Training (Overparametrized) Neural Networks in Near-Linear Time

• URL: http://arxiv.org/abs/2006.11648v2
• Date: Wed, 9 Dec 2020 02:02:10 GMT
• Title: Training (Overparametrized) Neural Networks in Near-Linear Time
• Authors: Jan van den Brand, Binghui Peng, Zhao Song, Omri Weinstein
• Abstract summary: We show how to speed up the algorithm of [CGH+1] for training (mildly overetrized) ReparamLU networks. The centerpiece of our algorithm is to reformulate the Gauss-Newton as an $ell$-recondition.
• Score: 21.616949485102342
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: The slow convergence rate and pathological curvature issues of first-order gradient methods for training deep neural networks, initiated an ongoing effort for developing faster $\mathit{second}$-$\mathit{order}$ optimization algorithms beyond SGD, without compromising the generalization error. Despite their remarkable convergence rate ($\mathit{independent}$ of the training batch size $n$), second-order algorithms incur a daunting slowdown in the $\mathit{cost}$ $\mathit{per}$ $\mathit{iteration}$ (inverting the Hessian matrix of the loss function), which renders them impractical. Very recently, this computational overhead was mitigated by the works of [ZMG19,CGH+19}, yielding an $O(mn^2)$-time second-order algorithm for training two-layer overparametrized neural networks of polynomial width $m$. We show how to speed up the algorithm of [CGH+19], achieving an $\tilde{O}(mn)$-time backpropagation algorithm for training (mildly overparametrized) ReLU networks, which is near-linear in the dimension ($mn$) of the full gradient (Jacobian) matrix. The centerpiece of our algorithm is to reformulate the Gauss-Newton iteration as an $\ell_2$-regression problem, and then use a Fast-JL type dimension reduction to $\mathit{precondition}$ the underlying Gram matrix in time independent of $M$, allowing to find a sufficiently good approximate solution via $\mathit{first}$-$\mathit{order}$ conjugate gradient. Our result provides a proof-of-concept that advanced machinery from randomized linear algebra -- which led to recent breakthroughs in $\mathit{convex}$ $\mathit{optimization}$ (ERM, LPs, Regression) -- can be carried over to the realm of deep learning as well.

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