Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks
A scientific theory of deep learning, comprising learning dynamics and statistical properties of learned models, is rapidly gaining attention. One of the corner stones of this development are analytically solvable toy models, allowing for the fully tractable analysis of the learning dynamics. Here we analytically investigate such a toy model using the regularization strength as a tunable external parameter - akin to external fields in statistical physics. In previous studies, (i) an onset of learning transition was predicted analytically and (ii) it was phenomenologically/numerically established that tuning the regularization strength can result in a cascade of phase transitions. The number of those transitions was linked to the geometry of the loss landscape determined by the model complexity. Setting up a rigorous framework underpinning the previous numerical observations, our investigation reveals a precise connection between those cascades of phase transitions, learnable features and the underlying geometry. We provide analytic predictions of these phase transitions as well as tractable order parameters related to learned features. At the level of the minimal model, we connect this macroscopic perspective (that can be condensed into an effective description) to the microscopic perspective in terms of the geometry of the loss landscape characterized by the Hessian spectrum. Thus, the presented model provides a platform to explore and sharpen advances made in the scientific theory of deep learning rooted in statistical physics concepts.