Geometric Learning Dynamics

2025-04-20

AI Summary

This paper addresses the heterogeneity of learning dynamics across physical, biological, and machine learning systems by establishing a unified framework grounded in differential geometry and stochastic dynamics. Methodologically, it identifies and rigorously derives a power-law relationship (G propto D^a) between the Riemannian metric tensor (G) and the noise covariance matrix (D). This yields three universal dynamical regimes: (i) classical equilibrium ((a = 0)), (ii) efficient learning ((a = 1/2)), newly revealed as the geometric origin of cross-scale emergence of biological complexity, and (iii) Schrödinger-type quantum evolution ((a = 1)). Integrating information geometry, symmetry analysis, and continuous-limit modeling, the framework achieves the first geometric unification of these three distinct dynamical classes. It provides a principled, physics-informed perspective on the physical nature and biological origins of learning behavior.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $a = frac{1}{2}$ 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 $a = frac{1}{2}$ is a key mechanism underlying the emergence of biological complexity.

Problem

• Model learning dynamics in diverse systems
• Identify three fundamental learning regimes
• Explain emergence of biological complexity

Innovation

• Unified geometric framework for learning dynamics
• Power-law relationship between metric and noise
• Three fundamental regimes: quantum, efficient, equilibration