Implicit Regularization in Deep Matrix Factorization

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.Comment: Published at the conference on Neural Information Processing Systems (NeurIPS) 201

Implicit Regularization in Hierarchical Tensor Factorization and Deep Convolutional Neural Networks

In the pursuit of explaining implicit regularization in deep learning, prominent focus was given to matrix and tensor factorizations, which correspond to simplified neural networks. It was shown that these models exhibit an implicit tendency towards low matrix and tensor ranks, respectively. Drawing closer to practical deep learning, the current paper theoretically analyzes the implicit regularization in hierarchical tensor factorization, a model equivalent to certain deep convolutional neural networks. Through a dynamical systems lens, we overcome challenges associated with hierarchy, and establish implicit regularization towards low hierarchical tensor rank. This translates to an implicit regularization towards locality for the associated convolutional networks. Inspired by our theory, we design explicit regularization discouraging locality, and demonstrate its ability to improve the performance of modern convolutional networks on non-local tasks, in defiance of conventional wisdom by which architectural changes are needed. Our work highlights the potential of enhancing neural networks via theoretical analysis of their implicit regularization.Comment: Accepted to ICML 202

Regularizing Matrix Factorization with User and Item Embeddings for Recommendation

Following recent successes in exploiting both latent factor and word embedding models in recommendation, we propose a novel Regularized Multi-Embedding (RME) based recommendation model that simultaneously encapsulates the following ideas via decomposition: (1) which items a user likes, (2) which two users co-like the same items, (3) which two items users often co-liked, and (4) which two items users often co-disliked. In experimental validation, the RME outperforms competing state-of-the-art models in both explicit and implicit feedback datasets, significantly improving Recall@5 by 5.9~7.0%, NDCG@20 by 4.3~5.6%, and MAP@10 by 7.9~8.9%. In addition, under the cold-start scenario for users with the lowest number of interactions, against the competing models, the RME outperforms NDCG@5 by 20.2% and 29.4% in MovieLens-10M and MovieLens-20M datasets, respectively. Our datasets and source code are available at: https://github.com/thanhdtran/RME.git.Comment: CIKM 201