# The Price equation reveals a universal force-metric-bias law of algorithmic learning and natural selection

arXiv: https://arxiv.org/abs/2507.18549 · HTML: https://mlsanity.com/paper/2507.18549/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 100 references that are in the map.
- First posted on arXiv 24 Jul 2025.

## Builds on (most cited first, 23 of 23)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Evolution Strategies as a Scalable Alternative to Reinforcement Learning](https://mlsanity.com/paper/1703.03864/) (2017)
- [BOHB: Robust and Efficient Hyperparameter Optimization at Scale](https://mlsanity.com/paper/1807.01774/) (2018)
- [Population Based Training of Neural Networks](https://mlsanity.com/paper/1711.09846/) (2017)
- [Lookahead Optimizer: k steps forward, 1 step back](https://mlsanity.com/paper/1907.08610/) (2019)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [An Empirical Model of Large-Batch Training](https://mlsanity.com/paper/1812.06162/) (2018)
- [A Bayesian Perspective on Generalization and Stochastic Gradient Descent](https://mlsanity.com/paper/1710.06451/) (2017)
- [Evolution-Guided Policy Gradient in Reinforcement Learning](https://mlsanity.com/paper/1805.07917/) (2018)
- [Meta-learning in natural and artificial intelligence](https://mlsanity.com/paper/2011.13464/) (2020)
- [Collaborative Evolutionary Reinforcement Learning](https://mlsanity.com/paper/1905.00976/) (2019)
- [Bridging Evolutionary Algorithms and Reinforcement Learning: A Comprehensive Survey on Hybrid Algorithms](https://mlsanity.com/paper/2401.11963/) (2024)
- [Meta-Learning by the Baldwin Effect](https://mlsanity.com/paper/1806.07917/) (2018)
- [Diagonal Acceleration for Covariance Matrix Adaptation Evolution Strategies](https://mlsanity.com/paper/1905.05885/) (2019)
- [The inductive theory of natural selection: summary and synthesis](https://mlsanity.com/paper/1412.1285/) (2014)

## Built on by (most cited first, 1 of 1)

- [Direct From Darwin: Deriving Advanced Optimizers From Evolutionary First Principles](https://mlsanity.com/paper/2605.05284/) (2026)

## Abstract (arXiv)

Diverse learning algorithms, optimization methods, and natural selection share a common mathematical structure, despite their apparent differences. Here I show that a simple notational partitioning of change by the Price equation reveals a universal force-metric-bias (FMB) law: Deltaθ = M,f + b + xi. The force f drives improvement in parameters, Deltaθ, in proportion to the slope of performance with respect to the parameters. The metric M rescales movement by inverse curvature. The bias b adds momentum or changes in the frame of reference. The noise xi enables exploration. This framework unifies natural selection, Bayesian updating, Newton's method, stochastic gradient descent, stochastic Langevin dynamics, Adam optimization, and most other algorithms as special cases of the same underlying process. The Price equation also reveals why Fisher information, Kullback-Leibler divergence, and d'Alembert's principle arise naturally in learning dynamics. By exposing this common structure, the FMB law provides a principled foundation for understanding, comparing, and designing learning algorithms across disciplines.