Variational methods for Learning Multilevel Genetic Algorithms using the Kantorovich Monad

TL;DR

This work derives an extended multilevel probabilistic version of Price's Equation via the Kantorovich Monad, and uses this to characterize regimes of parameter space within which selection acts antagonistically or cooperatively across levels.

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.

Figures (3)

• Figure 1: Multilevel evolutionary process for ant colonies. Selection may act at the cellular, individual or colony level, and fitness any of these levels may conflict with fitness at any other.
• Figure 2: Schematic of a Multi-level Evolutionary Process. The population consists of $L=3$ levels, with two genotypes (open and closed circles) at level $l=0$ and 8 individuals, 4 groups of 2 individuals at level $l=1$, and 2 meta-groups of 2 groups (4 individuals) at level $l=2$. Time $t$ flows down the page; at each time-step, a level is selected to reproduce, and the individuals (or groups) at that level which reproduce are highlighted in red. Mutations may occur during reproduction; when a group at level $l$ reproduces, the Wasserstein distance at level $l$ determines the probability for the state of the offspring group. The fitness of a group determines its probability of reproducing, and is determined by the fitness of the group members (here, the closed genotype has higher level 0 fitness than the open genotype), and the group's cohesion.
• Figure 3: Fitness across levels for 5 synthetic examples. Mean value of $f_l$ is shown across all individuals or groups at level $l$.