Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning
This work addresses a foundational problem in machine learning theory by elucidating mechanisms for efficient feature learning, with potential applications in improving network interpretability and training efficiency.
The paper tackled the problem of understanding how neural networks transition from lazy to rich feature learning regimes by deriving exact solutions for a minimal model, revealing that unbalanced initializations and learning rates influence feature learning through conserved quantities. The results show that unbalanced initializations accelerate rich learning in nonlinear networks, reducing sample complexity and time to grokking in experiments.
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.