Beyond Lazy Training for Over-parameterized Tensor Decomposition

Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an $l$-th order tensor in $(R^d)^{\otimes l}$ of rank $r$ (where $r\ll d$), can variants of gradient descent find a rank $m$ decomposition where $m > r$? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least $m = \Omega(d^{l-1})$, while a variant of gradient descent can find an approximate tensor when $m = O^*(r^{2. 5l}\log d)$. Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.

Authors

• Xiang Wang
• Chenwei Wu University of Michigan - Ann Arbor
• Jason D. Lee
• Tengyu Ma Dalian University of Technology
• Rong Ge

Keywords

No keywords are indexed for this paper.