On The Connection of Benford's Law and Neural Networks

Benford's law, also called Significant Digit Law, is observed in many naturally occurring data-sets. For instance, the physical constants such as Gravitational, Coulomb's Constant, etc., follow this law. In this paper, we define a score, $MLH$, for how closely a Neural Network's Weights match Benford's law. We show that Neural Network Weights follow Benford's Law regardless of the initialization method. We make a striking connection between Generalization and the $MLH$ of the network. We provide evidence that several architectures from AlexNet to ResNeXt trained on ImageNet, Transformers (BERT, Electra, etc.), and other pre-trained models on a wide variety of tasks have a strong correlation between their test performance and the $MLH$. We also investigate the influence of Data in the Weights to explain why NNs possibly follow Benford's Law. With repeated experiments on multiple datasets using MLPs, CNNs, and LSTMs, we provide empirical evidence that there is a connection between $MLH$ while training, overfitting. Understanding this connection between Benford's Law and Neural Networks promises a better comprehension of the latter.