Variational methods for Learning Multilevel Genetic Algorithms using the Kantorovich Monad

TL;DR

This paper introduces a category-theoretic framework for modeling multilevel evolutionary processes and genetic algorithms, providing new analytical tools and methods for understanding and optimizing these complex systems.

Contribution

It develops a unified mathematical framework using the Kantorovich Monad for multilevel evolution and genetic algorithms, including a multilevel Wright-Fisher process and Price's Equation extension.

Findings

Derived a multilevel Wright-Fisher process model

Extended Price's Equation using the Kantorovich Monad

Demonstrated how to fit multilevel GAs to simulated data

Abstract

Levels of selection and multilevel evolutionary processes are essential concepts in evolutionary theory, and yet there is a lack of common mathematical models for these core ideas. Here, we propose a unified mathematical framework for formulating and optimizing multilevel evolutionary processes and genetic algorithms over arbitrarily many levels based on concepts from category theory and population genetics. We formulate a multilevel version of the Wright-Fisher process using this approach, and we show that this model can be analyzed to clarify key features of multilevel selection. Particularly, we derive an extended multilevel probabilistic version of Price's Equation via the Kantorovich Monad, and we use this to characterize regimes of parameter space within which selection acts antagonistically or cooperatively across levels. Finally, we show how our framework can provide a unified setting for learning genetic algorithms (GAs), and we show how we can use a Variational Optimization and a multi-level analogue of coalescent analysis to fit multilevel GAs to simulated data.