A new initialization method based on normed statistical spaces in deep networks

Hongfei Yang; Xiaofeng Ding; Raymond Chan; Hui Hu; Yaxin Peng; Tieyong Zeng

doi:10.3934/ipi.2020045

Article Contents

2021, Volume 15, Issue 1: 147-158. Doi: 10.3934/ipi.2020045

This issue Previous Article Adversarial defense via the data-dependent activation, total variation minimization, and adversarial training Next Article Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory

A new initialization method based on normed statistical spaces in deep networks

1.
Department of Mathematics, Yeung Kin Man Academic Building, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong, China
2.
Department of Mathematics, School of Science, Shanghai University, Shanghai 200444, China
3.
HISILICON Technologies Co., Ltd., Huawei Base, Bantian, Longgang District, Shenzhen 518129, China
4.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China

^*Corresponding author: Tieyong Zeng (zeng@math.cuhk.edu.hk)
^*Corresponding author: Tieyong Zeng (zeng@math.cuhk.edu.hk)

Received: November 2019

Revised: April 2020

Early access: August 2020

Published: February 2021

Raymond Chan's research is supported by HKRGC Grants No. CUHK 14306316 and CUHK 14301718, CityU Grant 9380101, CRF Grant C1007-15G, AoE/M-05/12. Tieyong Zeng's research is supported by National Science Foundation of China No. 11671002, CUHK start-up and CUHK DAG 4053342, RGC 14300219, and NSFC/RGC N_CUHK 415/19

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Abstract

Training deep neural networks can be difficult. For classical neural networks, the initialization method by Xavier and Yoshua which is later generalized by He, Zhang, Ren and Sun can facilitate stable training. However, with the recent development of new layer types, we find that the above mentioned initialization methods may fail to lead to successful training. Based on these two methods, we will propose a new initialization by studying the parameter space of a network. Our principal is to put constrains on the growth of parameters in different layers in a consistent way. In order to do so, we introduce a norm to the parameter space and use this norm to measure the growth of parameters. Our new method is suitable for a wide range of layer types, especially for layers with parameter-sharing weight matrices.

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Mathematics Subject Classification: Primary: 68T01, 68T05; Secondary: 68Q32.

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Figure 1. (a) Plot of losses of network summarized in Table 1. (b) Plot of losses of network summarized in Table 2. (c) Plot of losses of network summarized in Table 3. (d) Plot of losses of network summarized in Table 4. Mean and std for the last of the smoothed loss values: Ours (a) $ 0.070\pm 0.005 $, (b) $ 0.111\pm 0.006 $, (c) $ 0.088\pm 0.003 $, (d) $ 0.083\pm 0.004 $; Xavier/He (a) $ 0.069\pm 0.001 $, (b) $ 0.206\pm 0.012 $, (c) $ 0.221\pm 0.016 $, (d) $ 0.164\pm 0.012 $. We also tested the evaluation accuracies on the test set with results: Ours versus Xavier/He (a) $ 98.06\% $, $ 98.13\% $, (b) $ 95.66\% $, $ 93.94\% $, (c) $ 97.17\% $, $ 95.07\% $, (d) $ 98.01\% $, $ 96.22\% $

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Table 2. Network structure of Figure 1(b). For the last convolution layer with kernel size $ 55\times 55 $ we use periodic padding on the input images to make sure the conditions on $ T $ in (6) are satisfied

Layer	Output channel	Number of parameters
Conv2d+Maxpool	$ 32 $	$ 3\times 3 \times 1 \times 32 $
Conv2d+Maxpool	$ 64 $	$ 3\times 3 \times 32\times 64 $
Reshape to $ 56\times 56 $
Conv2d	$ 1 $	$ 55\times 55 \times 1\times 1 $
Fc	$ 10 $	$ 3136\times 10 $

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Table 1. Network structure of Figure 1(a)

Layer	Output channel	Number of parameters
Conv2d+MaxPool	$ 32 $	$ 3\times 3 \times 1 \times 32 $
Conv2d+MaxPool	$ 64 $	$ 3\times 3 \times 32\times 64 $
Fc	$ 64 $	$ 3136\times 64 $
Fc	$ 10 $	$ 64\times 10 $

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Table 3. Network structure of Figure 1(c). For a compression ratio $ B> 1 $, we use circulant implementation with block size $ B $. The number of parameters for a CirCNN implementation should be divided by $ B $

Layer	Out channel	Number of parameters	Compression ratio
Conv2d+Maxpool	$ 32 $	$ 3\times 3 \times 1 \times 32 $	$ 1 $
Conv2d+Maxpool	$ 64 $	$ 3\times 3 \times 32\times 64 $	$ 1 $
Fc	$ 1568 $	$ 3136\times 1568 $	$ 1568 $
Fc	$ 10 $	$ 1568\times 10 $	$ 1 $

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Table 4. Network structure of Figure 1(d). For a compression ratio $ B> 1 $, we use circulant implementation with block size $ B $. The number of parameters for a CirCNN implementation should be divided by $ B $

Layer	Out channel	Number of parameters	Compression ratio
Conv2d+Maxpool	$ 32 $	$ 3\times 3 \times 1 \times 32 $	$ 1 $
Conv2d+Maxpool	$ 64 $	$ 3\times 3 \times 32\times 64 $	$ 1 $
Conv2d	$ 256 $	$ 3\times 3 \times 64 \times 256 $	$ 1 $
Conv2d	$ 256 $	$ 3\times 3 \times 256 \times 256 $	$ 256 $
Conv2d	$ 256 $	$ 3\times 3 \times 256 \times 256 $	$ 256 $
Fc	$ 64 $	$ 12544\times 64 $	$ 1 $
Fc	$ 10 $	$ 64\times 10 $	$ 1 $

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