统一学习与进化算法的数学本质，揭示其共性规律。
The Price equation reveals a universal force-metric-bias law of algorithmic learning and natural selection
- 用价格方程拆解参数更新，构建力-度量-偏差通用模型。
- 该模型涵盖梯度下降、自然选择等主流算法，解释其内在统一性。
- 适合跨学科研究者理解算法设计原理，指导新方法开发。
多种学习算法、优化方法和自然选择虽表面差异显著，但共享同一数学结构。本文通过价格方程对变化进行符号分解，揭示了一条普适的力-度量-偏差（FMB）定律：Δθ = M f + b + ξ。其中，力f推动参数改进，正比于性能对参数的梯度；度量M通过逆曲率调整步长；偏差b引入动量或参考系偏移；噪声ξ实现探索。该框架将自然选择、贝叶斯更新、牛顿法、随机梯度下降、随机朗之万动力学、Adam优化等均视为同一过程的特例。价格方程还阐明了费舍尔信息、KL散度和达朗贝尔原理为何在学习动态中自然出现。揭示这一共同结构，为跨领域理解、比较和设计学习算法提供了严谨基础。
原文摘要 · Abstract (English)
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.
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