Abstract
We introduce the concepts of a learning algorithm, an objective function, a recognition system, a class of patterns, a training set, a reward algorithm, a finitely convergent algorithm, an adaptive control system, a control goal, control tactics, adaptation time, etc., related to the problem of artificial intelligence in the processes of learning and adaptation. The general problem of self-learning (unsupervised learning)—about the separation of sets—in terms of the classical calculus of variations is posed. The generality of the problem is due to the introduction of an additional time variable into the analysis. The problem is solved by determining extremal conditions under which the minimization of the overall average risk functional is achieved. Problems corresponding to nonfixed and fixed time intervals are considered. For these two cases, expressions are found for calculating variations in cost functionals. Necessary conditions are indicated for determining the extremal values of the self-learning process (separation of classes of a set of patterns) in time.
REFERENCES
Fomin, V.N., Matematicheskaya teoriya obuchaemykh opoznayushchikh sistem (Mathematical Theory of Learning Recognition Systems), Leningrad: Izd. LGU, 1976.
Tsypkin, Ya.Z. and Kel’mans, G.K., On recurrent self-learning algorithms, Izv. Akad. Nauk SSSR. Tekh. Kibern., 1967, no. 5, pp. 78–87.
Yakubovich, V.A., On one problem of training expedient behavior, Autom. Remote Control, 1969, vol. 30, no. 8, pp. 1292–1310.
Yakubovich, V.A., Method of recurrent objective inequalities in the theory of adaptive systems, in Voprosy kibernetiki. Adaptivnye sistemy. Nauch. sovet po kompleksnoi probleme “Kibernetika” AN SSSR (Issues of Cybernetics. Adaptive Systems. Scientific Council on the Complex Problem “Cybernetics” of the Academy of Sciences of the USSR), Moscow, 1976, pp. 32–64.
Fomin, V.N., Fradkov, A.L., and Yakubovich, V.A., Adaptivnoe upravlenie dinamicheskimi ob”ektami (Adaptive Control of Dynamic Objects), Moscow: Nauka, 1981.
Tertychny-Dauri, V.Yu., Adaptive Mechanic, Dordrecht–Boston–London: Springer, 2002.
Gel’fand, I.M. and Fomin, S.V., Variatsionnoe ischislenie (Calculus of Variations), Moscow: Gos. Izd. Fiz.-Mat. Lit., 1961.
El’sgol’ts, L.E., Differentsial’nye uravneniya i variatsionnoe ischislenie (Differential Equations and Calculus of Variations), Moscow: Nauka, 1969.
Tertychny-Dauri, V.Yu., Solution of variational dynamic problems under parametric uncertainty, Probl. Inf. Transm., 2005, vol. 41, no. 1, pp. 45–58.
Tertychny-Dauri, V.Yu., Optimal stabilization in problems of adaptive nuclear kinetics, Differ. Equations, 2006, vol. 42, no. 3, pp. 400–411.
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Translated by V. Potapchouck
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Babushkin, M.V., Tertychny-Dauri, V.Y. Variational Methods for Solving Problems Associated with Artificial Intelligence. Diff Equat 59, 919–932 (2023). https://doi.org/10.1134/S0012266123070066
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DOI: https://doi.org/10.1134/S0012266123070066