Exponential-Quadratic-Logarithmic Composite Function Optimization In Positive Domains: Leveraging Multiplicative Calculus In Gradient Descent Algorithms

Year 2024, Volume: 10 Issue: 1, 209 - 227, 30.06.2024

Erkan Kıymık , Ali Emre Öztürk

https://doi.org/10.29132/ijpas.1467644

Abstract

This work investigates the integration of multiplicative calculus into gradient descent algorithms, including Adaptive Gradient algorithm (AdaGrad), Root Mean Squared Propagation (RMSProp), Nesterov Accelerated Gradient (NAG), and Momentum, to optimize exponential-quadratic-logarithmic composite functions with the positivity constrained. This research, conducted across five scenarios within the Constrained and Unconstrained Testing Environment (CUTEst), compares these multiplicative methods with their classical counterparts under a variety of constraints environments such as bounded, quadratic, and other types, and unconstrained environments. The results demonstrate the significant superiority of multiplicative-based algorithms, especially in unconstrained and bounded constrained scenarios, and demonstrate their potential for complex optimization tasks. Statistical analysis supports the observed performance advantages, indicating significant opportunities for optimization strate-gies in positive domains.

Keywords

Gradient Descent, Optimization, Multiplicative calculus, Machine Learning, Composite Functions, Non-newtonian Calculus

References

• Ruder, S. (2016). An overview of gradient descent optimization algorithms. 1–14, [Online]. Available: http://arxiv.org/abs/1609.04747.
• Baldi, P. (1995). Gradient Descent Learning Algorithm Overview: A General Dynamical Systems Perspective. IEEE Trans. Neural Networks, 6(1), 182–195.
• Zou, F., Shen, L., Jie, Z., Zhang, W. and Liu, W. (2019). A sufficient condition for convergences of adam and rmsprop. Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit., 2019(1), 11119–11127.
• Wilson, A. C., Roelofs, R., Stern, M., Srebro, N. and Recht, B. (2017). The marginal value of adaptive gradient methods in machine learning. Adv. Neural Inf. Process. Syst., 2017, 4149–4159.
• Qian, N. (1999). On the momentum term in gradient descent learning algorithms. Neural Networks, 12(1), 145–151.
• Gould, N. I. M., Orban, D. and Toint, P. L. (2015). CUTEst: a Constrained and Unconstrained Testing Environment with safe threads for mathematical optimization. Comput. Optim. Appl., 60(3),545–557.
• Zhang, J. (2019). Gradient Descent based Optimization Algorithms for Deep Learning Models Training. [Online]. Available: http://arxiv.org/abs/1903.03614.
• Polyak, B. T. (1964). Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys., 4(5), 1–17.
• Duchi, J. C., Bartlett, P. L.and Wainwright, M. J. (2012). Randomized smoothing for (parallel) stochastic optimization. Proc. IEEE Conf. Decis. Control, 12, 5442–5444.
• Bashirov, A. E., Kurpinar, E. M. and Özyapici, A. (2008). Multiplicative calculus and its applications. J. Math. Anal. Appl., 337(1), 36–48.
• Georgiev S. G. and Zennir, K. (2022). Multiplicative Differential Calculus, 1st ed., vol. 1, no. 1. Taylor & Francis.
• Gürefe Y. and Misirli, D. D. E.(2011). Product Calculi And Its Applications. J. Phys. A Math. Theor., 44(8), 1–22.
• Uzer, A. (2010). Multiplicative type complex calculus as an alternative to the classical calculus. Comput. Math. with Appl., 60(10), 2725–2737.
• Stanley, D. (1999). A multiplicative calculus. Primus, 9(4), 310–326.
• Özyapici A. and Misirli, A. P. D. E. (2009). Multiplicative Calculus And Its Aplications,” Ege University.
• Özyapıcı, A., Riza, M., Bilgehan, B. and Bashirov, A. E. (2014). On multiplicative and Volterra minimization methods. Numer. Algorithms, 67(3), 623–636.
• Filip, D., Piatecki, C. and Andrada Filip, D. (2014). A non-newtonian examination of the theory of exogenous economic growth. Work. Pap., X(XX), 2014, [Online]. Available: https://hal.archives-ouvertes.fr/hal-00945781.
• Florack L.and Van Assen, H. (2012). Multiplicative calculus in biomedical image analysis. J. Math. Imaging Vis., 42(1), 64–75.
• Riza M.and Aktöre, H. (2015). The Runge-Kutta method in geometric multiplicative calculus. LMS J. Comput. Math., 18(1), 539–554.
• Riza M. and Eminağa, B. (2014). Bigeometric Calculus and Runge Kutta Method. 1–19, [Online]. Available: http://arxiv.org/abs/1402.2877.
• Cubillos, M. (2018). Modelling wave propagation without sampling restrictions using the multiplicative calculus I: Theoretical considerations. 1–18, [Online]. Available: http://arxiv.org/abs/1801.03402.
• Stephen Boyd Lieven Vandenberghe, (2013). Convex Optimization 中文影印. Cambridge University Press.