Feature Learning in Infinite-Width Neural Networks

TL;DR

This paper introduces modified neural network parametrizations that enable feature learning in the infinite-width limit, surpassing NTK and finite-width models, with explicit formulas derived using Tensor Programs and validated on key tasks.

Contribution

It proposes simple modifications to standard parametrizations to allow feature learning at infinite width, deriving explicit formulas and demonstrating improved performance on canonical tasks.

Findings

Modified parametrizations enable feature learning in the infinite-width limit.

Explicit formulas for these limits are derived using Tensor Programs.

Infinite-width models with feature learning outperform NTK and finite-width networks.

Abstract

As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. given by the Neural Tangent Kernel (NTK)), if it is parametrized appropriately (e.g. the NTK parametrization). However, we show that the standard and NTK parametrizations of a neural network do not admit infinite-width limits that can learn features, which is crucial for pretraining and transfer learning such as with BERT. We propose simple modifications to the standard parametrization to allow for feature learning in the limit. Using the *Tensor Programs* technique, we derive explicit formulas for such limits. On Word2Vec and few-shot learning on Omniglot via MAML, two canonical tasks that rely crucially on feature learning, we compute these limits exactly. We find that they outperform both NTK baselines and finite-width networks, with the latter approaching the infinite-width feature learning performance as width increases. More generally, we classify a natural space of neural network parametrizations that generalizes standard, NTK, and Mean Field parametrizations. We show 1) any parametrization in this space either admits feature learning or has an infinite-width training dynamics given by kernel gradient descent, but not both; 2) any such infinite-width limit can be computed using the Tensor Programs technique. Code for our experiments can be found at github.com/edwardjhu/TP4.