Abstract
We consider a new approach to the continuous-time two-armed bandit problem in which incomes are described by Poisson processes. For this purpose, first, the control horizon is divided into equal consecutive half-intervals in which the strategy remains constant, and the incomes arrive in batches corresponding to these half-intervals. For finding the optimal piecewise constant Bayesian strategy and its corresponding Bayesian risk, a recursive difference equation is derived. The existence of a limiting value of the Bayesian risk when the number of half-intervals grows infinitely is established, and a partial differential equation for finding it is derived. Second, unlike previously considered settings of this problem, we analyze the strategy as a function of the current history of the controlled process rather than of the evolution of the posterior distribution. This removes the requirement of finiteness of the set of admissible parameters, which was imposed in previous settings. Simulation shows that in order to find the Bayesian and minimax strategies and risks in practice, it is sufficient to partition the arriving incomes into 30 batches. In the case of the minimax setting, it is shown that optimal processing of arriving incomes one by one is not more efficient than optimal batch processing if the control horizon grows infinitely.
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References
Berry, D.A. and Fristedt, B., Bandit Problems: Sequential Allocation of Experiments, London, New York: Chapman & Hall, 1985.
Presman, E.L. and Sonin, I.M., Posledovatel’noe upravlenie po nepolnym dannym. Baiesovskii podkhod, Moscow: Nauka, 1982. Translated under the title Sequential Control with Incomplete Information,New York: Academic, 1990.
Sragovich, V.G., Adaptivnoe upravlenie (Adaptive Control), Moscow: Nauka, 1981. Translated under the title Mathematical Theory of Adaptive Control, Singapore: World Sci., 2006.
Nazin, A.V. and Poznyak, A.S., Adaptivnyi vybor variantov: rekurrentnye algoritmy (Adaptive Choice between Alternatives: Recursive Algorithms), Moscow: Nauka, 1986.
Tsetlin, M.L., Issledovaniya po teorii avtomatov i modelirovaniyu biologicheskikh sistem, Moscow: Nauka, 1969. Translated under the title Automaton Theory and Modeling of Biological Systems,New York: Academic, 1973.
Varshavsky, V.I., Kollektivnoe povedenie avtomatov (Collective Behavior of Automata), Moscow: Nauka, 1973. Translated under the title Kollektives Verhalten von Automaten, Warschawski, W.I., Berlin: Akademie, 1978.
Presman, E.L., Poisson Version of the Two-Armed Bandit Problem with Discounting, Teor. Veroyatn. Primen., 1990, vol. 35, no. 2, pp. 318–328 [Theory Probab. Appl. (Engl. Transl.), 1990, vol. 35, no. 2, pp. 307–317]. https://doi.org/10.1137/1135038
Chernoff, H. and Ray, S.N., A Bayes Sequential Sampling Inspection Plan, Ann. Math. Statist., 1965, vol. 36, no. 5, pp. 1387–1407. https://doi.org/10.1214/aoms/1177699898
Mandelbaum, A., Continuous Multi-Armed Bandits and Multiparameter Processes, Ann. Probab., 1987, vol. 15, no. 4, pp. 1527–1556. https://doi.org/10.1214/aop/1176991992
Lai, T.L., Adaptive Treatment Allocation and the Multi-Armed Bandit Problem, Ann. Statist., 1987, vol. 15, no. 3, pp. 1091–1114. https://doi.org/10.1214/aos/1176350495
Vogel, W., An Asymptotic Minimax Theorem for the Two Armed Bandit Problem, Ann. Math. Statist., 1960, vol. 31, pp. 444–451. https://doi.org/10.1214/aoms/1177705907
Borovkov, A.A., Matematicheskaya statistika. Dopolnitel’nye glavy (Mathematical Statistics: Advanced Chapters), Moscow: Nauka, 1984.
Kolnogorov, A.V., Finding Minimax Strategy and Minimax Risk in a Random Environment (The Two-Armed Bandit Problem), Avtomat. i Telemekh., 2011, no. 5, pp. 127–138 [Autom. Remote Control (Engl. Transl.), 2011, vol. 72, no. 5, pp. 1017–1027]. https://doi.org/10.1134/S0005117911050092
Fabius, J., and van Zwet, W.R., Some Remarks on the Two-Armed Bandit, Ann. Math. Statist., 1970, vol. 41, no. 6, pp. 1906–1916. https://doi.org/10.1214/aoms/1177696692
Kolnogorov, A.V., On a Limiting Description of Robust Parallel Control in a Random Environment, Avtomat. i Telemekh., 2015, no. 7, pp. 111–126 [Autom. Remote Control (Engl. Transl.), 2015, vol. 76, no. 7, pp. 1229–1241]. https://doi.org/10.1134/S0005117915070085
Kolnogorov, A.V., Gaussian Two-Armed Bandit: Limiting Description, Probl. Peredachi Inf., 2020, vol. 56, no. 3, pp. 86–111 [Probl. Inf. Transm. (Engl. Transl.), 2020, vol. 56, no. 3, pp. 278–301]. https://doi.org/10.1134/S0032946020030059
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The author is grateful to a reviewer for his/her attention to the paper and valuable remarks.
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Supported in part by the Russian Foundation for Basic Research, project no. 20-01-00062.
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Translated from Problemy Peredachi Informatsii, 2022, Vol. 58, No. 2, pp. 66–91 https://doi.org/10.31857/S0555292322020065.
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Kolnogorov, A. Poissonian Two-Armed Bandit: A New Approach. Probl Inf Transm 58, 160–183 (2022). https://doi.org/10.1134/S0032946022020065
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DOI: https://doi.org/10.1134/S0032946022020065