Implicit Regularization in Hierarchical Tensor Factorization and Deep Convolutional Neural Networks
This work addresses the challenge of understanding and enhancing neural networks through theoretical insights into implicit regularization, offering a method to improve performance on non-local tasks without architectural changes.
The paper tackles the problem of explaining implicit regularization in deep learning by theoretically analyzing hierarchical tensor factorization, which is equivalent to certain deep convolutional neural networks, and establishes that it leads to low hierarchical tensor rank and locality. Inspired by this, they design explicit regularization to discourage locality and demonstrate performance improvements on non-local tasks for modern convolutional networks, with specific gains reported.
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.