Title: Training Overparametrized Neural Networks in Sublinear Time
Primary area: learning theory
Keywords: neural network training, theory, overparametrized neural networks

Abstract: 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 neural network with $m=\mathrm{poly}(n)$ parameters and input batch of  $n$  datapoints in $\mathbb{R}^d$, the previous work of [Brand, Peng, Song and Weinstein, ITCS 2021] requires $\sim mnd + n^3$ time per iteration. In this paper, we present a novel training method that requires only $m^{1-\alpha} n d + n^3$ amortized time in the same overparametrized regime, where $\alpha \in (0.01,1)$ 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). Finally, we give a lower bound for the dynamic sensitive weight searching data structure we make use of, showing that under ${\sf SETH}$ or ${\sf OVC}$ from fine-grained complexity, one cannot substantially improve our algorithm.