A unified theory of feature learning in RNNs and DNNs

TL;DR

This paper develops a unified mean-field theory for RNNs and DNNs, revealing how weight sharing in RNNs influences their functional properties and generalization in sequential tasks.

Contribution

It introduces a comprehensive theoretical framework linking RNN and DNN architectures through representational kernels and phase transitions in learning behavior.

Findings

Below threshold, RNNs and DNNs behave similarly.

Above threshold, RNNs develop correlated temporal representations.

Weight sharing in RNNs aids generalization in sequential tasks.

Abstract

Recurrent and deep neural networks (RNNs/DNNs) are cornerstone architectures in machine learning. Remarkably, RNNs differ from DNNs only by weight sharing, as can be shown through unrolling in time. How does this structural similarity fit with the distinct functional properties these networks exhibit? To address this question, we here develop a unified mean-field theory for RNNs and DNNs in terms of representational kernels, describing fully trained networks in the feature learning ($\mu$P) regime. This theory casts training as Bayesian inference over sequences and patterns, directly revealing the functional implications induced by the RNNs' weight sharing. In DNN-typical tasks, we identify a phase transition when the learning signal overcomes the noise due to randomness in the weights: below this threshold, RNNs and DNNs behave identically; above it, only RNNs develop correlated representations across timesteps. For sequential tasks, the RNNs' weight sharing furthermore induces an inductive bias that aids generalization by interpolating unsupervised time steps. Overall, our theory offers a way to connect architectural structure to functional biases.