Abstract
An important application of reinforcement learning (RL) is to finite-state control problems and one of the most difficult problems in learning for control is balancing the exploration/exploitation tradeoff. Existing theoretical results for RL give very little guidance on reasonable ways to perform exploration. In this paper, we examine the convergence of single-step on-policy RL algorithms for control. On-policy algorithms cannot separate exploration from learning and therefore must confront the exploration problem directly. We prove convergence results for several related on-policy algorithms with both decaying exploration and persistent exploration. We also provide examples of exploration strategies that can be followed during learning that result in convergence to both optimal values and optimal policies.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barto, A. G., Bradtke S. J., & Singh S. (1995). Learning to act using real-time dynamic programming. Artificial Intelligence, 72(1), 81–138.
Bellman, R. (1957). Dynamic Programming. Princeton, NJ: Princeton University Press.
Bertsekas, D. P. (1995). Dynamic Programming and Optimal Control. (Vol. 1 and 2). Belmont, Massachusetts: Athena Scientific.
Boyan, J. A. & Moore, A. W. (1995). Generalization in reinforcement learning: Safely approximating the value function. In G. Tesauro, D. S. Touretzky, & T. K. Leen (Eds.), Advances in neural information processing systems (Vol. 7, pp. 369–376). Cambridge, MA: The MIT Press.
Breiman, L. (1992).Probability. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics.
Crites, R. H. & Barto, A. G. (1996). Improving elevator performance using reinforcement learning. Advances in neural information processing systems (Vol. 8). MIT Press.
Dayan, P. (1992). The convergence of TD(λ) for general λ. Machine Learning, 8(3), 341–362.
Dayan, P. & Sejnowski, T. J. (1994). TD.(λ)converges with probability 1. Machine Learning, 14 (3).
Dayan, P. & Sejnowski, T. J. (1996). Exploration bonuses and dual control. Machine Learning, 25, 5–22.
Gullapalli, V. & Barto, A. G. (1994). Convergence of indirect adaptive asynchronous value iteration algorithms. In J. D. Cowan, G. Tesauro, & J. Alspector (Eds.), Advances in neural information processing systems (Vol. 6, pp. 695–702). San Mateo, CA: Morgan Kaufmann.
Jaakkola, T., Jordan, M. I., & Singh, S. (1994). On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6(6), 1185–1201.
John, G. H. (1994). When the best move isn't optimal: Q-learning with exploration. In Proceedings of the Twelfth National Conference on Artificial Intelligence, Seattle, WA, p. 1464.
John, G. H. (1995). When the best move isn't optimal: Q-learning with exploration. Unpublished manuscript, available at URL ftp://starry.stanford.edu/pub/gjohn/papers/rein-nips.ps.
Kaelbling, L. P., Littman, M. L., & Moore, A. W. (1996). Reinforcement learning: A survey. Journal of Artificial Intelligence Research, 4:237–285.
Kumar, P. R. & Varaiya, P. P. (1986). Stochastic systems: estimation, identification, and adaptive control. Englewood Cliffs, NJ: Prentice Hall.
Littman, M. L. & Szepesvári, C. (1996). A generalized reinforcement-learning model: Convergence and applications. In L. Saitta (Ed.), Proceedings of the Thirteenth International Conference on Machine Learning (pp. 310–318).
Littman, M. L. (1996). Algorithms for sequential decision making. Ph.D. Thesis, Department of Computer Science, Brown University, February. Also Technical Report CS–96–09.
Puterman, M. L. (1994). Markov decision processes—discrete stochastic dynamic programming. New York, NY: John Wiley & Sons, Inc.
Rummery, G. A. (1994). Problem solving with reinforcement learning. Ph.D. Thesis, Cambridge University Engineering Department.
Rummery, G. A. & Niranjan, M. (1994). On-line Q-learning using connectionist systems. Technical Report CUED/F-INFENG/TR 166, Cambridge University Engineering Department.
Singh, S. & Bertsekas, D. P. (1997). Reinforcement learning for dynamic channel allocation in cellular telephone systems. Advances in neural information processing systems (Vol. 9, pp. 974–980). MIT Press.
Singh, S. & Sutton, R. S. (1996). Reinforcement learning with replacing eligibility traces. Machine Learning, 22(1–3):123–158.
Singh, S. & Yee, R. C. (1994). An upper bound on the loss from approximate optimal-value functions. Machine Learning, 16, 227–233.
Sutton, R. S. & Barto, A. G. (1998). An introduction to reinforcement learning. The MIT Press.
Sutton, R. S. (1988). Learning to predict by the method of temporal differences. Machine Learning, 3(1): 9–44.
Sutton, R. S. (1996). Generalization in reinforcement learning: successful examples using sparse coarse coding. In D. S. Touretzky, M. C. Mozer, & M. E. Hasselmo (Eds.), Advances in neural information processing systems (Vol. 8). Cambridge, MA: The MIT Press.
Szepesvári, C. & Littman, M. L. (1996). Generalized Markov decision processes: dynamic-programming and reinforcement-learning algorithms. Technical Report CS–96–11, Brown University, Providence, RI.
Tesauro, G. J. (1995). Temporal difference learning and TD-gammon. Communications of the ACM, 38(3), 58–68.
Thrun, S. B. (1992). The role of exploration in learning control. In D. A. White & D. A. Sofge (Eds.), Handbook of intelligent control: neural, fuzzy, and adaptive approaches. New York, NY: Van Nostrand Reinhold.
Tsitsiklis, J. N. (1994). Asynchronous stochastic approximation and Q-learning. Machine Learning, 16(3):185–202,September 1994.
Tsitsiklis, J. N. & Van Roy, B. (1996). An analysis of temporal-difference learning with function approximation. Technical Report LIDS-P-2322, Massachusetts Institute of Technology, March. Available through http://web.mit.edu/bvr/www/td.ps. To appear in IEEE Transactions on Automatic Control.
Watkins, C. J. C. H. (1989). Learning from delayed rewards. Ph.D. Thesis, King's College, Cambridge, UK.
Watkins, C. J. C. H. & Dayan, P. (1992). Q-learning. Machine Learning, 8(3):279–292.
Williams, R. J. & Baird, L. C., III (1993). Tight performance bounds on greedy policies based on imperfect value functions. Technical Report NU-CCS–93–14, Northeastern University, College of Computer Science, Boston, MA.
Zhang, W. & Dietterich, T. G. (1995). High-performance job-shop scheduling with a time delay TD(λ) network. Advances in neural information processing systems (Vol. 8, pp. 1024–1030). MIT Press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Singh, S., Jaakkola, T., Littman, M.L. et al. Convergence Results for Single-Step On-Policy Reinforcement-Learning Algorithms. Machine Learning 38, 287–308 (2000). https://doi.org/10.1023/A:1007678930559
Issue Date:
DOI: https://doi.org/10.1023/A:1007678930559