From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks

TL;DR

This paper provides an exact analysis of how weight initialization affects learning dynamics in deep linear networks, bridging the gap between lazy and rich regimes and offering insights relevant to neuroscience and machine learning.

Contribution

It derives exact solutions for initialization-dependent learning dynamics, illuminating the transition between lazy and rich regimes in deep linear networks.

Findings

Exact solutions for weight initialization effects

Representation evolution across regimes

Implications for transfer and continual learning

Abstract

Biological and artificial neural networks develop internal representations that enable them to perform complex tasks. In artificial networks, the effectiveness of these models relies on their ability to build task specific representation, a process influenced by interactions among datasets, architectures, initialization strategies, and optimization algorithms. Prior studies highlight that different initializations can place networks in either a lazy regime, where representations remain static, or a rich/feature learning regime, where representations evolve dynamically. Here, we examine how initialization influences learning dynamics in deep linear neural networks, deriving exact solutions for lambda-balanced initializations-defined by the relative scale of weights across layers. These solutions capture the evolution of representations and the Neural Tangent Kernel across the spectrum from the rich to the lazy regimes. Our findings deepen the theoretical understanding of the impact of weight initialization on learning regimes, with implications for continual learning, reversal learning, and transfer learning, relevant to both neuroscience and practical applications.