Beyond Lazy Training for Over-parameterized Tensor Decomposition

• URL: http://arxiv.org/abs/2010.11356v1
• Date: Thu, 22 Oct 2020 00:32:12 GMT
• Title: Beyond Lazy Training for Over-parameterized Tensor Decomposition
• Authors: Xiang Wang, Chenwei Wu, Jason D. Lee, Tengyu Ma, Rong Ge
• Abstract summary: We show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data. 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.
• Score: 69.4699995828506
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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