# American Institute of Mathematical Sciences

doi: 10.3934/jdg.2020033

## A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions

 1 SBAI, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy 2 Dip. di Matematica e Fisica, Università degli Studi Roma Tre, Largo S. L. Murialdo 1, 00146 Roma, Italy 3 IConsulting, Via della Conciliazione 10, 00193 Roma, Italy

* Corresponding author: Fabio Camilli

Received  May 2020 Revised  November 2020 Published  December 2020

Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.

Citation: Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, doi: 10.3934/jdg.2020033
##### References:
 [1] L. Aquilanti, S. Cacace, F. Camilli and R. De Maio, A mean field games approach to cluster analysis, Applied Math. Optim., (2020). doi: 10.1007/s00245-019-09646-2.  Google Scholar [2] R. Bellman, Dynamic Programming, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1957.  Google Scholar [3] J. A. Bilmes, A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov model, CTIT Technical Reports Series, 1998. Google Scholar [4] C. M. Bishop, Pattern Recognition and Machine Learning, Information Science and Statistics, Springer, New York, 2006.  Google Scholar [5] A. Biswas, Mean Field Games with ergodic cost for discrete time Markov processes, preprint, arXiv: 1510.08968. Google Scholar [6] S. Cacace, F. Camilli and A. Goffi, A policy iteration method for Mean Field Games, preprint, arXiv: 2007.04818. Google Scholar [7] R. Carmona and M. Lauriere, Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: I – The ergodic case, preprint, arXiv: 1907.05980. Google Scholar [8] J. L. Coron, Quelques Exemples de Jeux à Champ Moyen, Ph.D. thesis, Université Paris-Dauphine, 2018. Available from: https://tel.archives-ouvertes.fr/tel-01705969/document. Google Scholar [9] W. E, J. Han and Q. Li, A mean-field optimal control formulation of deep learning, Res. Math. Sci., 6 (2019), 41pp. doi: 10.1007/s40687-018-0172-y.  Google Scholar [10] B. S. Everitt, S. Landau, M. Leese and D. Stahl, Cluster Analysis, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2011. doi: 10.1002/9780470977811.  Google Scholar [11] Fashion-MNIST., Available from: https://github.com/zalandoresearch/fashion-mnist. Google Scholar [12] W. H. Fleming, Some Markovian optimization problems, J. Math. Mech., 12 (1963), 131-140.   Google Scholar [13] D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308-328.  doi: 10.1016/j.matpur.2009.10.010.  Google Scholar [14] D. A. Gomes and J. Saúde, Mean field games models–A brief survey, Dyn. Games Appl., 4 (2014), 110-154.  doi: 10.1007/s13235-013-0099-2.  Google Scholar [15] R. A. Howard, Dynamic Programming and Markov Processes, The Technology Press of MIT, Cambridge, Mass.; John Wiley & Sons, Inc., New York-London, 1960. doi: 10.1126/science.132.3428.667.  Google Scholar [16] M. Huang, R. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar [17] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar [18] G. McLachlan and D. Peel, Finite Mixture Models, Wiley Series in Probability and Statistics: Applied Probability and Statistics, Wiley-Interscience, New York, 2000. doi: 10.1002/0471721182.  Google Scholar [19] The MNIST Database of Handwritten Digits., Available from: http://yann.lecun.com/exdb/mnist/. Google Scholar [20] K. Pearson, Contributions to the mathematical theory of evolution, Philosophical Trans. Roy. Soc., 185 (1894), 71-110.  doi: 10.1098/rsta.1894.0003.  Google Scholar [21] S. Pequito, A. Pedro Aguiar, B. Sinopoli and D. A. Gomes, Unsupervised learning of finite mixture models using mean field games, 49$^th$ Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, 2011. doi: 10.1109/Allerton.2011.6120185.  Google Scholar [22] M. L. Puterman, On the convergence of policy iteration for controlled diffusions, J. Optim. Theory Appl., 33 (1981), 137-144.  doi: 10.1007/BF00935182.  Google Scholar [23] M. L. Puterman and S. L. Brumelle, On the convergence of policy iteration in stationary dynamic programming, Math. Oper. Res., 4 (1979), 60-69.  doi: 10.1287/moor.4.1.60.  Google Scholar [24] M. E. Tarter and M. D. Lock, Model-Free Curve Estimation, Monographs on Statistics and Applied Probability, 56, Chapman & Hall, New York, 1993.  Google Scholar [25] D. M. Titterington, A. F. M. Smith and U. E. Makov, Statistical Analysis of Finite Mixture Distributions, Wiley Series Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1985.  Google Scholar [26] M. Wedel and W. A. Kamakura, Market Segmentation: Conceptual and Methodological Foundations, International Series in Quantitative Marketing, 8, Springer, Boston, MA, 2000. doi: 10.1007/978-1-4615-4651-1.  Google Scholar

show all references

##### References:
 [1] L. Aquilanti, S. Cacace, F. Camilli and R. De Maio, A mean field games approach to cluster analysis, Applied Math. Optim., (2020). doi: 10.1007/s00245-019-09646-2.  Google Scholar [2] R. Bellman, Dynamic Programming, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1957.  Google Scholar [3] J. A. Bilmes, A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov model, CTIT Technical Reports Series, 1998. Google Scholar [4] C. M. Bishop, Pattern Recognition and Machine Learning, Information Science and Statistics, Springer, New York, 2006.  Google Scholar [5] A. Biswas, Mean Field Games with ergodic cost for discrete time Markov processes, preprint, arXiv: 1510.08968. Google Scholar [6] S. Cacace, F. Camilli and A. Goffi, A policy iteration method for Mean Field Games, preprint, arXiv: 2007.04818. Google Scholar [7] R. Carmona and M. Lauriere, Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: I – The ergodic case, preprint, arXiv: 1907.05980. Google Scholar [8] J. L. Coron, Quelques Exemples de Jeux à Champ Moyen, Ph.D. thesis, Université Paris-Dauphine, 2018. Available from: https://tel.archives-ouvertes.fr/tel-01705969/document. Google Scholar [9] W. E, J. Han and Q. Li, A mean-field optimal control formulation of deep learning, Res. Math. Sci., 6 (2019), 41pp. doi: 10.1007/s40687-018-0172-y.  Google Scholar [10] B. S. Everitt, S. Landau, M. Leese and D. Stahl, Cluster Analysis, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2011. doi: 10.1002/9780470977811.  Google Scholar [11] Fashion-MNIST., Available from: https://github.com/zalandoresearch/fashion-mnist. Google Scholar [12] W. H. Fleming, Some Markovian optimization problems, J. Math. Mech., 12 (1963), 131-140.   Google Scholar [13] D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308-328.  doi: 10.1016/j.matpur.2009.10.010.  Google Scholar [14] D. A. Gomes and J. Saúde, Mean field games models–A brief survey, Dyn. Games Appl., 4 (2014), 110-154.  doi: 10.1007/s13235-013-0099-2.  Google Scholar [15] R. A. Howard, Dynamic Programming and Markov Processes, The Technology Press of MIT, Cambridge, Mass.; John Wiley & Sons, Inc., New York-London, 1960. doi: 10.1126/science.132.3428.667.  Google Scholar [16] M. Huang, R. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar [17] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar [18] G. McLachlan and D. Peel, Finite Mixture Models, Wiley Series in Probability and Statistics: Applied Probability and Statistics, Wiley-Interscience, New York, 2000. doi: 10.1002/0471721182.  Google Scholar [19] The MNIST Database of Handwritten Digits., Available from: http://yann.lecun.com/exdb/mnist/. Google Scholar [20] K. Pearson, Contributions to the mathematical theory of evolution, Philosophical Trans. Roy. Soc., 185 (1894), 71-110.  doi: 10.1098/rsta.1894.0003.  Google Scholar [21] S. Pequito, A. Pedro Aguiar, B. Sinopoli and D. A. Gomes, Unsupervised learning of finite mixture models using mean field games, 49$^th$ Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, 2011. doi: 10.1109/Allerton.2011.6120185.  Google Scholar [22] M. L. Puterman, On the convergence of policy iteration for controlled diffusions, J. Optim. Theory Appl., 33 (1981), 137-144.  doi: 10.1007/BF00935182.  Google Scholar [23] M. L. Puterman and S. L. Brumelle, On the convergence of policy iteration in stationary dynamic programming, Math. Oper. Res., 4 (1979), 60-69.  doi: 10.1287/moor.4.1.60.  Google Scholar [24] M. E. Tarter and M. D. Lock, Model-Free Curve Estimation, Monographs on Statistics and Applied Probability, 56, Chapman & Hall, New York, 1993.  Google Scholar [25] D. M. Titterington, A. F. M. Smith and U. E. Makov, Statistical Analysis of Finite Mixture Distributions, Wiley Series Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1985.  Google Scholar [26] M. Wedel and W. A. Kamakura, Market Segmentation: Conceptual and Methodological Foundations, International Series in Quantitative Marketing, 8, Springer, Boston, MA, 2000. doi: 10.1007/978-1-4615-4651-1.  Google Scholar
Samples of hand-written digits from the MNIST database
Different samples of hand-written digits from the MNIST database
Clusterization histogram for digits $\mathbf{1},\mathbf{3}$ and the corresponding Bernoulli parameters
Clusterization histogram for digits $\mathbf{3},\mathbf{5}$ and the corresponding Bernoulli parameters
Clusterization histogram for even digits and the corresponding Bernoulli parameters
Samples of fashion products from the Fashion-MNIST database
Averaged categorical distributions for the Fashion-MNIST database
Clusterization histogram for types T-shirt, Trouser and the corresponding categorical parameters
Clusterization histogram for types Dress, Sneaker, Bag, Boot and the corresponding categorical parameters
 [1] Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111 [2] Kalikinkar Mandal, Guang Gong. On ideal $t$-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020125 [3] Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 [4] Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025 [5] Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 [6] Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167 [7] Aihua Fan, Jörg Schmeling, Weixiao Shen. $L^\infty$-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363 [8] Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005 [9] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 [10] Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 [11] Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 [12] Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389 [13] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [14] Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010 [15] Urszula Ledzewicz, Heinz Schättler. On the role of pharmacometrics in mathematical models for cancer treatments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 483-499. doi: 10.3934/dcdsb.2020213 [16] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 [17] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [18] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [19] Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158 [20] Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

Impact Factor:

## Tools

Article outline

Figures and Tables