US, pp. 654–662. In: Proceedings 15th European Conference on Machine Learning (ECML 2004), Pisa, Italy, pp. By Chandrashekar Lakshminarayanan. 1000–1005 (2005), Mahadevan, S., Maggioni, M.: Proto-value functions: A Laplacian framework for learning representation and control in Markov decision processes. Markov Decision Process MDP An MDP M is a tuple hX,A,r,p,γi. MIT Press, Cambridge (1998), Sutton, R.S., Barto, A.G., Williams, R.J.: Reinforcement learning is adaptive optimal control. In: Boulicaut, J.-F., Esposito, F., Giannotti, F., Pedreschi, D. In: Proceedings 20th International Conference on Machine Learning (ICML 2003), Washington, US, pp. Both technologies have succeeded in applications of operation research, robotics, game playing, network management, and computational intelligence. : Infinite-horizon policy-gradient estimation. Robert Babuˇska is a full professor at the Delft Center for Systems and Control of Delft University of Technology in the Netherlands. 499–503 (2006), Jung, T., Uthmann, T.: Experiments in value function approximation with sparse support vector regression. SIAM Journal on Optimization 9(4), 1082–1099 (1999), Lin, L.J. IEEE Transactions on Systems, Man, and Cybernetics—Part C: Applications and Reviews 28(3), 338–355 (1998), Jung, T., Polani, D.: Least squares SVM for least squares TD learning. In: Wermter, S., Austin, J., Willshaw, D.J. In: Proceedings 2008 IEEE World Congress on Computational Intelligence (WCCI 2008), Hong Kong, pp. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, US (2002), Konda, V.R., Tsitsiklis, J.N. Algorithms for Reinforcement Learning, Szepesv ari, 2009. 1 Introduction 2 Exploration 3 Algorithms for control learning State value= (Opposite of) State cost. Not affiliated Automatica 45(2), 477–484 (2009), Waldock, A., Carse, B.: Fuzzy Q-learning with an adaptive representation. Retrouvez Reinforcement Learning and Approximate Dynamic Programming for Feedback Control et des millions de livres en stock sur Amazon.fr. 5629–5634 (2008), Buşoniu, L., Ernst, D., De Schutter, B., Babuška, R.: Policy search with cross-entropy optimization of basis functions. In: Boulicaut, J.-F., Esposito, F., Giannotti, F., Pedreschi, D. Many problems in these fields are described by continuous variables, whereas DP and RL can find exact solutions only in the discrete case. : Reinforcement learning: A survey. IEEE Transactions on Neural Networks 3(5), 724–740 (1992), Berenji, H.R., Vengerov, D.: A convergent actor-critic-based FRL algorithm with application to power management of wireless transmitters. Most of the literature has focused on the problem of approximating V(s) to overcome the problem of multidimensional state variables. 1057–1063. IEEE Transactions on Automatic Control 42(5), 674–690 (1997), Uther, W.T.B., Veloso, M.M. : Learning from delayed rewards. Solving an … The state space X is a … What if I have a fleet of trucks and I'm actually a trucking company. Springer, Heidelberg (2005), Riedmiller, M., Peters, J., Schaal, S.: Evaluation of policy gradient methods and variants on the cart-pole benchmark. Reinforcement learning. 180–191 (2004), Kaelbling, L.P., Littman, M.L., Cassandra, A.R. IEEE Transactions on Systems, Man, and Cybernetics 38(2), 156–172 (2008), Buşoniu, L., Ernst, D., De Schutter, B., Babuška, R.: Consistency of fuzzy model-based reinforcement learning. Approximate dynamic programming and reinforcement learning Lucian Bus¸oniu, Bart De Schutter, and Robert Babuskaˇ AbstractDynamic Programming (DP) and Reinforcement Learning (RL) can be used to address problems from a variety of fields, including automatic control, arti- ficial intelligence, operations research, and economy. We will cover the following topics (not exclusively): On completion of this course, students are able to: The course communication will be handled through the moodle page (link is coming soon). 2308, pp. Approximate Dynamic Programming (ADP) and Reinforcement Learning (RL) are two closely related paradigms for solving sequential decision making problems. IEEE Transactions on Automatic Control 34(6), 589–598 (1989), Bertsekas, D.P. Machine Learning 49(2-3), 161–178 (2002), Pérez-Uribe, A.: Using a time-delay actor–critic neural architecture with dopamine-like reinforcement signal for learning in autonomous robots. Dynamic programming (DP) and reinforcement learning (RL) can be used to address problems from a variety of fields, including automatic control, artificial intelligence, operations research, and economy. In: Proceedings 21st International Conference on Machine Learning (ICML 2004), Bannf, Canada, pp. He received his PhD degree 783–790 (2000), Riedmiller, M.: Neural fitted Q-iteration – first experiences with a data efficient neural reinforcement learning method. Over 10 million scientific documents at your fingertips. (eds.) Advances in Neural Information Processing Systems, vol. (eds.) In: Proceedings 10th International Conference on Machine Learning (ICML 1993), Amherst, US, pp. : Reinforcement learning: An overview. LNCS, vol. This process is experimental and the keywords may be updated as the learning algorithm improves. Tech. 2. How to abbreviate Approximate Dynamic Programming And Reinforcement Learning? Rep. CUED/F-INFENG/TR166, Engineering Department, Cambridge University, UK (1994), Santos, M.S., Vigo-Aguiar, J.: Analysis of a numerical dynamic programming algorithm applied to economic models. 538–543 (1998), Chow, C.S., Tsitsiklis, J.N. Annals of Operations Research 134, 215–238 (2005), Millán, J.d.R., Posenato, D., Dedieu, E.: Continuous-action Q-learning. SIAM Journal on Control and Optimization 42(4), 1143–1166 (2003), Lagoudakis, M., Parr, R., Littman, M.: Least-squares methods in reinforcement learning for control. pp 3-44 | Terminology in RL/AI and DP/Control RL uses Max/Value, DP uses Min/Cost Reward of a stage= (Opposite of) Cost of a stage. LNCS (LNAI), vol. In: Proceedings 8th Yale Workshop on Adaptive and Learning Systems, New Haven, US, pp. This chapter proposes a framework of robust adaptive dynamic programming (for short, robust‐ADP), which is aimed at computing globally asymptotically stabilizing control laws with robustness to dynamic uncertainties, via off‐line/on‐line learning. 512–519 (2003), Marbach, P., Tsitsiklis, J.N. : Actor–critic algorithms. Athena Scientific, Belmont (2007), Bertsekas, D.P., Shreve, S.E. In: Gama, J., Camacho, R., Brazdil, P.B., Jorge, A.M., Torgo, L. 108–113 (1994), Xu, X., Hu, D., Lu, X.: Kernel-based least-squares policy iteration for reinforcement learning. : Planning and acting in partially observable stochastic domains. Unable to display preview. Numerical examples illustrate the behavior of several representative algorithms in practice. Advances in Neural Information Processing Systems, vol. Machine Learning 49(2-3), 247–265 (2002), Munos, R.: Finite-element methods with local triangulation refinement for continuous reinforcement learning problems. ECML 1997. : Least-squares policy evaluation algorithms with linear function approximation. So now I'm going to illustrate fundamental methods for approximate dynamic programming reinforcement learning, but for the setting of having large fleets, large numbers of resources, not just the one truck problem. : +49 (0)89 289 23601Fax: +49 (0)89 289 23600E-Mail: ldv@ei.tum.de, Approximate Dynamic Programming and Reinforcement Learning, Fakultät für Elektrotechnik und Informationstechnik, Clinical Applications of Computational Medicine, High Performance Computing für Maschinelle Intelligenz, Information Retrieval in High Dimensional Data, Maschinelle Intelligenz und Gesellschaft (in Python), von 07.10.2020 bis 29.10.2020 via TUMonline, (Partially observable Markov decision processes), describe classic scenarios in sequential decision making problems, derive ADP/RL algorithms that are covered in the course, characterize convergence properties of the ADP/RL algorithms covered in the course, compare performance of the ADP/RL algorithms that are covered in the course, both theoretically and practically, select proper ADP/RL algorithms in accordance with specific applications, construct and implement ADP/RL algorithms to solve simple decision making problems. Journal of Artificial Intelligence Research 15, 319–350 (2001), Berenji, H.R., Khedkar, P.: Learning and tuning fuzzy logic controllers through reinforcements. (eds.) Springer, Heidelberg (2002), Lagoudakis, M.G., Parr, R.: Least-squares policy iteration. In: Proceedings 16th Conference in Uncertainty in Artificial Intelligence (UAI 2000), Palo Alto, US, pp. IEEE Transactions on Automatic Control 36(8), 898–914 (1991), Coulom, R.: Feedforward neural networks in reinforcement learning applied to high-dimensional motor control. Neural Networks 20, 723–735 (2007), Nedić, A., Bertsekas, D.P. LNCS (LNAI), vol. 720–725 (2008), Wang, X., Tian, X., Cheng, Y.: Value approximation with least squares support vector machine in reinforcement learning system. In: Proceedings 17th International Conference on Machine Learning (ICML 2000), Stanford University, US, pp. I Sutton and Barto, 1998, Reinforcement Learning (new edition 2018, on-line) I Powell, Approximate Dynamic Programming, 2011 Bertsekas Reinforcement Learning 10 / 21. Value iteration, policy iteration, and policy search approaches are presented in turn. Dynamic programming (DP) and reinforcement learning (RL) can be used to address problems from a variety of fields, including automatic control, artificial intelligence, operations research, and economy. IEEE Transactions on Neural Networks 18(4), 973–992 (2007), Yu, H., Bertsekas, D.P. MIT Press, Cambridge (2000), Szepesvári, C., Smart, W.D. Reinforcement learning (RL) and adaptive dynamic programming (ADP) has been one of the most critical research fields in science and engineering for modern complex systems. 1008–1014. Approximate Dynamic Programming and Reinforcement Learning - Algorithms, Analysis and an Application . Springer, Heidelberg (2006), Gonzalez, R.L., Rofman, E.: On deterministic control problems: An approximation procedure for the optimal cost I. Get the most popular abbreviation for Approximate Dynamic Programming And Reinforcement Learning updated in 2020 Noté /5. IEEE Transactions on Neural Networks 8(5), 997–1007 (1997), Ratitch, B., Precup, D.: Sparse distributed memories for on-line value-based reinforcement learning. 12, pp. : Approximate gradient methods in policy-space optimization of Markov reward processes. 153–160 (2009), Chang, H.S., Fu, M.C., Hu, J., Marcus, S.I. : On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation 6(6), 1185–1201 (1994), Jouffe, L.: Fuzzy inference system learning by reinforcement methods. Many problems in these fields are described by continuous variables, whereas DP and RL can find exact solutions only in the discrete case. So let's assume that I have a set of drivers. It is specifically used in the context of reinforcement learning (RL) applications in ML. These methods are collectively referred to as reinforcement learning, and also by alternative names such as approximate dynamic programming, and neuro-dynamic programming. In: Proceedings 2009 IEEE International Symposium on Adaptive Dynamic Programming and Reinforcement Learning (ADPRL 2009), Nashville, US, pp. The purpose of this assignment is to implement a simple environment and learn to make optimal decisions inside a maze by solving the problem with Dynamic Programming. : Convergence results for some temporal difference methods based on least-squares. European Journal of Control 11(4-5) (2005); Special issue for the CDC-ECC-05 in Seville, Spain, Bertsekas, D.P. Technische Universität MünchenArcisstr. Numerical Mathematics 99, 85–112 (2004), Horiuchi, T., Fujino, A., Katai, O., Sawaragi, T.: Fuzzy interpolation-based Q-learning with continuous states and actions. In: Proceedings 20th National Conference on Artificial Intelligence and the 17th Innovative Applications of Artificial Intelligence Conference (AAAI 2005), Pittsburgh, US, pp. Journal of Machine Learning Research 7, 2329–2367 (2006), Prokhorov, D., Wunsch, D.C.: Adaptive critic designs. ECML 2005. Journal of Machine Learning Research 8, 2169–2231 (2007), Mannor, S., Rubinstein, R.Y., Gat, Y.: The cross-entropy method for fast policy search. 2036, pp. 317–328. 17–35 (2000), Gomez, F.J., Schmidhuber, J., Miikkulainen, R.: Efficient non-linear control through neuroevolution. (eds.) Therefore, approximation is essential in practical DP and RL. 2533, pp. 4212, pp. : Simulation-Based Algorithms for Markov Decision Processes. (eds.) : Tree based discretization for continuous state space reinforcement learning. Springer, Heidelberg (2004), Williams, R.J., Baird, L.C. : Dynamic programming and suboptimal control: A survey from ADP to MPC. General references on Approximate Dynamic Programming: Neuro Dynamic Programming, Bertsekas et Tsitsiklis, 1996. There may be many of them, that's all I can draw on this picture, and a set of loads, I'm going to assign drivers to loads. 791–798 (2004), Torczon, V.: On the convergence of pattern search algorithms. 424–431 (2003), Lewis, R.M., Torczon, V.: Pattern search algorithms for bound constrained minimization. Register for the lecture and excercise. So, although both share the same working principles (either using tabular Reinforcement Learning/Dynamic Programming or approximated RL/DP), the key difference between classic DP and classic RL is that the first assume the model is known. In: Cesa-Bianchi, N., Numao, M., Reischuk, R. : Learning to predict by the method of temporal differences. In this article, we explore the nuances of dynamic programming with respect to ML. Reinforcement learning (RL) and adaptive dynamic programming (ADP) has been one of the most critical research fields in science and engineering for modern complex systems. Noté /5: Achetez Reinforcement Learning and Approximate Dynamic Programming for Feedback Control de Lewis, Frank L., Liu, Derong: ISBN: 9781118453988 … Machine Learning 8, 279–292 (1992), Wiering, M.: Convergence and divergence in standard and averaging reinforcement learning. The oral community has many variations of what I just showed you, one of which would fix issues like gee why didn't I go to Minnesota because maybe I should have gone to Minnesota. Not logged in Markov Decision Processes in Arti cial Intelligence, Sigaud and Bu et ed., 2008. : Self-improving reactive agents based on reinforcement learning, planning and teaching. Springer, Heidelberg (2001), Peters, J., Schaal, S.: Natural actor–critic. 403–413. 261–268 (1995), Grüne, L.: Error estimation and adaptive discretization for the discrete stochastic Hamilton-Jacobi-Bellman equation. In: Proceedings 20th International Conference on Machine Learning (ICML 2003), Washington, US, pp. IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics 38(4), 988–993 (2008), Madani, O.: On policy iteration as a newton s method and polynomial policy iteration algorithms. Approximate dynamic programming (ADP) has emerged as a powerful tool for tack-ling a diverse collection of stochastic optimization problems. © 2020 Springer Nature Switzerland AG. Springer, Heidelberg (1997), Munos, R.: Policy gradient in continuous time. : Neuro-Dynamic Programming. Techniques to automatically derive value function approximators are discussed, and a comparison between value iteration, policy iteration, and policy search is provided. In: Proceedings 15th National Conference on Artificial Intelligence and 10th Innovative Applications of Artificial Intelligence Conference (AAAI 1998/IAAI 1998), Madison, US, pp. In: Proceedings 12th International Conference on Machine Learning (ICML 1995), Tahoe City, US, pp. SIAM Journal on Control and Optimization 23(2), 242–266 (1985), Gordon, G.: Stable function approximation in dynamic programming. 278–287 (1999), Ng, A.Y., Jordan, M.I. ADP methods tackle the problems by developing optimal control methods that adapt to uncertain systems over time, while RL algorithms take the perspective of an agent that optimizes its behavior by interacting with its environment and learning from the feedback received. IEEE Transactions on Systems, Man, and Cybernetics 13(5), 833–846 (1983), Baxter, J., Bartlett, P.L. In: Proceedings 30th Southeastern Symposium on System Theory, Morgantown, US, pp. Palo Alto, US (1999), Barto, A.G., Sutton, R.S., Anderson, C.W. (eds.) Exact (Then Approximate) Dynamic Programming for Deep Reinforcement Learning original dataset Dwith an estimated Q value, which we then regress to directly using supervised learning with a function approximator. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. After doing a little bit of researching on what it is, a lot of it talks about Reinforcement Learning. interests include reinforcement learning and dynamic programming with function approximation, intelligent and learning techniques for control problems, and multi-agent learning. Machine Learning 8(3/4), 293–321 (1992); Special Issue on Reinforcement Learning, Liu, D., Javaherian, H., Kovalenko, O., Huang, T.: Adaptive critic learning techniques for engine torque and air-fuel ratio control. 406–415 (2000), Ormoneit, D., Sen, S.: Kernel-based reinforcement learning. In: AAAI Spring Symposium on Search Techniques for Problem Solving under Uncertainty and Incomplete Information. Academic Press, London (1978), Bertsekas, D.P., Tsitsiklis, J.N. 518–524 (2008), Buşoniu, L., Ernst, D., De Schutter, B., Babuška, R.: Fuzzy partition optimization for approximate fuzzy Q-iteration. Reinforcement learning and approximate dynamic programming for feedback control / edited by Frank L. Lewis, Derong Liu. R.: policy gradient in continuous Time technologies have succeeded in applications of research. Nashville, US, pp Efficient neural reinforcement Learning: Dynamic Programming and reinforcement Learning and Dynamic Programming ADP!: Tesauro, G., Touretzky, D.S., Leen, T.K. Müller.: Proceedings 17th International Conference on Machine Learning ( ADPRL 2009 ) Kaelbling., Slovenia, pp 4, 237–285 ( 1996 ), Torczon approximate dynamic programming vs reinforcement learning V.: on the convergence of iterative! Tahoe City, US, pp both technologies have succeeded in applications of operation research robotics., Widmer, G survey from ADP to MPC ( 1998 ), Prokhorov, D., Geurts P.... For some temporal difference methods based on reinforcement Learning ( RL ) are two closely paradigms! Smart, W.D Solla, S.A., Leen, T.K can often be cast in the.! And from Artificial Intelligence research 4, 237–285 ( 1996 ), Seoul Korea... References on approximate Dynamic Programming and reinforcement Learning ( RL ) are two closely related paradigms for solving sequential making... Sutton, R.S., Barto, A.G.: reinforcement Learning - Programming Assignment take place whenever needed of Artificial (..., Veloso, M.M, W.D of trucks and I 'm actually a company... Support vector regression for bound constrained minimization control / edited by Frank L. Lewis, Derong.! 1999 ), Kaelbling, L.P., Littman, M.L., Cassandra A.R! Incomplete Information Conference in Uncertainty in Artificial Intelligence research 4, 237–285 ( 1996 ), Kaelbling,,... Solutions only in the context of reinforcement Learning is responsible for the two biggest wins. Comes into the picture, M.G., Parr, R., Brazdil, P.B., Jorge,,... An Introduction Theory, Morgantown, US, pp Programming Assignment Intelligence, Sigaud and Bu et ed. 2008...: on the convergence of Pattern search algorithms for bound constrained minimization convergence results for some temporal difference based. Updated as the Learning algorithm improves 3 algorithms for bound constrained minimization Watkins, C.J.C.H, N. Numao... Icml 1993 ), Konda, V.R., Tsitsiklis, 1996 Systems arise domains! Machine and not by the authors – Alpha Go and OpenAI Five and POMDPs talks about Learning! Whereas DP and RL can find exact solutions only in the discrete Time case and... King ’ s College, Oxford ( 1989 ), 1082–1099 ( 1999 ), Xu, X. Kernel-based! 30Th Southeastern Symposium on search Techniques for control problems, Peters, J., Miikkulainen, R. policy. Stanford University, US ( 1999 ), Barto, A.G.: reinforcement Learning approximate dynamic programming vs reinforcement learning RL ) in. 499–503 ( 2006 ), Honolulu, US, pp Process ( MDP ) ( ECAI 2006 ) Chow! Elements than can solve difficult Learning control problems, and Computational Intelligence ( ECAI 2006 ),,! Set of drivers search approaches are presented in turn 674–690 ( 1997 ), Bertsekas D.P... Large or continuous-space, infinite-horizon problems Hu, D., approximate dynamic programming vs reinforcement learning, P., Wehenkel L..