Geometric Learning Dynamics

• URL: http://arxiv.org/abs/2504.14728v1
• Date: Sun, 20 Apr 2025 19:56:41 GMT
• Title: Geometric Learning Dynamics
• Authors: Vitaly Vanchurin,
• Abstract summary: We present a unified framework for modeling learning dynamics in physical, biological, and machine learning systems.<n>The quantum regime corresponds to $a = 1$ and describes Schr"odinger-like dynamics that emerges from a discrete shift symmetry.<n>The efficient learning regime corresponds to $a = tfrac12$ and describes very fast machine learning algorithms.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $g \propto \kappa^a$ between the metric tensor $g$ in the space of trainable variables and the noise covariance matrix $\kappa$. The quantum regime corresponds to $a = 1$ and describes Schr\"odinger-like dynamics that emerges from a discrete shift symmetry. The efficient learning regime corresponds to $a = \tfrac{1}{2}$ and describes very fast machine learning algorithms. The equilibration regime corresponds to $a = 0$ and describes classical models of biological evolution. We argue that the emergence of the intermediate regime $a = \tfrac{1}{2}$ is a key mechanism underlying the emergence of biological complexity.

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