Implicit Regularization in Deep Matrix Factorization
This work addresses the generalization mystery in deep learning for researchers, suggesting that implicit regularization may be more complex than previously thought, though it is incremental in building on existing theories.
The paper investigates implicit regularization in deep matrix factorization for matrix completion and sensing, finding that increased depth enhances low-rank solutions and improves recovery accuracy, while challenging the view that this regularization can be captured by simple mathematical norms.
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion and sensing, a model referred to as deep matrix factorization. Our first finding, supported by theory and experiments, is that adding depth to a matrix factorization enhances an implicit tendency towards low-rank solutions, oftentimes leading to more accurate recovery. Secondly, we present theoretical and empirical arguments questioning a nascent view by which implicit regularization in matrix factorization can be captured using simple mathematical norms. Our results point to the possibility that the language of standard regularizers may not be rich enough to fully encompass the implicit regularization brought forth by gradient-based optimization.