Implicit Bias and Loss of Plasticity in Matrix Completion: Depth Promotes Low-Rankness
arXiv:2603.04703v1 Announce Type: new Abstract: We study matrix completion via deep matrix factorization (a.k.a. deep linear neural networks) as a simplified testbed to examine how network depth influences training dynamics. Despite the simplicity and importance of the problem, prior theory largely focuses on shallow (depth-2) models and does not fully explain the implicit low-rank bias observed in deeper networks. We identify coupled dynamics as a key mechanism behind this bias and show that it intensifies with increasing depth. Focusing on gradient flow under block-diagonal observations, we prove: (a) networks of depth $\geq 3$ exhibit coupling unless initialized diagonally, and (b) convergence to rank-1 occurs if and only if the dynamics is coupled -- resolving an open question by Menon (2024) for a family of initializations. We also revisit the loss of plasticity phenomenon in matrix completion (Kleinman et al., 2024), where pre-training on few observations and resuming with more
arXiv:2603.04703v1 Announce Type: new Abstract: We study matrix completion via deep matrix factorization (a.k.a. deep linear neural networks) as a simplified testbed to examine how network depth influences training dynamics. Despite the simplicity and importance of the problem, prior theory largely focuses on shallow (depth-2) models and does not fully explain the implicit low-rank bias observed in deeper networks. We identify coupled dynamics as a key mechanism behind this bias and show that it intensifies with increasing depth. Focusing on gradient flow under block-diagonal observations, we prove: (a) networks of depth $\geq 3$ exhibit coupling unless initialized diagonally, and (b) convergence to rank-1 occurs if and only if the dynamics is coupled -- resolving an open question by Menon (2024) for a family of initializations. We also revisit the loss of plasticity phenomenon in matrix completion (Kleinman et al., 2024), where pre-training on few observations and resuming with more degrades performance. We show that deep models avoid plasticity loss due to their low-rank bias, whereas depth-2 networks pre-trained under decoupled dynamics fail to converge to low-rank, even when resumed training (with additional data) satisfies the coupling condition -- shedding light on the mechanism behind this phenomenon.
Executive Summary
This article delves into the relationship between network depth and the implicit low-rank bias observed in matrix completion via deep matrix factorization. The authors identify coupled dynamics as a key mechanism behind this bias, which intensifies with increasing depth. They prove that networks of depth ≥3 exhibit coupling unless initialized diagonally and demonstrate that deep models avoid plasticity loss due to their low-rank bias. This research contributes to the understanding of the training dynamics of deep neural networks and sheds light on the phenomenon of loss of plasticity in matrix completion. The findings have significant implications for the design and implementation of deep learning models, particularly in applications where low-rankness is a desirable property.
Key Points
- ▸ The authors identify coupled dynamics as a key mechanism behind the implicit low-rank bias observed in deep matrix factorization.
- ▸ Networks of depth ≥3 exhibit coupling unless initialized diagonally.
- ▸ Deep models avoid plasticity loss due to their low-rank bias.
Merits
Methodological Innovation
The authors employ a novel approach to study the training dynamics of deep matrix factorization, providing a deeper understanding of the relationship between network depth and low-rank bias.
Theoretical Contributions
The authors provide rigorous theoretical results, resolving an open question by Menon (2024) and shedding light on the mechanism behind the loss of plasticity phenomenon in matrix completion.
Demerits
Limited Generalizability
The authors focus on a simplified testbed (matrix completion via deep matrix factorization), which may limit the generalizability of their findings to more complex applications.
Experimental Evaluation
While the authors provide theoretical contributions, they may benefit from experimental evaluation to further validate their findings and demonstrate the practical implications of their research.
Expert Commentary
The article presents a rigorous and insightful analysis of the relationship between network depth and the implicit low-rank bias observed in matrix completion via deep matrix factorization. The authors' identification of coupled dynamics as a key mechanism behind this bias is a significant contribution to the field. However, the limited generalizability of their findings and the lack of experimental evaluation are notable limitations. Nevertheless, the research has significant implications for the design and implementation of deep learning models, particularly in applications where low-rankness is a desirable property. As such, it is an important contribution to the field of deep learning and will likely spark further research and debate.
Recommendations
- ✓ Future research should aim to extend the findings of this study to more complex applications and evaluate the practical implications of the authors' research through experimental evaluation.
- ✓ The authors' identification of coupled dynamics as a key mechanism behind the implicit low-rank bias observed in deep matrix factorization has significant implications for the development of deep learning models, particularly in applications where low-rankness is a desirable property.