Width Provably Matters in Optimization for Deep Linear Neural Networks

We prove that for an L-layer fully-connected linear neural network, if the width of every hidden layer is Ω(L · r · dₒᵤₜ · κ³ ), where r and $κ$ are the rank and the condition number of the input data, and dₒᵤₜ is the output dimension, then gradient descent with Gaussian random initialization converges to a global minimum at a linear rate. The number of iterations to find an $ε$-suboptimal solution is O(κlog(1ε)). Our polynomial upper bound on the total running time for wide deep linear networks and the exp(Ω(L)) lower bound for narrow deep linear neural networks [Shamir, 2018] together demonstrate that wide layers are necessary for optimizing deep models.