Fixed Point Theory in Generalized Metric Vector Spaces and their applications in Machine Learning and Optimization Algorithms
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Abstract
This study introduces a novel formulation of fixed-point theory within Generalized metric spaces, with an emphasis on applications in machine learning optimization and high-dimensional data analysis. Recall on the concept of complete G-metric spaces, we define a generalized contraction condition tailored for operators representing iterative updates in machine learning algorithms. The proposed framework is exemplified through gradient descent with regularization, demonstrating convergence within a non-Euclidean, high-dimensional setting. Results reveal that our approach not only strengthens convergence properties in iterative algorithms but also complements modern regularization techniques, supporting sparsity and robustness in high-dimensional spaces. These findings underscore the relevance of G-metric spaces and auxiliary functions within fixed-point theory, highlighting their potential to advance adaptive optimization methods. Future work will explore further applications across machine learning paradigms, addressing challenges such as sparse data representation and scalability in complex data environments.
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How to Cite
Vinsensia, D., & Utami, Y. (2024). Fixed Point Theory in Generalized Metric Vector Spaces and their applications in Machine Learning and Optimization Algorithms. International Journal of Basic and Applied Science, 13(2), 112–122. https://doi.org/10.35335/ijobas.v13i2.504
References
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A. G. Aksoy and M. A. Khamsi, Nonstandard methods in fixed point theory. Springer Science & Business Media, 2012.
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S. Sachdeva, “Fixed point and approximate fixed point theorems with applications,” College of Engineering, University of Petroleum & Energy Studies, 2014.
A. Mishra and H. Raja, “Study Of Fixed-Point Results In G-Metric Space And G-Cone Metric Space and Its Application,” in 2024 OPJU International Technology Conference (OTCON) on Smart Computing for Innovation and Advancement in Industry 4.0, IEEE, 2024, pp. 1–9. doi: https://doi.org/10.1109/OTCON60325.2024.10687678.
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P. M. Bajracharya and N. Adhikari, “A study on fixed point theory in M-metric space,” Sci. World, vol. 13, no. 13, pp. 62–68, 2020.
N. Abbas, “Fixed point theorems for some contractions in Rectangular b-metric space,” A thesis submitted in partial fulfillment for the degree of Master of …, 2017.
J. A. Jiddah, H. Işık, M. Alansari, and M. S. Shagari, “A Study of Fixed Point Results in G-Metric Space via New Contractions with Applications,” in Recent Developments in Fixed-Point Theory: Theoretical Foundations and Real-World Applications, Springer, 2024, pp. 241–263. doi: https://doi.org/10.1007/978-981-99-9546-2_10.
C. Jin, P. Netrapalli, R. Ge, S. M. Kakade, and M. I. Jordan, “On nonconvex optimization for machine learning: Gradients, stochasticity, and saddle points,” J. ACM, vol. 68, no. 2, pp. 1–29, 2021, doi: https://doi.org/10.1145/3418526.
Y.-P. Hsieh, P. Mertikopoulos, and V. Cevher, “The limits of min-max optimization algorithms: Convergence to spurious non-critical sets,” in International Conference on Machine Learning, PMLR, 2021, pp. 4337–4348. doi: https://proceedings.mlr.press/v139/hsieh21a.html.
R. Sun, “Optimization for deep learning: theory and algorithms,” arXiv Prepr. arXiv1912.08957, vol. 1912, no. 08957, p. 1912.08957, 2019.
R. Carmona and M. Laurière, “Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games I: The ergodic case,” SIAM J. Numer. Anal., vol. 59, no. 3, pp. 1455–1485, 2021, doi: https://doi.org/10.1137/19M1274377.
C. T. Kelley, Iterative methods for optimization. SIAM, 1999. [Online]. Available: https://epubs.siam.org/doi/pdf/10.1137/1.9781611970920.bm
M. Danilova et al., “Recent theoretical advances in non-convex optimization,” in High-Dimensional Optimization and Probability: With a View Towards Data Science, Springer, 2022, pp. 79–163. doi: https://doi.org/10.1007/978-3-031-00832-0_3.
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H. Snyder, “Literature review as a research methodology: An overview and guidelines,” J. Bus. Res., vol. 104, no. 11, pp. 333–339, 2019, doi: https://doi.org/10.1016/j.jbusres.2019.07.039.
J. W. Knopf, “Doing a literature review,” PS Polit. Sci. Polit., vol. 39, no. 1, pp. 127–132, 2006, doi: https://doi.org/10.1017/S1049096506060264.
M. Iqbal, A. Batool, A. Hussain, and H. Alsulami, “Fuzzy Fixed Point Theorems in S-Metric Spaces: Applications to Navigation and Control Systems,” Axioms, vol. 13, no. 9, p. 650, 2024, doi: https://doi.org/10.3390/axioms13090650.
B. Samet, C. Vetro, and F. Vetro, “Remarks on G‐Metric Spaces,” Int. J. Anal., vol. 2013, no. 1, p. 917158, 2013, doi: https://doi.org/10.1155/2013/917158.
Z. Mustafa and H. Obiedat, “A fixed point theorem of Reich in G-metric spaces,” Cubo, vol. 12, no. 1, pp. 83–93, 2010, doi: http://dx.doi.org/10.4067/S0719-06462010000100008.
T. Van An, N. Van Dung, and V. T. Le Hang, “A new approach to fixed point theorems on G-metric spaces,” Topol. Appl., vol. 160, no. 12, pp. 1486–1493, 2013, doi: https://doi.org/10.1016/j.topol.2013.05.027.
Y. U. Gaba, “Fixed point theorems in G-metric spaces,” J. Math. Anal. Appl., vol. 455, no. 1, pp. 528–537, 2017, doi: https://doi.org/10.1016/j.jmaa.2017.05.062.
S. Kasahara, “On some generalizations of the Banach contraction theorem,” Publ. Res. Inst. Math. Sci., vol. 12, no. 2, pp. 427–437, 1976, doi: https://doi.org/10.2977/prims/1195190723.
P. R. Meyers, “A converse to Banach’s contraction theorem,” J. Res. Nat. Bur. Stand. Sect. B, vol. 71, no. 2, pp. 73–76, 1967.
J. Shawe-Taylor and S. Sun, “A review of optimization methodologies in support vector machines,” Neurocomputing, vol. 74, no. 17, pp. 3609–3618, 2011, doi: https://doi.org/10.1016/j.neucom.2011.06.026.
F. Gieseke, A. Airola, T. Pahikkala, and O. Kramer, “Fast and simple gradient-based optimization for semi-supervised support vector machines,” Neurocomputing, vol. 123, no. 10, pp. 23–32, 2014, doi: https://doi.org/10.1016/j.neucom.2012.12.056.
W. A. Kirk, “Geodesic geometry and fixed point theory,” in Seminar of mathematical analysis (Malaga/Seville, 2002/2003), 2003, pp. 195–225.
R. Espınola and W. A. Kirk, “Set-valued contractions and fixed points,” Nonlinear Anal. Theory, Methods Appl., vol. 54, no. 3, pp. 485–494, 2003, doi: https://doi.org/10.1016/S0362-546X(03)00107-X.
T. Bian and Z.-P. Jiang, “Reinforcement learning and adaptive optimal control for continuous-time nonlinear systems: A value iteration approach,” IEEE Trans. neural networks Learn. Syst., vol. 33, no. 7, pp. 2781–2790, 2021, doi: https://doi.org/10.1109/TNNLS.2020.3045087.
D. Bertsekas, “Multiagent value iteration algorithms in dynamic programming and reinforcement learning,” Results Control Optim., vol. 1, no. 12, p. 100003, 2020, doi: https://doi.org/10.1016/j.rico.2020.100003.
H. Bonnel, A. N. Iusem, and B. F. Svaiter, “Proximal methods in vector optimization,” SIAM J. Optim., vol. 15, no. 4, pp. 953–970, 2005, doi: https://doi.org/10.1137/S1052623403429093.
R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM J. Control Optim., vol. 14, no. 5, pp. 877–898, 1976, doi: https://doi.org/10.1137/0314056.
G. C. Bento, O. P. Ferreira, and J. G. Melo, “Iteration-complexity of gradient, subgradient and proximal point methods on Riemannian manifolds,” J. Optim. Theory Appl., vol. 173, no. 2, pp. 548–562, 2017, doi: https://doi.org/10.1007/s10957-017-1093-4.
J. N. Tsitsiklis, “Problems in decentralized decision making and computation,” 1984, Massachusetts Institute of Technology.
F. G. Mohammadi, M. H. Amini, and H. R. Arabnia, “Evolutionary computation, optimization, and learning algorithms for data science,” Optim. Learn. Control Interdependent Complex Networks, vol. 1123, no. 3, pp. 37–65, 2020, doi: https://doi.org/10.1007/978-3-030-34094-0_3.
I. H. Sarker, “Machine learning: Algorithms, real-world applications and research directions,” SN Comput. Sci., vol. 2, no. 3, p. 160, 2021, doi: https://doi.org/10.1007/s42979-021-00592-x.
A. Cegielski, Iterative methods for fixed point problems in Hilbert spaces, vol. 2057. Springer, 2012.
V. I. Istratescu, Fixed point theory: an introduction, vol. 7. Springer Science & Business Media, 2001.
M. A. Khamsi and W. M. Kozlowski, Fixed point theory in modular function spaces. Birkhäuser, 2015.
W. Blum and M. Niss, “Applied mathematical problem solving, modelling, applications, and links to other subjects—State, trends and issues in mathematics instruction,” Educ. Stud. Math., vol. 22, no. 1, pp. 37–68, 1991, doi: https://doi.org/10.1007/BF00302716.
K. Zikan, The theory and applications of algebraic metric spaces. Stanford University, 1990.
A. G. Aksoy and M. A. Khamsi, Nonstandard methods in fixed point theory. Springer Science & Business Media, 2012.
V. Pata, Fixed point theorems and applications, vol. 116. Springer, 2019. doi: https://doi.org/10.1007/978-3-030-19670-7.
W. C. Rheinboldt, “A unified convergence theory for a class of iterative processes,” SIAM J. Numer. Anal., vol. 5, no. 1, pp. 42–63, 1968, doi: https://doi.org/10.1137/0705003.
G. R. Hjaltason and H. Samet, “Properties of embedding methods for similarity searching in metric spaces,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 25, no. 5, pp. 530–549, 2003, doi: https://doi.org/10.1109/TPAMI.2003.1195989.
B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific J. Math, vol. 10, no. 1, pp. 313–334, 1960, [Online]. Available: https://msp.org/pjm/1960/10-1/pjm-v10-n1-s.pdf#page=315
K.-T. Sturm, “On the geometry of metric measure spaces,” vol. 196, no. 1, pp. 65–131, 2006, doi: 10.1007/s11511-006-0002-8.
Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” J. Nonlinear Convex Anal., vol. 7, no. 2, p. 289, 2006.
N. Freymuth et al., “Iterative Sizing Field Prediction for Adaptive Mesh Generation From Expert Demonstrations,” arXiv Prepr. arXiv2406.14161, vol. 2, no. 3, p. 2406.14161 Focus to learn more, 2024, doi: https://doi.org/10.48550/arXiv.2406.14161.
M. Alfuraidan and Q. Ansari, Fixed point theory and graph theory: Foundations and integrative approaches. Academic Press, 2016.
S. Sachdeva, “Fixed point and approximate fixed point theorems with applications,” College of Engineering, University of Petroleum & Energy Studies, 2014.
A. Mishra and H. Raja, “Study Of Fixed-Point Results In G-Metric Space And G-Cone Metric Space and Its Application,” in 2024 OPJU International Technology Conference (OTCON) on Smart Computing for Innovation and Advancement in Industry 4.0, IEEE, 2024, pp. 1–9. doi: https://doi.org/10.1109/OTCON60325.2024.10687678.
A. Cataldo, E. Lee, X. Liu, E. Matsikoudis, and H. Zheng, “Discrete-event systems: Generalizing metric spaces and fixed point semenatics,” Citeseer, 2005.
J.-J. Minana and O. Valero, “Are fixed point theorems in G-metric spaces an authentic generalization of their classical counterparts?,” J. Fixed Point Theory Appl., vol. 21, no. 2, p. 70, 2019, doi: https://doi.org/10.1007/s11784-019-0705-z.
B. F. Azevedo, A. M. A. C. Rocha, and A. I. Pereira, “Hybrid approaches to optimization and machine learning methods: a systematic literature review,” Mach. Learn., vol. 113, no. 33, pp. 1–43, 2024, doi: https://doi.org/10.1007/s10994-023-06467-x Share this article.
S. Chrétien, M. Cucuringu, G. Lecué, and L. Neirac, “Learning with semi-definite programming: statistical bounds based on fixed point analysis and excess risk curvature,” J. Mach. Learn. Res., vol. 22, no. 230, pp. 1–64, 2021, [Online]. Available: https://www.jmlr.org/papers/v22/21-0021.html
D. H. Ackley, G. E. Hinton, and T. J. Sejnowski, “A learning algorithm for Boltzmann machines,” Cogn. Sci., vol. 9, no. 1, pp. 147–169, 1985, doi: https://doi.org/10.1016/S0364-0213(85)80012-4.
S. C. Pinto, S. Welikala, S. B. Andersson, J. M. Hendrickx, and C. G. Cassandras, “Minimax Multi-Agent Persistent Monitoring of a Network System,” arXiv Prepr. arXiv2201.06607, vol. 11, no. 1, p. 2201.06607, 2022, doi: https://doi.org/10.48550/arXiv.2201.06607.
Y. Liu, T. Yu, Q. Meng, and Q. Liu, “Flow optimization strategies in data center networks: A survey,” J. Netw. Comput. Appl., vol. 226, no. 6, p. 103883, 2024, doi: https://doi.org/10.1016/j.jnca.2024.103883.
T. Van An, N. Van Dung, Z. Kadelburg, and S. Radenović, “Various generalizations of metric spaces and fixed point theorems,” Rev. la Real Acad. Ciencias Exactas, Fis. y Nat. Ser. A. Mat., vol. 109, no. 109, pp. 175–198, 2015, doi: https://doi.org/10.1007/s13398-014-0173-7.
P. M. Bajracharya and N. Adhikari, “A study on fixed point theory in M-metric space,” Sci. World, vol. 13, no. 13, pp. 62–68, 2020.
N. Abbas, “Fixed point theorems for some contractions in Rectangular b-metric space,” A thesis submitted in partial fulfillment for the degree of Master of …, 2017.
J. A. Jiddah, H. Işık, M. Alansari, and M. S. Shagari, “A Study of Fixed Point Results in G-Metric Space via New Contractions with Applications,” in Recent Developments in Fixed-Point Theory: Theoretical Foundations and Real-World Applications, Springer, 2024, pp. 241–263. doi: https://doi.org/10.1007/978-981-99-9546-2_10.
C. Jin, P. Netrapalli, R. Ge, S. M. Kakade, and M. I. Jordan, “On nonconvex optimization for machine learning: Gradients, stochasticity, and saddle points,” J. ACM, vol. 68, no. 2, pp. 1–29, 2021, doi: https://doi.org/10.1145/3418526.
Y.-P. Hsieh, P. Mertikopoulos, and V. Cevher, “The limits of min-max optimization algorithms: Convergence to spurious non-critical sets,” in International Conference on Machine Learning, PMLR, 2021, pp. 4337–4348. doi: https://proceedings.mlr.press/v139/hsieh21a.html.
R. Sun, “Optimization for deep learning: theory and algorithms,” arXiv Prepr. arXiv1912.08957, vol. 1912, no. 08957, p. 1912.08957, 2019.
R. Carmona and M. Laurière, “Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games I: The ergodic case,” SIAM J. Numer. Anal., vol. 59, no. 3, pp. 1455–1485, 2021, doi: https://doi.org/10.1137/19M1274377.
C. T. Kelley, Iterative methods for optimization. SIAM, 1999. [Online]. Available: https://epubs.siam.org/doi/pdf/10.1137/1.9781611970920.bm
M. Danilova et al., “Recent theoretical advances in non-convex optimization,” in High-Dimensional Optimization and Probability: With a View Towards Data Science, Springer, 2022, pp. 79–163. doi: https://doi.org/10.1007/978-3-031-00832-0_3.
P. Jain and P. Kar, “Non-convex optimization for machine learning,” Found. Trends® Mach. Learn., vol. 10, no. 3–4, pp. 142–363, 2017, doi: http://dx.doi.org/10.1561/2200000058.
H. Snyder, “Literature review as a research methodology: An overview and guidelines,” J. Bus. Res., vol. 104, no. 11, pp. 333–339, 2019, doi: https://doi.org/10.1016/j.jbusres.2019.07.039.
J. W. Knopf, “Doing a literature review,” PS Polit. Sci. Polit., vol. 39, no. 1, pp. 127–132, 2006, doi: https://doi.org/10.1017/S1049096506060264.
M. Iqbal, A. Batool, A. Hussain, and H. Alsulami, “Fuzzy Fixed Point Theorems in S-Metric Spaces: Applications to Navigation and Control Systems,” Axioms, vol. 13, no. 9, p. 650, 2024, doi: https://doi.org/10.3390/axioms13090650.
B. Samet, C. Vetro, and F. Vetro, “Remarks on G‐Metric Spaces,” Int. J. Anal., vol. 2013, no. 1, p. 917158, 2013, doi: https://doi.org/10.1155/2013/917158.
Z. Mustafa and H. Obiedat, “A fixed point theorem of Reich in G-metric spaces,” Cubo, vol. 12, no. 1, pp. 83–93, 2010, doi: http://dx.doi.org/10.4067/S0719-06462010000100008.
T. Van An, N. Van Dung, and V. T. Le Hang, “A new approach to fixed point theorems on G-metric spaces,” Topol. Appl., vol. 160, no. 12, pp. 1486–1493, 2013, doi: https://doi.org/10.1016/j.topol.2013.05.027.
Y. U. Gaba, “Fixed point theorems in G-metric spaces,” J. Math. Anal. Appl., vol. 455, no. 1, pp. 528–537, 2017, doi: https://doi.org/10.1016/j.jmaa.2017.05.062.
S. Kasahara, “On some generalizations of the Banach contraction theorem,” Publ. Res. Inst. Math. Sci., vol. 12, no. 2, pp. 427–437, 1976, doi: https://doi.org/10.2977/prims/1195190723.
P. R. Meyers, “A converse to Banach’s contraction theorem,” J. Res. Nat. Bur. Stand. Sect. B, vol. 71, no. 2, pp. 73–76, 1967.
J. Shawe-Taylor and S. Sun, “A review of optimization methodologies in support vector machines,” Neurocomputing, vol. 74, no. 17, pp. 3609–3618, 2011, doi: https://doi.org/10.1016/j.neucom.2011.06.026.
F. Gieseke, A. Airola, T. Pahikkala, and O. Kramer, “Fast and simple gradient-based optimization for semi-supervised support vector machines,” Neurocomputing, vol. 123, no. 10, pp. 23–32, 2014, doi: https://doi.org/10.1016/j.neucom.2012.12.056.
W. A. Kirk, “Geodesic geometry and fixed point theory,” in Seminar of mathematical analysis (Malaga/Seville, 2002/2003), 2003, pp. 195–225.
R. Espınola and W. A. Kirk, “Set-valued contractions and fixed points,” Nonlinear Anal. Theory, Methods Appl., vol. 54, no. 3, pp. 485–494, 2003, doi: https://doi.org/10.1016/S0362-546X(03)00107-X.
T. Bian and Z.-P. Jiang, “Reinforcement learning and adaptive optimal control for continuous-time nonlinear systems: A value iteration approach,” IEEE Trans. neural networks Learn. Syst., vol. 33, no. 7, pp. 2781–2790, 2021, doi: https://doi.org/10.1109/TNNLS.2020.3045087.
D. Bertsekas, “Multiagent value iteration algorithms in dynamic programming and reinforcement learning,” Results Control Optim., vol. 1, no. 12, p. 100003, 2020, doi: https://doi.org/10.1016/j.rico.2020.100003.
H. Bonnel, A. N. Iusem, and B. F. Svaiter, “Proximal methods in vector optimization,” SIAM J. Optim., vol. 15, no. 4, pp. 953–970, 2005, doi: https://doi.org/10.1137/S1052623403429093.
R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM J. Control Optim., vol. 14, no. 5, pp. 877–898, 1976, doi: https://doi.org/10.1137/0314056.
G. C. Bento, O. P. Ferreira, and J. G. Melo, “Iteration-complexity of gradient, subgradient and proximal point methods on Riemannian manifolds,” J. Optim. Theory Appl., vol. 173, no. 2, pp. 548–562, 2017, doi: https://doi.org/10.1007/s10957-017-1093-4.
J. N. Tsitsiklis, “Problems in decentralized decision making and computation,” 1984, Massachusetts Institute of Technology.
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