用几何框架统一解释物理、生物和机器学习的演化规律。
Geometric Learning Dynamics
- 基于度量张量与噪声协方差的幂律关系建模学习动态。
- 发现三种基本演化模式，对应α=1、½、0的不同动力学行为。
- 揭示生物复杂性涌现的关键机制，适合跨学科研究者。
我们提出一个统一的几何框架，用于建模物理、生物及机器学习系统中的学习动态。该理论揭示了三种基本演化模式，均源于可训练变量空间中度量张量 $g$ 与噪声协方差矩阵 $κ$ 之间的幂律关系 $g /propto κ^α$。当 $α=1$ 时为量子态，呈现类薛定谔动力学，源于离散平移对称性；当 $α=\tfrac{1}{2}$ 时为高效学习态，描述极快速的机器学习算法；当 $α=0$ 时为平衡态，对应经典生物演化模型。我们认为中间态 $α=\tfrac{1}{2}$ 的出现是生物复杂性涌现的关键机制。
原文摘要 · Abstract (English)
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 $g \propto κ^α$ between the metric tensor $g$ in the space of trainable variables and the noise covariance matrix $κ$. The quantum regime corresponds to $α= 1$ and describes Schrödinger-like dynamics that emerges from a discrete shift symmetry. The efficient learning regime corresponds to $α= \tfrac{1}{2}$ 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 $α= \tfrac{1}{2}$ is a key mechanism underlying the emergence of biological complexity.
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