Semi-Lipschitz functions and machine learning for discrete dynamical systems on graphs

• Published in: Machine learning Vol. 111; no. 5; pp. 1765 - 1797
• Main Authors: Falciani, H., Sánchez-Pérez, E. A.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2022; Springer Nature B.V
• Subjects: Artificial Intelligence; Asymmetry; Computer Science; Control; Dynamical systems; Graphs; Machine Learning; Mechatronics; Metric space; Natural Language Processing (NLP); Polytrees; Robotics; Simulation and Modeling; Trees (mathematics); Graph distance; Lipschitz function; quasi-pseudo-metric; bifurcation metric; reinforcement learning
• ISSN: 0885-6125; 1573-0565
• DOI: 10.1007/s10994-022-06130-x

Consider a directed tree U and the space of all finite walks on it endowed with a quasi-pseudo-metric—the space of the strategies S on the graph,—which represent the possible changes in the evolution of a dynamical system over time. Consider a reward function acting in a subset S 0 ⊂ S which measures the success. Using well-known facts of the theory of semi-Lipschitz functions in quasi-pseudo-metric spaces, we extend the reward function to the whole space S . We obtain in this way an oracle function, which gives a forecast of the reward function for the elements of S , that is, an estimate of the degree of success for any given strategy. After explaining the fundamental properties of a specific quasi-pseudo-metric that we define for the (graph) trees (the bifurcation quasi-pseudo-metric), we focus our attention on analyzing how this structure can be used to represent dynamical systems on graphs. We begin the explanation of the method with a simple example, which is proposed as a reference point for which some variants and successive generalizations are consecutively shown. The main objective is to explain the role of the lack of symmetry of quasi-metrics in our proposal: the irreversibility of dynamical processes is reflected in the asymmetry of their definition.