Many machine learning models based on neural networks exhibit scaling laws:
their performance scales as power laws with respect to the sizes of the model
and training data set. We use large-N field theory methods to solve a model
recently proposed by Maloney, Roberts and Sully which provides a simplified
setting to study neural scaling laws. Our solution extends the result in this
latter paper to general nonzero values of the ridge parameter, which are
essential to regularize the behavior of the model. In addition to obtaining new
and more precise scaling laws, we also uncover a duality transformation at the
diagrams level which explains the symmetry between model and training data set
sizes. The same duality underlies recent efforts to design neural networks to
simulate quantum field theories.

基于大 N 场论方法，我们解决了一个由 Maloney，Roberts 和 Sully 提出的模型，该模型为研究神经网络的缩放定律提供了一个简化的环境。我们的解决方案将这个模型的结果推广到了一般非零的岭参数值，这对于规范模型的行为是至关重要的。除了获得新的和更精确的缩放定律，我们还揭示了在图表级别上的对偶变换，该变换解释了模型和训练数据集大小之间的对称性。相同的对偶变换也支持最近设计神经网络来模拟量子场论的努力。