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Training (Overparametrized) Neural Networks in Near-Linear Time

Authors Jan van den Brand, Binghui Peng, Zhao Song, Omri Weinstein

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Author Details

Jan van den Brand
  • KTH Royal Institute of Technology, Stockholm, Sweden
Binghui Peng
  • Columbia University, New York, NY, USA
Zhao Song
  • Princeton University and Institute for Advanced Study, NJ, USA
Omri Weinstein
  • Columbia University, New York, NY, USA

Funding

  • van den Brand, Jan: This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program under grant agreement No 715672. Partially supported by the Google PhD Fellowship Program.
  • Peng, Binghui: Research supported by NSF IIS-1838154, NSF CCF-1703925 and NSF CCF-1763970.
  • Song, Zhao: Research supported by Special Year on Optimization, Statistics, and Theoretical Machine Learning (being led by Sanjeev Arora) at Institute for Advanced Study.
  • Weinstein, Omri: Research supported by NSF CAREER award CCF-1844887.

Acknowledgements

The author would like to thank David Woodruff for telling us the tensor trick for computing kernel matrices and helping us improve the presentation of the paper.

Cite AsGet BibTex

Jan van den Brand, Binghui Peng, Zhao Song, and Omri Weinstein. Training (Overparametrized) Neural Networks in Near-Linear Time. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.63

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Nonconvex optimization
Keywords
  • Deep learning theory
  • Nonconvex optimization

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