From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks
Clémentine Carla Juliette Dominé Nicolas Anguita Alexandra Maria Proca Lukas Braun Daniel Kunin Pedro A. M. Mediano Andrew M Saxe
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.
Deep learning, Learning theory, Learning Regime, Rich, Lazy
The paper offers explicit solutions for gradient flow in two-layer linear networks under various initializations, modeling the shift between lazy and rich learning, with applications to neuroscience and machine learning.