Exchangeability and Kernel Invariance in Trained MLPs

TL;DR

This paper investigates the exchangeability of weights in trained MLPs and demonstrates that the layer-wise kernel remains approximately invariant during training, revealing a phase transition related to weight covariance.

Contribution

It establishes the exchangeability property of MLP weights and links it to kernel invariance, providing new insights into the training dynamics of neural networks.

Findings

Layer-wise kernels stay approximately constant during training.

A phase transition occurs when weight covariance shifts from zero.

Exchangeability explains certain invariances in trained MLPs.

Abstract

In the analysis of machine learning models, it is often convenient to assume that the parameters are IID. This assumption is not satisfied when the parameters are updated through training processes such as SGD. A relaxation of the IID condition is a probabilistic symmetry known as exchangeability. We show the sense in which the weights in MLPs are exchangeable. This yields the result that in certain instances, the layer-wise kernel of fully-connected layers remains approximately constant during training. We identify a sharp change in the macroscopic behavior of networks as the covariance between weights changes from zero.