Beyond Lazy Training for Over-parameterized Tensor Decomposition
This addresses a theoretical bottleneck in optimization for machine learning, offering insights into how over-parametrization can exploit low-rank structures beyond lazy training regimes.
The paper tackles the problem of tensor decomposition via over-parametrized gradient descent, showing that lazy training requires at least m = Ω(d^{l-1}) parameters, while a variant achieves approximate decomposition with m = O^*(r^{2.5l} log d).
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 $m = Ω(d^{l-1})$, 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.