Width Provably Matters in Optimization for Deep Linear Neural Networks
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We prove that for an L-layer fully-connected linear neural network, if the width of every hidden layer is Ω̃ (L · r · d_out·κ^3 ), where r and κ are the rank and the condition number of the input data, and d_out 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(κ(1/ϵ)). Our polynomial upper bound on the total running time for wide deep linear networks and the (Ω(L)) lower bound for narrow deep linear neural networks [Shamir, 2018] together demonstrate that wide layers are necessary for optimizing deep models.
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