The Price equation reveals a universal force-metric-bias law of algorithmic learning and natural selection
This provides a foundational framework for understanding and designing learning algorithms across disciplines, though it is incremental in unifying existing concepts.
The paper tackles the problem of unifying diverse learning algorithms and natural selection by deriving a universal force-metric-bias law from the Price equation, showing that algorithms like stochastic gradient descent and natural selection are special cases of this framework.
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: $Δ\mathbfθ = \mathbf{M}\,\mathbf{f} + \mathbf{b} + \mathbfξ$. The force $\mathbf{f}$ drives improvement in parameters, $Δ\mathbfθ$, in proportion to the slope of performance with respect to the parameters. The metric $\mathbf{M}$ rescales movement by inverse curvature. The bias $\mathbf{b}$ adds momentum or changes in the frame of reference. The noise $\mathbfξ$ 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.