Geometric Learning Dynamics

We present a unified geometric framework for modeling learning dynamics in physical, biological, and machine learning systems. The theory reveals three fundamental regimes, each emerging from the power-law relationship g propto kappaα between the metric tensor g in the space of trainable variables and the noise covariance matrix kappa. The quantum regime corresponds to α = 1 and describes Schr"odinger-like dynamics that emerges from a discrete shift symmetry. The efficient learning regime corresponds to α = tfrac12 and describes very fast machine learning algorithms. The equilibration regime corresponds to α = 0 and describes classical models of biological evolution. We argue that the emergence of the intermediate regime α = tfrac12 is a key mechanism underlying the emergence of biological complexity.