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  • Open Access

Fitness and Overfitness: Implicit Regularization in Evolutionary Dynamics

Hagai Rappeport*

Mor Nitzan†

  • *Contact author: Hagai.Rappeport@mail.huji.ac.il
  • †Contact author: Mor.Nitzan@mail.huji.ac.il
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PRX Life 4, 023028 – Published 17 June, 2026

DOI: https://doi.org/10.1103/zrqb-cj47

Abstract

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.

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PRX Life Collection on adaptation and learning

PRX Life presents a special collection on ‘Biological systems that learn’, curated by the organizers of the National Institute for Theory and Mathematics in Biology (NITMB) workshop together with early-career researchers. The papers in this collection illustrate how living systems acquire functions, showing it is useful to describe them as tunable matter with many individually adjustable degrees of freedom characterizing microscopic properties. Drawing inspiration from AI and non-biological systems that can acquire desired functions, the featured research highlights the timeliness of the topic and its potential to drive unifying frameworks for emergent collective function in living tunable matter.

Every article in this collection is subject to a rigorous peer review process, upholding the high standards applied to all papers in the journal. The PRX Life editorial team manages peer review and makes all editorial decisions.

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Outline

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PRX Life 4, 023028– Published 17 June, 2026
  • Received 31 July 2025
  • Accepted 12 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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