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 second-order optimization algorithms beyond SGD, without compromising the generalization error. Despite their remarkable convergence rate (independent of the training batch size n), second-order algorithms incur a daunting slowdown in the cost per iteration (inverting the Hessian matrix of the loss function), which renders them impractical. Very recently, this computational overhead was mitigated by the works of [Zhang et al., 2019; Cai et al., 2019], yielding an O(mn²)-time second-order algorithm for training two-layer overparametrized neural networks of polynomial width m.

We show how to speed up the algorithm of [Cai et al., 2019], achieving an Õ(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 𝓁₂-regression problem, and then use a Fast-JL type dimension reduction to precondition the underlying Gram matrix in time independent of M, allowing to find a sufficiently good approximate solution via first-order conjugate gradient. Our result provides a proof-of-concept that advanced machinery from randomized linear algebra - which led to recent breakthroughs in convex optimization (ERM, LPs, Regression) - can be carried over to the realm of deep learning as well.