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 "excerpt": "A learning automaton is a stochastic decision-making device that updates action probabilities from a reinforcement signal, a foundation of reinforcement learning introduced by Tsetlin in 1961.",
 "snippet": "A learning automaton is a stochastic decision-making device that updates action probabilities from a reinforcement signal, a foundation of reinforcement learning introduced by Tsetlin in 1961.",
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 "markdown": "# Learning automata\n\nA learning automaton is a stochastic decision-making device that chooses an action from a probability distribution over a fixed action set and updates that distribution using only a reinforcement signal from an unknown random environment, with the goal of identifying the action most likely to be rewarded. The approach is arguably a foundation of reinforcement learning and remains in use for adaptive control, routing, and stochastic optimization.\n\n| Key fact | Detail |\n|---|---|\n| What is updated | The action-probability vector \\( p(n) \\), via an operator of the form \\( p(n+1) = T[p(n), \\alpha(n), \\beta(n)] \\), where \\( \\alpha(n) \\) is the chosen action and \\( \\beta(n) \\) the environment response <sup>[1](https://www.ias.ac.in/article/fulltext/sadh/015/04-05/0263-0281)</sup> |\n| Environment models | P-model (binary response), Q-model (finite set of values in [0, 1]), S-model (continuous response in [0, 1]) <sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> |\n| Convergence target | The action with maximum reward probability \\( d_i \\); ε-optimal schemes make the probability of any suboptimal action arbitrarily small <sup>[3](https://eprints.iisc.ac.in/5011/1/varieties.pdf)</sup> |\n| Canonical schemes | LR-I, LR-P, LR-εP, Multiple Response schemes, estimator (pursuit) algorithms <sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> |\n| Speed of pursuit | Typically 10 to 50 times faster than LR-I, at the cost of extra computation and memory <sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> |\n| Origin | Tsetlin's 1961 deterministic automata in random media; variable-structure stochastic automata from 1963 <sup>[4](https://pure.iiasa.ac.at/id/file/231720)</sup> |\n| Modern uses | Telephone and network routing, cloud and edge task assignment, offloading, and load balancing <sup>[5](https://link.springer.com/article/10.1007/s11227-024-06292-6)</sup> |\n\n## How it works\n\nThe automaton and environment form a feedback loop. At each instant \\( n \\) the automaton samples an action \\( \\alpha(n) \\) from \\( p(n) \\); the environment responds with \\( \\beta(n) \\); and the update operator \\( T \\) revises the probability vector accordingly.<sup>[1](https://www.ias.ac.in/article/fulltext/sadh/015/04-05/0263-0281)</sup> The environment is characterized by reward probabilities \\( d_i = P\\{\\beta(n)=1 \\mid \\alpha(n)=\\alpha_i\\} \\), with penalty probabilities \\( c_i = 1 - d_i \\); the optimal action is the one with the largest \\( d_i \\).<sup>[1](https://www.ias.ac.in/article/fulltext/sadh/015/04-05/0263-0281)</sup> In S-models, where responses are continuous, the reward strength \\( s_i = E[\\beta(n) \\mid \\alpha(n)=\\alpha_i] \\) plays the role of the reward probability.<sup>[1](https://www.ias.ac.in/article/fulltext/sadh/015/04-05/0263-0281)</sup>\n\nPerformance norms are graded. A scheme is expedient if the expected penalty stays bounded; it is optimal if it converges to the best action; and it is ε-optimal if, given any \\( \\epsilon > 0 \\), parameters can be chosen so that \\( \\liminf E[G(k)] > d_m - \\epsilon \\), where \\( G(k) = \\sum_i d_i p_i(k) \\) is the average reward.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> Two convergence behaviors are distinguished: ergodic algorithms, such as LR-P and LR-εP, converge in distribution and lose their initial \\( p(0) \\), while absorbing algorithms, such as LR-I, converge with probability one to absorbing states.<sup>[1](https://www.ias.ac.in/article/fulltext/sadh/015/04-05/0263-0281)</sup>\n\n## How it is done\n\nA practitioner defines the action set \\( \\alpha \\), the input set \\( \\beta \\), the initial probability vector \\( p \\), and the update operator \\( T \\); this quadruple defines a variable-structure automaton.<sup>[5](https://link.springer.com/article/10.1007/s11227-024-06292-6)</sup> The most common update is the linear reward-inaction scheme, LR-I:\n\n\\[ p(k+1) = p(k) + \\lambda \\left( e_i - p(k) \\right) \\beta(k), \\]\n\nwhere \\( 0 < \\lambda < 1 \\) is the stepsize, \\( e_i \\) is the unit probability vector on the chosen action, and probabilities are left unchanged when the response indicates penalty.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> LR-I is obtained by setting \\( b = 0 \\) in the more general LR-P scheme and is called a \"benevolent\" scheme because no updating occurs under penalty.<sup>[1](https://www.ias.ac.in/article/fulltext/sadh/015/04-05/0263-0281)</sup> In LR-P, if the reward parameter \\( a \\) exceeds the penalty parameter \\( b \\) and one action has zero penalty probability, the automaton reaches the pure optimal strategy.<sup>[6](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/3c1b5d24-e34c-4365-9be8-f95ea062fdee/content)</sup> Termination in applications is typically declared when an action probability crosses a threshold (95% in one cloud-scheduling deployment), when an objective such as makespan stops changing, or when an iteration limit is reached.<sup>[5](https://link.springer.com/article/10.1007/s11227-024-06292-6)</sup>\n\n## Origin\n\nTsetlin introduced learning automata in 1961 in Avtomatika i Telemekhanika, studying deterministic automata operating in an unknown random environment and showing they are asymptotically optimal under some conditions.<sup>[4](https://pure.iiasa.ac.at/id/file/231720)</sup> Variable-structure stochastic automata update action probabilities directly and need fewer states.<sup>[4](https://pure.iiasa.ac.at/id/file/231720)</sup> Precursors include Bush and Mosteller's 1958 mathematical-psychology studies and Tsypkin's 1960s reduction of learning to stochastic hill climbing over parameters; the two-armed bandit problem in statistics is a close relative.<sup>[1](https://www.ias.ac.in/article/fulltext/sadh/015/04-05/0263-0281)</sup><sup> • </sup><sup>[7](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/2b645ea4-89e5-4145-bf92-0b05195578a9/content)</sup> The LR-I scheme, described by Bush and Mosteller, was independently rediscovered and introduced with proper emphasis by Shapiro and Narendra in 1969.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup><sup> • </sup><sup>[8](https://doi.org/10.1109/tssc.1969.300228)</sup> The term \"learning automata\" was publicized in a survey <sup>[9](https://doi.org/10.1109/tsmc.1974.5408453)</sup><sup> • </sup><sup>[3](https://eprints.iisc.ac.in/5011/1/varieties.pdf)</sup>, and their 1989 book surveys the field through the end of that decade.<sup>[7](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/2b645ea4-89e5-4145-bf92-0b05195578a9/content)</sup>\n\n## Variants\n\nThe finite action-set learning automaton (FALA) is the original form. Estimator algorithms accelerate it: the pursuit algorithm of Thathachar and Sastry maintains reward estimates \\( \\hat{d}_i(k) = B_i(k)/Z_i(k) \\) and moves probability toward the estimated best action, \\( p(k+1) = p(k) + \\lambda (e_H - p(k)) \\), converging 10 to 50 times faster than LR-I while working for any bounded reinforcement set.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup><sup> • </sup><sup>[10](https://doi.org/10.1109/tsmc.1985.6313407)</sup> The Generalized Pursuit Algorithm pursues all actions whose estimates exceed the chosen action's, converging faster still.<sup>[11](https://people.scs.carleton.ca/~oommen/papers/GenPurCn.PDF)</sup> Discretizing the probability space was introduced to raise the slow convergence rate of variable-structure automata, producing variants such as DPRI, CPRI, CPRP, and DPRP.<sup>[11](https://people.scs.carleton.ca/~oommen/papers/GenPurCn.PDF)</sup> Multiple Response automata use distinct reward and penalty rates for different responses, with LR-P, LR-I, and LR-εP as special cases.<sup>[12](https://conta.uom.gr/conta/publications/PDF/Multiple%20Response%20Learning%20Automata.pdf)</sup> Parameterized (PLA), generalized (GLA), and continuous action-set (CALA) automata extend the framework: CALA uses a normal action distribution \\( N(\\mu(k), \\sigma(k)) \\) with a floor on \\( \\sigma(k) \\) so the algorithm does not get stuck at a nonoptimal point.<sup>[3](https://eprints.iisc.ac.in/5011/1/varieties.pdf)</sup><sup> • </sup><sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> Teams of automata using LR-I converge to a [Nash equilibrium](https://www.edgepedia.org/nash-equilibrium), and pursuit-based teams reach globally optimal action sets even for non-unimodal games.<sup>[3](https://eprints.iisc.ac.in/5011/1/varieties.pdf)</sup><sup> • </sup><sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> Larger structures include hierarchical systems, distributed learning automata <sup>[13](https://doi.org/10.1142/s0218488506004217)</sup>, and cellular learning automata, which place an automaton in each cell of a cellular space.<sup>[14](https://doi.org/10.1142/s0219525904000202)</sup> The Tsetlin Machine, a rule-based learner built from two-action Tsetlin automata with stochastic Reward, Penalty, and Inaction feedback, carries the framework into pattern recognition.<sup>[15](https://arxiv.org/html/2607.20124v1)</sup>\n\n## Applications\n\nNarendra, Wright, and Mason applied learning automata to telephone traffic routing in 1977, showing by comparative simulation significantly lower blocking probability than fixed-rule alternate routing under selected overload conditions when spare capacity existed elsewhere in the network.<sup>[16](https://doi.org/10.1109/tsmc.1977.4309623)</sup> In connection-oriented data networks, automata at source nodes route virtual calls dynamically, and a \"virtual link length\" combining packet and virtual-call counts outperformed minimum-packet-delay and shortest-queue link lengths in simulation.<sup>[17](https://www.conta.uom.gr/conta/publications/PDF/LEARNING%20AUTOMATA%20ROUTING%20IN%20CONNECTION-ORIENTED%20NETWORKS.pdf)</sup> Recent work applies automata to cloud and edge computing: LATA maps DAG tasks to heterogeneous virtual machines using a game of variable-structure automata in an S-model environment, optimizing makespan, load, and energy <sup>[5](https://link.springer.com/article/10.1007/s11227-024-06292-6)</sup>; a mobile edge offloading scheme combines a two-action automaton (OffCloud/OffEdge, initialized at 0.5 each) with LSTM prediction and A3C-based fuzzy auto-scaling, improving CPU consumption, execution time, and energy over LAF and LAQ baselines <sup>[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC10772128/)</sup>; ALBLA distributes real-time tasks between edge and cloud servers using a Service Time Measurement metric <sup>[19](https://dl.acm.org/doi/10.1007/s00607-024-01380-0)</sup>; and ICLA-ARO hybridizes irregular cellular learning automata with a metaheuristic for fog task scheduling.<sup>[20](https://link.springer.com/article/10.1007/s10586-026-06038-4)</sup>\n\n## Limitations and alternatives\n\nLR-I is ε-optimal in all stationary random environments and absolutely expedient, meaning the average penalty is a supermartingale <sup>[21](https://tarlton.info/09-Citations/thathacharLearningAutomataChanging1987_Amedia/Thathachar_Harita_1987_Learning-automata-with-changing-number-of-actions.pdf)</sup>; a stepsize \\( \\lambda^* \\) exists such that for all \\( \\lambda \\le \\lambda^* \\) the optimal-action probability eventually exceeds \\( 1 - \\epsilon \\) with large probability.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> But smaller stepsizes buy accuracy at the price of slower convergence, and LR-I can converge rather slowly.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> In decentralized teams of many automata, LR-I finds only local minima.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)</sup> Most classical analysis assumes a stationary, single-teacher environment; extensions to nonstationary multi-teacher S-model settings, such as the MGAE scheme, are proven ε-optimal only under stated conditions.<sup>[4](https://pure.iiasa.ac.at/id/file/231720)</sup> Fixed action sets are another classical restriction, addressed by automata with a changing number of actions using a modified LR-I <sup>[21](https://tarlton.info/09-Citations/thathacharLearningAutomataChanging1987_Amedia/Thathachar_Harita_1987_Learning-automata-with-changing-number-of-actions.pdf)</sup> and by CALA for large or continuous action spaces.<sup>[22](https://doi.org/10.1016/0016-0032%2894%2990039-6)</sup> On the theory side, practical pursuit implementations fix the tuning parameter, while ε-optimality in earlier proofs required a vanishing sequence of tuning parameters; formal ε-optimality proofs for discretized and absorbing continuous pursuit algorithms appeared only in 2014 and 2015.<sup>[23](https://www.cambridge.org/core/journals/journal-of-applied-probability/article/on-optimality-of-the-pursuit-learning-algorithm/ADB5180F9A789385D839D8DBF920B819)</sup><sup> • </sup><sup>[24](https://doi.org/10.1007/s10489-015-0670-1)</sup><sup> • </sup><sup>[25](https://doi.org/10.1007/s10489-014-0541-1)</sup> Published sources describe learning automata as arguably the foundation of reinforcement learning <sup>[26](https://ieeexplore.ieee.org/document/9999274)</sup>, and recent structural work continues along other lines, such as the ε-optimal hierarchical discretized pursuit automaton of 2024.<sup>[26](https://ieeexplore.ieee.org/document/9999274)</sup>\n\n## References\n\n1. [Stochastic automata and learning systems (Thathachar, Sādhanā 1990)](https://www.ias.ac.in/article/fulltext/sadh/015/04-05/0263-0281)\n2. [Learning automata algorithms for pattern classification (Sastry & Thathachar, Sādhanā 1999)](https://www.ias.ac.in/article/fulltext/sadh/024/04-05/0261-0292)\n3. [Varieties of learning automata: an overview (Thathachar & Sastry, IEEE Trans. SMC-B, 2002)](https://eprints.iisc.ac.in/5011/1/varieties.pdf)\n4. [Learning behaviors of stochastic automata under nonstationary multi-teacher environments (Baba et al., IIASA)](https://pure.iiasa.ac.at/id/file/231720)\n5. [LATA: learning automata-based task assignment on heterogeneous cloud computing platform (Journal of Supercomputing, 2024)](https://link.springer.com/article/10.1007/s11227-024-06292-6)\n6. [Intelligent Navigation of Autonomous Vehicles in an Automated Highway System (Ünsal, Virginia Tech thesis, 1998)](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/3c1b5d24-e34c-4365-9be8-f95ea062fdee/content)\n7. [Stochastic Learning Automata (Virginia Tech thesis)](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/2b645ea4-89e5-4145-bf92-0b05195578a9/content)\n8. [I. Shapiro, Kumpati Narendra (1969). Use of Stochastic Automata for Parameter Self-Optimization with Multimodal Performance Criteria. IEEE Transactions on Systems Science and Cybernetics.](https://doi.org/10.1109/tssc.1969.300228)\n9. [Kumpati S. Narendra, M. A. L. Thathachar (1974). Learning Automata - A Survey. IEEE Transactions on Systems Man and Cybernetics.](https://doi.org/10.1109/tsmc.1974.5408453)\n10. [M. A. L. Thathachar, P. S. Sastry (1985). A new approach to the design of reinforcement schemes for learning automata. IEEE Transactions on Systems Man and Cybernetics.](https://doi.org/10.1109/tsmc.1985.6313407)\n11. [Continuous and Discretized Generalized Pursuit Learning Schemes (Oommen & Agache)](https://people.scs.carleton.ca/~oommen/papers/GenPurCn.PDF)\n12. [Multiple Response Learning Automata (IEEE Trans. SMC correspondence)](https://conta.uom.gr/conta/publications/PDF/Multiple%20Response%20Learning%20Automata.pdf)\n13. [HAMID BEIGY, M. R. MEYBODI (2006). UTILIZING DISTRIBUTED LEARNING AUTOMATA TO SOLVE STOCHASTIC SHORTEST PATH PROBLEMS. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems.](https://doi.org/10.1142/s0218488506004217)\n14. [HAMID BEIGY, M. R. MEYBODI (2004). A MATHEMATICAL FRAMEWORK FOR CELLULAR LEARNING AUTOMATA. Advances in Complex Systems.](https://doi.org/10.1142/s0219525904000202)\n15. [Autonomous Collaborative Learning Among an Ensemble of Tsetlin Machines with Consensus-Based Inference (preprint)](https://arxiv.org/html/2607.20124v1)\n16. [Application of Learning Automata to Telephone Traffic Routing and Control (Narendra, Wright, Mason, IEEE TSMC 1977)](https://doi.org/10.1109/tsmc.1977.4309623)\n17. [Learning automata routing in connection-oriented networks (Papadimitriou et al.)](https://www.conta.uom.gr/conta/publications/PDF/LEARNING%20AUTOMATA%20ROUTING%20IN%20CONNECTION-ORIENTED%20NETWORKS.pdf)\n18. [A decision-making mechanism for task offloading using learning automata and deep learning in mobile edge networks](https://pmc.ncbi.nlm.nih.gov/articles/PMC10772128/)\n19. [ALBLA: an adaptive load balancing approach in edge-cloud networks utilizing learning automata (Computing, 2024)](https://dl.acm.org/doi/10.1007/s00607-024-01380-0)\n20. [A novel hybrid task scheduling method in fog computing using reinforcement learning and metaheuristic algorithms (Cluster Computing)](https://link.springer.com/article/10.1007/s10586-026-06038-4)\n21. [Learning automata with changing number of actions (Thathachar & Harita, 1987)](https://tarlton.info/09-Citations/thathacharLearningAutomataChanging1987_Amedia/Thathachar_Harita_1987_Learning-automata-with-changing-number-of-actions.pdf)\n22. [Continuous action set learning automata for stochastic optimization (Journal of the Franklin Institute, 1994)](https://doi.org/10.1016/0016-0032%2894%2990039-6)\n23. [On ε-Optimality of the Pursuit Learning Algorithm (Journal of Applied Probability, Cambridge Core)](https://www.cambridge.org/core/journals/journal-of-applied-probability/article/on-optimality-of-the-pursuit-learning-algorithm/ADB5180F9A789385D839D8DBF920B819)\n24. [Xuan Zhang and colleagues (2015). A formal proof of the 𝜖-optimality of discretized pursuit algorithms. Applied Intelligence.](https://doi.org/10.1007/s10489-015-0670-1)\n25. [Xuan Zhang and colleagues (2014). A formal proof of the ε-optimality of absorbing continuous pursuit algorithms using the theory of regular functions. Applied Intelligence.](https://doi.org/10.1007/s10489-014-0541-1)\n26. [The Hierarchical Discrete Pursuit Learning Automaton (Omslandseter, Jiao, Zhang, Yazidi, Oommen, IEEE TNNLS 2024)](https://ieeexplore.ieee.org/document/9999274)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Machine learning and neural computation › Machine learning methods*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026; Oct 21, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n",
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