Why Do Big Data and Machine Learning Entail the Fractional Dynamics?

Chapter 2 explores the fundamental question of why big data and machine learning inherently involve fractional dynamics. The investigation unfolds through an exploration of fractional calculus (FC) and fractional-order thinking (FOT), shedding light on their relevance in understanding the intricate dynamics of complex systems. The chapter delves into the concept of complexity and inverse power laws (IPLs), establishing a connection between heavy-tailed distributions and fractional dynamics. Various heavy-tailed distributions are examined in the context of their implications for machine learning in diverse applications. The discussion extends to the interplay between big data, variability, and fractional calculus, incorporating topics such as the Hurst parameter, fractional Gaussian noise (fGn), etc. The chapter progresses to elucidate the concept of optimal machine learning and optimal randomness, distinguishing between derivative-free methods and gradient-based methods. Additionally, it explores the contributions of the control community to machine learning. A detailed case study on optimal randomness for Stochastic Configuration Network (SCN) with heavy-tailed distributions is presented, encompassing an introduction, SCN with heavy-tailed probability density functions (PDFs), a regression model, parameter tuning, and a practical application involving MNIST handwritten digit classification.