Implicit Regularization in Tensor Factorization
This work addresses the fundamental problem of explaining generalization in deep learning for researchers, offering a novel theoretical step beyond matrix factorization but still incremental in practical impact.
The authors tackled the problem of understanding implicit regularization in deep learning by analyzing tensor factorization, showing that gradient descent induces a greedy low tensor rank search, which they proved theoretically and demonstrated empirically. They found that tensor rank captures dataset complexity and may explain how implicit regularization leads to generalization in neural networks.
Recent efforts to unravel the mystery of implicit regularization in deep learning have led to a theoretical focus on matrix factorization -- matrix completion via linear neural network. As a step further towards practical deep learning, we provide the first theoretical analysis of implicit regularization in tensor factorization -- tensor completion via certain type of non-linear neural network. We circumvent the notorious difficulty of tensor problems by adopting a dynamical systems perspective, and characterizing the evolution induced by gradient descent. The characterization suggests a form of greedy low tensor rank search, which we rigorously prove under certain conditions, and empirically demonstrate under others. Motivated by tensor rank capturing the implicit regularization of a non-linear neural network, we empirically explore it as a measure of complexity, and find that it captures the essence of datasets on which neural networks generalize. This leads us to believe that tensor rank may pave way to explaining both implicit regularization in deep learning, and the properties of real-world data translating this implicit regularization to generalization.