Skip to main content

Reinforcement Learning in Games

  • Chapter

Part of the Adaptation, Learning, and Optimization book series (ALO,volume 12)

Abstract

Reinforcement learning and games have a long and mutually beneficial common history. From one side, games are rich and challenging domains for testing reinforcement learning algorithms. From the other side, in several games the best computer players use reinforcement learning. The chapter begins with a selection of games and notable reinforcement learning implementations.Without any modifications, the basic reinforcement learning algorithms are rarely sufficient for high-level gameplay, so it is essential to discuss the additional ideas, ways of inserting domain knowledge, implementation decisions that are necessary for scaling up. These are reviewed in sufficient detail to understand their potentials and their limitations. The second part of the chapter lists challenges for reinforcement learning in games, together with a review of proposed solution methods. While this listing has a game-centric viewpoint, and some of the items are specific to games (like opponent modelling), a large portion of this overview can provide insight for other kinds of applications, too. In the third part we review how reinforcement learning can be useful in game development and find its way into commercial computer games. Finally, we provide pointers for more in-depth reviews of specific games and solution approaches.

Keywords

  • Computer Game
  • Reinforcement Learning
  • Reinforcement Learning Algorithm
  • Temporal Difference Learning
  • Reinforcement Learning Approach

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-27645-3_17
  • Chapter length: 39 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   299.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-27645-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   379.99
Price excludes VAT (USA)
Hardcover Book
USD   379.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aha, D.W., Molineaux, M., Ponsen, M.: Learning to win: Case-based plan selection in a real-time strategy game. Case-Based Reasoning Research and Development, 5–20 (2005)

    Google Scholar 

  • Amit, A., Markovitch, S.: Learning to bid in bridge. Machine Learning 63(3), 287–327 (2006)

    CrossRef  Google Scholar 

  • Andrade, G., Santana, H., Furtado, A., Leitão, A., Ramalho, G.: Online adaptation of computer games agents: A reinforcement learning approach. Scientia 15(2) (2004)

    Google Scholar 

  • Auer, P., Cesa-Bianchi, N., Fischer, P.: Finite-time analysis of the multiarmed bandit problem. Machine Learning 47, 235–256 (2002)

    CrossRef  Google Scholar 

  • Bartók, G., Szepesvári, C., Zilles, S.: Models of active learning in group-structured state spaces. Information and Computation 208, 364–384 (2010)

    MathSciNet  CrossRef  Google Scholar 

  • Baxter, J., Tridgell, A., Weaver, L.: Learning to play chess using temporal-differences. Machine learning 40(3), 243–263 (2000)

    CrossRef  Google Scholar 

  • Baxter, J., Tridgell, A., Weaver, L.: Reinforcement learning and chess. In: Machines that learn to play games, pp. 91–116. Nova Science Publishers, Inc. (2001)

    Google Scholar 

  • Beal, D., Smith, M.C.: Learning piece values using temporal differences. ICCA Journal 20(3), 147–151 (1997)

    Google Scholar 

  • Bertsekas, D.P., Tsitsiklis, J.N.: Neuro-Dynamic Programming. Athena Scientific (1996)

    Google Scholar 

  • Billings, D., Davidson, A., Schauenberg, T., Burch, N., Bowling, M., Holte, R.C., Schaeffer, J., Szafron, D.: Game-Tree Search with Adaptation in Stochastic Imperfect-Information Games. In: van den Herik, H.J., Björnsson, Y., Netanyahu, N.S. (eds.) CG 2004. LNCS, vol. 3846, pp. 21–34. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  • Björnsson, Y., Finnsson, H.: Cadiaplayer: A simulation-based general game player. IEEE Transactions on Computational Intelligence and AI in Games 1(1), 4–15 (2009)

    CrossRef  Google Scholar 

  • Böhm, N., Kókai, G., Mandl, S.: Evolving a heuristic function for the game of tetris. In: Proc. Lernen, Wissensentdeckung und Adaptivität LWA, pp. 118–122 (2004)

    Google Scholar 

  • Boumaza, A.: On the evolution of artificial Tetris players. In: IEEE Symposium on Computational Intelligence and Games (2009)

    Google Scholar 

  • Bouzy, B., Helmstetter, B.: Monte Carlo Go developments. In: Advances in Computer Games, pp. 159–174 (2003)

    Google Scholar 

  • Bowling, M.: Convergence and no-regret in multiagent learning. In: Neural Information Processing Systems, pp. 209–216 (2004)

    Google Scholar 

  • Buro, M.: From simple features to sophisticated evaluation functions. In: International Conference on Computers and Games, pp. 126–145 (1998)

    Google Scholar 

  • Buro, M., Furtak, T.: RTS games as test-bed for real-time research. JCIS, 481–484 (2003)

    Google Scholar 

  • Buro, M., Lanctot, M., Orsten, S.: The second annual real-time strategy game AI competition. In: GAME-ON NA (2007)

    Google Scholar 

  • Chaslot, G., Winands, M., Herik, H., Uiterwijk, J., Bouzy, B.: Progressive strategies for monte-carlo tree search. New Mathematics and Natural Computation 4(3), 343 (2008)

    MathSciNet  CrossRef  Google Scholar 

  • Chaslot, G., Fiter, C., Hoock, J.B., Rimmel, A., Teytaud, O.: Adding Expert Knowledge and Exploration in Monte-Carlo Tree Search. In: van den Herik, H.J., Spronck, P. (eds.) ACG 2009. LNCS, vol. 6048, pp. 1–13. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  • Chatriot, L., Gelly, S., Jean-Baptiste, H., Perez, J., Rimmel, A., Teytaud, O.: Including expert knowledge in bandit-based Monte-Carlo planning, with application to computer-Go. In: European Workshop on Reinforcement Learning (2008)

    Google Scholar 

  • Coquelin, P.A., Munos, R.: Bandit algorithms for tree search. In: Uncertainty in Artificial Intelligence (2007)

    Google Scholar 

  • Coulom, R.: Efficient Selectivity and Backup Operators in Monte-carlo Tree Search. In: van den Herik, H.J., Ciancarini, P., Donkers, H.H.L.M(J.) (eds.) CG 2006. LNCS, vol. 4630, pp. 72–83. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  • Coulom, R.: Computing Elo ratings of move patterns in the game of go. ICGA Journal 30(4), 198–208 (2007)

    Google Scholar 

  • Dahl, F.A.: Honte, a Go-playing program using neural nets. In: Machines that learn to play games, pp. 205–223. Nova Science Publishers (2001)

    Google Scholar 

  • Davidson, A.: Opponent modeling in poker: Learning and acting in a hostile and uncertain environment. Master’s thesis, University of Alberta (2002)

    Google Scholar 

  • Diuk, C., Cohen, A., Littman, M.L.: An object-oriented representation for efficient reinforcement learning. In: International Conference on Machine Learning, pp. 240–247 (2008)

    Google Scholar 

  • Droste, S., Fürnkranz, J.: Learning of piece values for chess variants. Tech. Rep. TUD–KE–2008-07, Knowledge Engineering Group, TU Darmstadt (2008)

    Google Scholar 

  • Džeroski, S., Raedt, L.D., Driessens, K.: Relational reinforcement learning. Machine Learning 43(1-2), 7–52 (2001)

    CrossRef  Google Scholar 

  • Epstein, S.L.: Toward an ideal trainer. Machine Learning 15, 251–277 (1994)

    Google Scholar 

  • Farias, V.F., van Roy, B.: Tetris: A Study of Randomized Constraint Sampling. In: Probabilistic and Randomized Methods for Design Under Uncertainty. Springer, UK (2006)

    Google Scholar 

  • Fawcett, T., Utgoff, P.: Automatic feature generation for problem solving systems. In: International Conference on Machine Learning, pp. 144–153 (1992)

    Google Scholar 

  • Finkelstein, L., Markovitch, S.: Learning to play chess selectively by acquiring move patterns. ICCA Journal 21, 100–119 (1998)

    Google Scholar 

  • Fudenberg, D., Levine, D.K.: The theory of learning in games. MIT Press (1998)

    Google Scholar 

  • Fürnkranz, J.: Machine learning in games: a survey. In: Machines that Learn to Play Games, pp. 11–59. Nova Science Publishers (2001)

    Google Scholar 

  • Fürnkranz, J.: Recent advances in machine learning and game playing. Tech. rep., TU Darmstadt (2007)

    Google Scholar 

  • Galway, L., Charles, D., Black, M.: Machine learning in digital games: a survey. Artificial Intelligence Review 29(2), 123–161 (2008)

    CrossRef  Google Scholar 

  • Gelly, S., Silver, D.: Achieving master-level play in 9x9 computer go. In: AAAI, pp. 1537–1540 (2008)

    Google Scholar 

  • Gelly, S., Wang, Y., Munos, R., Teytaud, O.: Modification of UCT with patterns in Monte-Carlo go. Tech. rep., INRIA (2006)

    Google Scholar 

  • Gherrity, M.: A game-learning machine. PhD thesis, University of California, San Diego, CA (1993)

    Google Scholar 

  • Ghory, I.: Reinforcement learning in board games. Tech. rep., Department of Computer Science, University of Bristol (2004)

    Google Scholar 

  • Gilgenbach, M.: Fun game AI design for beginners. In: AI Game Programming Wisdom, vol. 3. Charles River Media, Inc. (2006)

    Google Scholar 

  • Gilpin, A., Sandholm, T.: Lossless abstraction of imperfect information games. Journal of the ACM 54(5), 25 (2007)

    MathSciNet  CrossRef  Google Scholar 

  • Gilpin, A., Sandholm, T., Sørensen, T.B.: Potential-aware automated abstraction of sequential games, and holistic equilibrium analysis of Texas Hold’em poker. In: AAAI, vol. 22, pp. 50–57 (2007)

    Google Scholar 

  • Ginsberg, M.L.: Gib: Imperfect information in a computationally challenging game. Journal of Artificial Intelligence Research 14, 313–368 (2002)

    Google Scholar 

  • Gould, J., Levinson, R.: Experience-based adaptive search. Tech. Rep. UCSC-CRL-92-10, University of California at Santa Cruz (1992)

    Google Scholar 

  • Günther, M.: Automatic feature construction for general game playing. PhD thesis, Dresden University of Technology (2008)

    Google Scholar 

  • Hagelbäck, J., Johansson, S.J.: Measuring player experience on runtime dynamic difficulty scaling in an RTS game. In: International Conference on Computational Intelligence and Games (2009)

    Google Scholar 

  • Hartley, T., Mehdi, Q., Gough, N.: Online learning from observation for interactive computer games. In: International Conference on Computer Games: Artificial Intelligence and Mobile Systems, pp. 27–30 (2005)

    Google Scholar 

  • van den Herik, H.J., Uiterwijk, J.W.H.M., van Rijswijck, J.: Games solved: Now and in the future. Artificial Intelligence 134, 277–311 (2002)

    CrossRef  Google Scholar 

  • Hsu, F.H.: Behind Deep Blue: Building the Computer that Defeated the World Chess Champion. Princeton University Press, Princeton (2002)

    Google Scholar 

  • Hunicke, R., Chapman, V.: AI for dynamic difficult adjustment in games. In: Challenges in Game AI Workshop (2004)

    Google Scholar 

  • Kakade, S.: A natural policy gradient. In: Advances in Neural Information Processing Systems, vol. 14, pp. 1531–1538 (2001)

    Google Scholar 

  • Kalles, D., Kanellopoulos, P.: On verifying game designs and playing strategies using reinforcement learning. In: ACM Symposium on Applied Computing, pp. 6–11 (2001)

    Google Scholar 

  • Kerbusch, P.: Learning unit values in Wargus using temporal differences. BSc thesis (2005)

    Google Scholar 

  • Kocsis, L., Szepesvári, C.: Bandit Based Monte-Carlo Planning. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. (eds.) ECML 2006. LNCS (LNAI), vol. 4212, pp. 282–293. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  • Kocsis, L., Szepesvári, C., Winands, M.H.M.: RSPSA: Enhanced Parameter Optimization in Games. In: van den Herik, H.J., Hsu, S.-C., Hsu, T.-s., Donkers, H.H.L.M(J.) (eds.) CG 2005. LNCS, vol. 4250, pp. 39–56. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  • Kok, E.: Adaptive reinforcement learning agents in RTS games. Master’s thesis, University of Utrecht, The Netherlands (2008)

    Google Scholar 

  • Koza, J.: Genetic programming: on the programming of computers by means of natural selection. MIT Press (1992)

    Google Scholar 

  • Kuhlmann, G.J.: Automated domain analysis and transfer learning in general game playing. PhD thesis, University of Texas at Austin (2010)

    Google Scholar 

  • Lagoudakis, M.G., Parr, R., Littman, M.L.: Least-Squares Methods in Reinforcement Learning for Control. In: Vlahavas, I.P., Spyropoulos, C.D. (eds.) SETN 2002. LNCS (LNAI), vol. 2308, pp. 249–260. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  • Laursen, R., Nielsen, D.: Investigating small scale combat situations in real time strategy computer games. Master’s thesis, University of Aarhus (2005)

    Google Scholar 

  • Levinson, R., Weber, R.: Chess Neighborhoods, Function Combination, and Reinforcement Learning. In: Marsland, T., Frank, I. (eds.) CG 2001. LNCS, vol. 2063, pp. 133–150. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  • Lorenz, U.: Beyond Optimal Play in Two-Person-Zerosum Games. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 749–759. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  • Mańdziuk, J.: Knowledge-Free and Learning-Based Methods in Intelligent Game Playing. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  • Marthi, B., Russell, S., Latham, D.: Writing Stratagus-playing agents in concurrent alisp. In: IJCAI Workshop on Reasoning, Representation, and Learning in Computer Games, pp. 67–71 (2005)

    Google Scholar 

  • McGlinchey, S.J.: Learning of AI players from game observation data. In: GAME-ON, pp. 106–110 (2003)

    Google Scholar 

  • Molineaux, M., Aha, D.W., Ponsen, M.: Defeating novel opponents in a real-time strategy game. In: IJCAI Workshop on Reasoning, Representation, and Learning in Computer Games, pp. 72–77 (2005)

    Google Scholar 

  • Moriarty, D.E., Miikkulainen, R.: Discovering complex Othello strategies through evolutionary neural networks. Connection Science 7, 195–209 (1995)

    Google Scholar 

  • Müller, M.: Position evaluation in computer go. ICGA Journal 25(4), 219–228 (2002)

    Google Scholar 

  • Naddaf, Y.: Game-independent AI agents for playing Atari 2600 console games. Master’s thesis, University of Alberta (2010)

    Google Scholar 

  • Pollack, J.B., Blair, A.D.: Why did TD-Gammon work? In: Neural Information Processing Systems, vol. 9, pp. 10–16 (1997)

    Google Scholar 

  • Ponsen, M., Spronck, P.: Improving adaptive game AI with evolutionary learning. In: Computer Games: Artificial Intelligence, Design and Education (2004)

    Google Scholar 

  • Ponsen, M., Muñoz-Avila, H., Spronck, P., Aha, D.W.: Automatically acquiring adaptive real-time strategy game opponents using evolutionary learning. In: Proceedings of the 17th Innovative Applications of Artificial Intelligence Conference (2005)

    Google Scholar 

  • Ponsen, M., Spronck, P., Tuyls, K.: Hierarchical reinforcement learning in computer games. In: Adaptive Learning Agents and Multi-Agent Systems, pp. 49–60 (2006)

    Google Scholar 

  • Ponsen, M., Taylor, M.E., Tuyls, K.: Abstraction and Generalization in Reinforcement Learning: A Summary and Framework. In: Taylor, M.E., Tuyls, K. (eds.) ALA 2009. LNCS, vol. 5924, pp. 1–33. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  • Ramanujan, R., Sabharwal, A., Selman, B.: Adversarial search spaces and sampling-based planning. In: International Conference on Automated Planning and Scheduling (2010)

    Google Scholar 

  • Risk, N., Szafron, D.: Using counterfactual regret minimization to create competitive multiplayer poker agents. In: International Conference on Autonomous Agents and Multiagent Systems, pp. 159–166 (2010)

    Google Scholar 

  • Rubin, J., Watson, I.: Computer poker: A review. Artificial Intelligence 175(5-6), 958–987 (2011)

    MathSciNet  CrossRef  Google Scholar 

  • Schaeffer, J.: The games computers (and people) play. In: Zelkowitz, M. (ed.) Advances in Computers, vol. 50, pp. 89–266. Academic Press (2000)

    Google Scholar 

  • Schaeffer, J., Hlynka, M., Jussila, V.: Temporal difference learning applied to a high-performance game-playing program. In: International Joint Conference on Artificial Intelligence, pp. 529–534 (2001)

    Google Scholar 

  • Schnizlein, D., Bowling, M., Szafron, D.: Probabilistic state translation in extensive games with large action sets. In: International Joint Conference on Artificial Intelligence, pp. 278–284 (2009)

    Google Scholar 

  • Schraudolph, N.N., Dayan, P., Sejnowski, T.J.: Learning to evaluate go positions via temporal difference methods. In: Computational Intelligence in Games. Studies in Fuzziness and Soft Computing, ch. 4, vol. 62, pp. 77–98. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  • Scott, B.: The illusion of intelligence. In: AI Game Programming Wisdom, pp. 16–20. Charles River Media (2002)

    Google Scholar 

  • Shapiro, A., Fuchs, G., Levinson, R.: Learning a Game Strategy Using Pattern-Weights and Self-Play. In: Schaeffer, J., Müller, M., Björnsson, Y. (eds.) CG 2002. LNCS, vol. 2883, pp. 42–60. Springer, Heidelberg (2003)

    CrossRef  Google Scholar 

  • Sharifi, A.A., Zhao, R., Szafron, D.: Learning companion behaviors using reinforcement learning in games. In: AIIDE (2010)

    Google Scholar 

  • Sharma, S., Kobti, Z., Goodwin, S.: General game playing: An overview and open problems. In: International Conference on Computing, Engineering and Information, pp. 257–260 (2009)

    Google Scholar 

  • Silver, D., Tesauro, G.: Monte-carlo simulation balancing. In: International Conference on Machine Learning (2009)

    Google Scholar 

  • Silver, D., Sutton, R., Mueller, M.: Sample-based learning and search with permanent and transient memories. In: ICML (2008)

    Google Scholar 

  • Spronck, P., Sprinkhuizen-Kuyper, I., Postma, E.: Difficulty scaling of game AI. In: GAME-ON 2004: 5th International Conference on Intelligent Games and Simulation (2004)

    Google Scholar 

  • Spronck, P., Ponsen, M., Sprinkhuizen-Kuyper, I., Postma, E.: Adaptive game AI with dynamic scripting. Machine Learning 63(3), 217–248 (2006)

    CrossRef  Google Scholar 

  • Stanley, K.O., Bryant, B.D., Miikkulainen, R.: Real-time neuroevolution in the NERO video game. IEEE Transactions on Evolutionary Computation 9(6), 653–668 (2005)

    CrossRef  Google Scholar 

  • Sturtevant, N., White, A.: Feature construction for reinforcement learning in Hearts. In: Advances in Computers and Games, pp. 122–134 (2007)

    Google Scholar 

  • Szczepański, T., Aamodt, A.: Case-based reasoning for improved micromanagement in real-time strategy games. In: Workshop on Case-Based Reasoning for Computer Games, 8th International Conference on Case-Based Reasoning, pp. 139–148 (2009)

    Google Scholar 

  • Szita, I., Lőrincz, A.: Learning Tetris using the noisy cross-entropy method. Neural Computation 18(12), 2936–2941 (2006a)

    CrossRef  Google Scholar 

  • Szita, I., Lőrincz, A.: Learning to play using low-complexity rule-based policies: Illustrations through Ms. Pac-Man. Journal of Articial Intelligence Research 30, 659–684 (2006b)

    Google Scholar 

  • Szita, I., Szepesvári, C.: Sz-tetris as a benchmark for studying key problems of rl. In: ICML 2010 Workshop on Machine Learning and Games (2010)

    Google Scholar 

  • Szita, I., Chaslot, G., Spronck, P.: Monte-Carlo Tree Search in Settlers of Catan. In: van den Herik, H.J., Spronck, P. (eds.) ACG 2009. LNCS, vol. 6048, pp. 21–32. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  • Tesauro, G.: Practical issues in temporal difference learning. Machine Learning 8, 257–277 (1992)

    Google Scholar 

  • Tesauro, G.: Temporal difference learning and TD-gammon. Communications of the ACM 38(3), 58–68 (1995)

    CrossRef  Google Scholar 

  • Tesauro, G.: Comments on co-evolution in the successful learning of backgammon strategy’. Machine Learning 32(3), 241–243 (1998)

    CrossRef  Google Scholar 

  • Tesauro, G.: Programming backgammon using self-teaching neural nets. Artificial Intelligence 134(1-2), 181–199 (2002)

    CrossRef  Google Scholar 

  • Thiery, C., Scherrer, B.: Building controllers for Tetris. ICGA Journal 32(1), 3–11 (2009)

    Google Scholar 

  • Thrun, S.: Learning to play the game of chess. In: Neural Information Processing Systems, vol. 7, pp. 1069–1076 (1995)

    Google Scholar 

  • Utgoff, P.: Feature construction for game playing. In: Fürnkranz, J., Kubat, M. (eds.) Machines that Learn to Play Games, pp. 131–152. Nova Science Publishers (2001)

    Google Scholar 

  • Utgoff, P., Precup, D.: Constructive function approximation. In: Liu, H., Motoda, H. (eds.) Feature Extraction, Construction and Selection: A Data Mining Perspective, vol. 453, pp. 219–235. Kluwer Academic Publishers (1998)

    Google Scholar 

  • Veness, J., Silver, D., Uther, W., Blair, A.: Bootstrapping from game tree search. In: Neural Information Processing Systems, vol. 22, pp. 1937–1945 (2009)

    Google Scholar 

  • Weber, B.G., Mateas, M.: Case-based reasoning for build order in real-time strategy games. In: Artificial Intelligence and Interactive Digital Entertainment, pp. 1313–1318 (2009)

    Google Scholar 

  • Wender, S., Watson, I.: Using reinforcement learning for city site selection in the turn-based strategy game Civilization IV. In: Computational Intelligence and Games, pp. 372–377 (2009)

    Google Scholar 

  • Wiering, M.A.: Self-play and using an expert to learn to play backgammon with temporal difference learning. Journal of Intelligent Learning Systems and Applications 2, 57–68 (2010)

    CrossRef  Google Scholar 

  • Zinkevich, M., Johanson, M., Bowling, M., Piccione, C.: Regret minimization in games with incomplete information. In: Neural Information Processing Systems, pp. 1729–1736 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to István Szita .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Szita, I. (2012). Reinforcement Learning in Games. In: Wiering, M., van Otterlo, M. (eds) Reinforcement Learning. Adaptation, Learning, and Optimization, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27645-3_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27645-3_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27644-6

  • Online ISBN: 978-3-642-27645-3

  • eBook Packages: EngineeringEngineering (R0)