Connectivity Shapes Implicit Regularization in Matrix Factorization Models for Matrix Completion

Matrix factorization models have been extensively studied as a valuable test-bed for understanding the implicit biases of overparameterized models. Although both low nuclear norm and low rank regularization have been studied for these models, a unified understanding of when, how, and why they achieve different implicit regularization effects remains elusive. In this work, we systematically investigate the implicit regularization of matrix factorization for solving matrix completion problems. We empirically discover that the connectivity of observed data plays a crucial role in the implicit bias, with a transition from low nuclear norm to low rank as data shifts from disconnected to connected with increased observations. We identify a hierarchy of intrinsic invariant manifolds in the loss landscape that guide the training trajectory to evolve from low-rank to higher-rank solutions. Based on this finding, we theoretically characterize the training trajectory as following the hierarchical invariant manifold traversal process, generalizing the characterization of Li et al. (2020) to include the disconnected case. Furthermore, we establish conditions that guarantee minimum nuclear norm, closely aligning with our experimental findings, and we provide a dynamics characterization condition for ensuring minimum rank. Our work reveals the intricate interplay between data connectivity, training dynamics, and implicit regularization in matrix factorization models.

Disentangle Sample Size and Initialization Effect on Perfect Generalization for Single-Neuron Target

Overparameterized models like deep neural networks have the intriguing ability to recover target functions with fewer sampled data points than parameters (see arXiv:2307.08921). To gain insights into this phenomenon, we concentrate on a single-neuron target recovery scenario, offering a systematic examination of how initialization and sample size influence the performance of two-layer neural networks. Our experiments reveal that a smaller initialization scale is associated with improved generalization, and we identify a critical quantity called the "initial imbalance ratio" that governs training dynamics and generalization under small initialization, supported by theoretical proofs. Additionally, we empirically delineate two critical thresholds in sample size--termed the "optimistic sample size" and the "separation sample size"--that align with the theoretical frameworks established by (see arXiv:2307.08921 and arXiv:2309.00508). Our results indicate a transition in the model's ability to recover the target function: below the optimistic sample size, recovery is unattainable; at the optimistic sample size, recovery becomes attainable albeit with a set of initialization of zero measure. Upon reaching the separation sample size, the set of initialization that can successfully recover the target function shifts from zero to positive measure. These insights, derived from a simplified context, provide a perspective on the intricate yet decipherable complexities of perfect generalization in overparameterized neural networks.