Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning

TL;DR

This paper derives exact solutions to understand how unbalanced initializations in neural networks promote rapid feature learning, revealing the underlying mechanisms and their impact on learning regimes and efficiency.

Contribution

It provides a theoretical framework connecting unbalanced initializations with accelerated feature learning, extending analysis to complex models and validating findings through experiments.

Findings

Unbalanced initializations can accelerate feature learning in neural networks.

Balanced initializations are necessary for rapid feature learning in linear networks.

Unbalanced initializations improve interpretability and reduce sample complexity in deep networks.

Abstract

While the impressive performance of modern neural networks is often attributed to their capacity to efficiently extract task-relevant features from data, the mechanisms underlying this rich feature learning regime remain elusive, with much of our theoretical understanding stemming from the opposing lazy regime. In this work, we derive exact solutions to a minimal model that transitions between lazy and rich learning, precisely elucidating how unbalanced layer-specific initialization variances and learning rates determine the degree of feature learning. Our analysis reveals that they conspire to influence the learning regime through a set of conserved quantities that constrain and modify the geometry of learning trajectories in parameter and function space. We extend our analysis to more complex linear models with multiple neurons, outputs, and layers and to shallow nonlinear networks with piecewise linear activation functions. In linear networks, rapid feature learning only occurs from balanced initializations, where all layers learn at similar speeds. While in nonlinear networks, unbalanced initializations that promote faster learning in earlier layers can accelerate rich learning. Through a series of experiments, we provide evidence that this unbalanced rich regime drives feature learning in deep finite-width networks, promotes interpretability of early layers in CNNs, reduces the sample complexity of learning hierarchical data, and decreases the time to grokking in modular arithmetic. Our theory motivates further exploration of unbalanced initializations to enhance efficient feature learning.