Linear networks provide valuable insight into the workings of neural networks in general.
In this paper, we improve the state of the art in (Bah et al., 2019) by identifying conditions under which gradient flow successfully trains a linear network, in spite of the non-strict saddle points present in the optimization landscape.
We also improve the state of the art for computational complexity of training linear networks in (Arora et al., 2018a) by establishing non-local linear convergence rates for gradient flow.
Crucially, these new results are not in the lazy training regime, cautioned against in (Chizat et al., 2019; Yehudai & Shamir, 2019).
Our results require the network to have a layer with one neuron, which corresponds to the popular spiked covariance model in statistics, and subsumes the important case of networks with a scalar output. Extending these results to all linear networks remains an open problem.