Training Overparametrized Neural Networks in Sublinear Time

The success of deep learning comes at a tremendous computational and energy cost, and the scalability of training massively overparametrized neural networks is becoming a real barrier to the progress of artificial intelligence (AI). Despite the popularity and low cost-per-iteration of traditional backpropagation via gradient decent, stochastic gradient descent (SGD) has prohibitive convergence rate in non-convex settings, both in theory and practice. To mitigate this cost, recent works have proposed to employ alternative (Newton-type) training methods with much faster convergence rate, albeit with higher cost-per-iteration. For a typical neural network with m=poly(n) parameters and input batch of n datapoints in ℝ^d, the previous work of [Brand, Peng, Song, and Weinstein, ITCS'2021] requires ∼ mnd + n^3 time per iteration. In this paper, we present a novel training method that requires only m^1-α n d + n^3 amortized time in the same overparametrized regime, where α∈ (0.01,1) is some fixed constant. This method relies on a new and alternative view of neural networks, as a set of binary search trees, where each iteration corresponds to modifying a small subset of the nodes in the tree. We believe this view would have further applications in the design and analysis of deep neural networks (DNNs).