  Recent research has been focused on two different approaches to studying
neural networks training in the limit of infinite width (1) a mean-field (MF)
and (2) a constant neural tangent kernel (NTK) approximations. These two
approaches have different scaling of hyperparameters with the width of a
network layer and as a result, different infinite-width limit models. We
propose a general framework to study how the limit behavior of neural models
depends on the scaling of hyperparameters with network width. Our framework
allows us to derive scaling for existing MF and NTK limits, as well as an
uncountable number of other scalings that lead to a dynamically stable limit
behavior of corresponding models. However, only a finite number of distinct
limit models are induced by these scalings. Each distinct limit model
corresponds to a unique combination of such properties as boundedness of logits
and tangent kernels at initialization or stationarity of tangent kernels.
Existing MF and NTK limit models, as well as one novel limit model, satisfy
most of the properties demonstrated by finite-width models. We also propose a
novel initialization-corrected mean-field limit that satisfies all properties
noted above, and its corresponding model is a simple modification for a
finite-width model.
