Abstract
In this paper, we develop an efficient sketch-based empirical natural gradient method (SENG) for large-scale deep learning problems. The empirical Fisher information matrix is usually low-rank since the sampling is only practical on a small amount of data at each iteration. Although the corresponding natural gradient direction lies in a small subspace, both the computational cost and memory requirement are still not tractable due to the high dimensionality. We design randomized techniques for different neural network structures to resolve these challenges. For layers with a reasonable dimension, sketching can be performed on a regularized least squares subproblem. Otherwise, since the gradient is a vectorization of the product between two matrices, we apply sketching on the low-rank approximations of these matrices to compute the most expensive parts. A distributed version of SENG is also developed for extremely large-scale applications. Global convergence to stationary points is established under mild assumptions and a fast linear convergence is analyzed under the neural tangent kernel (NTK) case. Extensive experiments on convolutional neural networks show the competitiveness of SENG compared with the state-of-the-art methods. On the task ResNet50 with ImageNet-1k, SENG achieves 75.9% Top-1 testing accuracy within 41 epochs. Experiments on the distributed large-batch training Resnet50 with ImageNet-1k show that the scaling efficiency is quite reasonable.
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Data Availability
Enquiries about data availability should be directed to the authors.
Notes
When \(\kappa \) is large enough, the approximation (3.10) can be obtained by computing a partial SVD of \({{\hat{G}}}_i\) or \({{\hat{A}}}_i\) regarding to their sizes.
For simplicity, we consider one-dimension output and fix the second layer. However, it is easy to extend the analysis to the multi-dimensional output or the case for jointly training both layers.
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Acknowledgements
The authors are grateful to the AE and two anonymous referees for their valuable comments and suggestions.
Funding
M. Yang, D. Xu and Z. Wen are supported in part by Key-Area Research and Development Program of Guangdong Province (No.2019B121204008), the NSFC grants 11831002 and Beijing Academy of Artificial Intelligence.
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Yang, M., Xu, D., Wen, Z. et al. Sketch-Based Empirical Natural Gradient Methods for Deep Learning. J Sci Comput 92, 94 (2022). https://doi.org/10.1007/s10915-022-01911-x
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DOI: https://doi.org/10.1007/s10915-022-01911-x