Fitness and Overfitness: Implicit Regularization in Evolutionary Dynamics

A common underlying assumption in evolutionary thought is that adaptation generally drives an increase in biological complexity. However, the rules governing the evolution of complexity appear more nuanced, and the forces that drive the increase or decrease in organismal complexity are not fully understood. Evolution is deeply connected to learning, where complexity, as well as its origins and consequences, are much better understood, with established results on the optimal complexity appropriate for a given learning task in various settings. In this work, we suggest a mathematical framework for studying the relationship between organismal complexity and the complexity of the environment in which the organisms evolve by leveraging an existing mathematical isomorphism between evolutionary dynamics and learning theory, specifically an isomorphism between the replicator equation and sequential Bayesian learning, with evolving types corresponding to competing hypotheses, and fitness in a given environment corresponding to the likelihood of observed evidence. In Bayesian learning, implicit regularization prevents overfitting and drives the inference of hypotheses whose complexity matches the learning challenge. We show how these results naturally carry over to the evolutionary setting, where they are interpreted as organism complexity evolving to match the complexity of the environment, with organisms that are too complex or too simple for their environment suffering from effects we term overfitness and underfitness, respectively. Other aspects, peculiar to evolution and not to learning, reveal additional trends. One such trend is that frequently changing environments decrease selected complexity, a result with potential implications in both evolution and learning. Together, our results suggest that the balance between complex organisms that may overadapt to transient environmental features, and simple organisms that may be insufficiently flexible in responding to environmental challenges, drives the emergence of optimal complexity that reflects environmental structure. This mathematical framework offers new ways to think about biological complexity, and it suggests new potential causes for an increase or decrease in complexity in different selective environments.