Implicit Regularization in Deep Matrix Factorization (1905.13655v3)

Abstract: 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.

• The paper demonstrates that increasing depth in matrix factorization promotes a stronger low-rank bias, resulting in more accurate matrix recovery in sparse data scenarios.
• It challenges traditional nuclear norm explanations by revealing that gradient descent in deep networks induces implicit regularization effects beyond standard norms.
• Empirical evaluations confirm that deeper factorizations outperform shallow models by dynamically attenuating smaller singular values during optimization.

Implicit Regularization in Deep Matrix Factorization

The paper "Implicit Regularization in Deep Matrix Factorization" explores the implicit regularization introduced by gradient-based optimization in deep neural networks, focusing particularly on deep matrix factorization for tasks such as matrix completion and matrix sensing. It challenges the understanding that such implicit regularization can solely be explained by traditional mathematical norms, like the nuclear norm, especially as depth in the model increases.

Main Contributions

The authors present two main findings:

1. Depth Enhances Low-rank Bias: The research shows, both theoretically and empirically, that increasing the depth in matrix factorization enhances the tendency towards low-rank solutions. This bias often results in more accurate recovery of the original matrix, particularly in data-poor scenarios where few observations are available.
2. Limitations of Norm-based Explanations: The work questions the prevailing view that implicit regularization can be captured by minimizing known mathematical norms, such as the nuclear norm. By showing disparities between nuclear norm minimization and the solutions obtained from deep matrix factorizations, the authors suggest that standard regularizers do not fully encompass the implicit regularization effects induced by gradient descent in these scenarios.

Theoretical Framework

The authors build upon earlier work by Gunasekar et al., extending the theoretical analysis to deeper matrix factorizations. They illustrate that while previous analyses have associated nuclear norm minimization with implicit regularization in shallow networks, such explanations fall short for deeper networks. This prompts the hypothesis that no simple norm may adequately capture the implicit dynamics at play.

Empirical Studies

The paper supports its theoretical claims with extensive empirical evaluations, demonstrating that:

• Shallow vs. Deep Factorizations: When the number of observed entries is inadequate for effective recovery, deeper factorizations consistently outperform the nuclear norm solution. This result is measured through improved performance in matrix completion tasks, highlighting a stronger bias towards low rank with increased depth.
• Dynamical Behavior: The dynamics of singular value evolution during optimization reveal an enhanced attenuation of smaller singular values in deeper factorizations, a behavior not observed in shallow networks. This influences the convergence behavior, resulting in solutions that inherently possess lower effective rank.

Implications and Future Work

The findings suggest important implications for the theoretical understanding and practical application of deep learning models. By shifting the focus from static norm-based models to a dynamical approach, the paper advocates for a more nuanced understanding of the optimization paths taken by gradient descent.

Future directions may include deeper explorations into the specific roles of initialization and optimization dynamics across varying data types and structures. Additionally, extending these insights to non-linear neural networks and uncovering hidden biases in their implicit regularization patterns remains an open and potentially fruitful area of research.

In sum, this paper provides critical insights into how depth influences the implicit regularization in matrix factorization, prompting a reconsideration of the sufficiency of current theoretical models in capturing the complexities of gradient-based optimization in deep learning contexts.