Does Preprocessing Help Training Over-parameterized Neural Networks?
This work addresses the fundamental challenge of efficient training for deep neural networks, offering theoretical speedups that could benefit machine learning practitioners dealing with large-scale models.
The paper tackles the problem of reducing the computational cost of training over-parameterized neural networks by proposing two preprocessing methods: one on initial weights to achieve cost per iteration of O~(m^(1-Θ(1/d)) n d) and another on input data to achieve O~(m^(4/5) n d), bypassing the classical Ω(m n d) barrier.
Deep neural networks have achieved impressive performance in many areas. Designing a fast and provable method for training neural networks is a fundamental question in machine learning. The classical training method requires paying $Ω(mnd)$ cost for both forward computation and backward computation, where $m$ is the width of the neural network, and we are given $n$ training points in $d$-dimensional space. In this paper, we propose two novel preprocessing ideas to bypass this $Ω(mnd)$ barrier: $\bullet$ First, by preprocessing the initial weights of the neural networks, we can train the neural network in $\widetilde{O}(m^{1-Θ(1/d)} n d)$ cost per iteration. $\bullet$ Second, by preprocessing the input data points, we can train the neural network in $\widetilde{O} (m^{4/5} nd )$ cost per iteration. From the technical perspective, our result is a sophisticated combination of tools in different fields, greedy-type convergence analysis in optimization, sparsity observation in practical work, high-dimensional geometric search in data structure, concentration and anti-concentration in probability. Our results also provide theoretical insights for a large number of previously established fast training methods. In addition, our classical algorithm can be generalized to the Quantum computation model. Interestingly, we can get a similar sublinear cost per iteration but avoid preprocessing initial weights or input data points.