Junk-neuron-deletion strategy for hyperparameter optimization of neural networks

Huang Ying; Gu Chang-Gui; Yang Hui-Jie

doi:10.7498/aps.71.20220436

Abstract

With the complexity of problems in reality increasing, the sizes of deep learning neural networks, including the number of layers, neurons, and connections, are increasing in an explosive way. Optimizing hyperparameters to improve the prediction performance of neural networks has become an important task. In literatures, the methods of finding optimal parameters, such as sensitivity pruning and grid search, are complicated and cost a large amount of computation time. In this paper, a hyperparameter optimization strategy called junk neuron deletion is proposed. A neuron with small mean weight in the weight matrix can be ignored in the prediction, and is defined subsequently as a junk neuron. This strategy is to obtain a simplified network structure by deleting the junk neurons, to effectively shorten the computation time and improve the prediction accuracy and model the generalization capability. The LSTM model is used to train the time series data generated by Logistic, Henon and Rossler dynamical systems, and the relatively optimal parameter combination is obtained by grid search with a certain step length. The partial weight matrix that can influence the model output is extracted under this parameter combination, and the neurons with smaller mean weights are eliminated with different thresholds. It is found that using the weighted mean value of 0.1 as the threshold, the identification and deletion of junk neurons can significantly improve the prediction efficiency. Increasing the threshold accuracy will gradually fall back to the initial level, but with the same prediction effect, more operating costs will be saved. Further reduction will result in prediction ability lower than the initial level due to lack of fitting. Using this strategy, the prediction performance of LSTM model for several typical chaotic dynamical systems is improved significantly.

Keywords:

Authors and contacts

Business School, University of Shanghai for Science and Technology, Shanghai 200093, China

Corresponding author: Yang Hui-Jie, hjyang@usst.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11875042, 11505114).

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References

[1]	邓帅 2019 计算机应用研究 36 1984 Deng S 2019 Appl. Res. Comput. 36 1984
[2]	邵恩泽, 吴正勇, 王灿 2020 工业控制计算机 33 11 Google Scholar Shao E Z, Wu Z Y, Wang C 2020 Ind. Contrl. Comput. 33 11 Google Scholar
[3]	乔俊飞, 樊瑞元, 韩红桂, 阮晓钢 2010 控制理论与应用 27 111 Qiao J F, Fan R Y, Han H G, Ruan X G 2010 Contl. Theor. Appl. 27 111
[4]	陈国茗, 于腾腾, 刘新为 2021 数值计算与计算机应用 42 215 Google Scholar Chen G M, Yu T T, Liu X W 2021 J. Num. Method. Comp. Appl. 42 215 Google Scholar
[5]	魏德志, 陈福集, 郑小雪 2015 物理学报 64 110503 Google Scholar Wei D Z, Chen F J, Zheng X X 2015 Acta Phys. Sin. 64 110503 Google Scholar
[6]	王新迎, 韩敏 2015 物理学报 64 070504 Google Scholar Wang X Y, Han M 2015 Acta Phys. Sin 64 070504 Google Scholar
[7]	黄伟建, 李永涛, 黄远 2021 物理学报 70 010501 Google Scholar Huang W J, Li Y T, Huang Y 2021 Acta Phys. Sin. 70 010501 Google Scholar
[8]	Yamaguti Y, Tsuda I 2021 Chaos 31 013137 Google Scholar
[9]	Graves A 2013 arXiv: 1308.0850 [cs. NE]
[10]	Johnston D E 1978 Proc 8 th BHRA Int Conf Fluid Sealing Durham, UK, 1978 pC1-1
[11]	Sezer O B, Gudelek M U, Ozbayoglu A M 2020 Appl. Soft Comput. J. 90 106181 Google Scholar
[12]	甘文娟, 陈永红, 韩静, 王亚飞 2020 计算机系统应用 29 212 Google Scholar Gan W J, Chen Y H, Han J, Wang Y F 2020 Comput. Syst. Appl. 29 212 Google Scholar
[13]	Farmelo G 2002 It Must Be Beautiful: Great Equations of Modern Science (London: Granta Publications) pp28–45
[14]	Grassberger P, Procaccia I 1983 Physica D 9 189 Google Scholar
[15]	Nauenberg M 1983 Ann. N. Y. Acad. Sci. 410 317 Google Scholar
[16]	张中华, 丁华福 2009 计算机技术与发展 19 185 Google Scholar Zhang Z H, Ding H F 2009 Comput. Technol. Dev. 19 185 Google Scholar
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[18]	Liu C, Yin S Q, Zhang M, Zeng Y, Liu J Y 2014 Appl. Mech. Mater. 644-650 2216
[19]	Bao Y K, Liu Z T 2006 LNCS 4224 504
[20]	Ou Y Y, Chen G H, Oyang Y J 2006 LNCS 4099 1017

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图 1 LSTM神经网络　(a) LSTM模型网络结构; (b)单元内部运行逻辑

Figure 1. LSTM neural network: (a) network structure of LSTM; (b) run logic inside the cell.

DownLoad: Full-Size Img PowerPoint

图 2 混沌时间序列　(a) Logistic系统, μ$ =3.9 $; (b) Henon系统; (c) Rossler系统

Figure 2. Chaotic time series: (a) Logistic system, μ$ =3.9 $; (b) Henon system; (c) Rossler system

DownLoad: Full-Size Img PowerPoint

图 3 隐藏层输入神经元对输出神经元的权重热图

Figure 3. Heat map of weight of input neuron to output neuron in hidden layer.

DownLoad: Full-Size Img PowerPoint

图 4 隐藏层各输入神经元对输出神经元的权重均值热图

Figure 4. Heat map of weights’ mean value of input neurons to output neurons in hidden layer.

DownLoad: Full-Size Img PowerPoint

图 5 权重均值折线图

Figure 5. Line graph of the weights’ mean.

DownLoad: Full-Size Img PowerPoint

图 6 不同混沌状态对应的最优权重阈值变化

Figure 6. The change of optimal weight threshold corresponding to different chaotic states.

DownLoad: Full-Size Img PowerPoint

图 7 不同混沌状态对应的预测准确率变化

Figure 7. The change of prediction accuracy of different chaotic states.

DownLoad: Full-Size Img PowerPoint

表 1 模型参数及结果

Table 1. Parameters and results of the models.

模型	train	test	win	L₁	L₂	D₁	D₂	准确率
Logistic (μ = 3.6)	5000	15	22	16	4	4	1	82.90％
Logistic (μ = 3.7)	5000	15	22	16	4	4	1	70.70％
Logistic (μ = 3.8)	5000	15	2	20	4	4	1	68.60％
Logistic (μ = 3.9)	5000	15	2	16	4	2	1	60.00％
Logistic (μ = 3.99)	5000	15	2	16	4	4	1	57.10％
Henon	5000	15	2	22	2	2	1	67.10％
Rossler	5000	15	2	16	4	4	1	77.10％

DownLoad: CSV

表 2 权重结构拆分

Table 2. Weight structure resolution.

结构	起始点	终点
input_gate	0	units
forget_gate	units	2×units
cell	2×units	3×units
output_gate	3×units	4×units

DownLoad: CSV

表 3 输出门权重矩阵图

Table 3. Heat diagram of output door’s weights.

	w1	w2	w3	w4	w5	w6	w7	w8	w9	w10	w11	w12	w13	w14	w15	w16
x1	0.032129	0.144328	0.007293	0.003557	0.070465	0.011652	0.11142	0.048287	0.105727	0.039649	0.010162	0.119949	0.0771	0.126937	0.001947	0.089396
x2	0.070867	0.013232	0.054503	0.073496	0.005734	0.064071	0.145835	0.083217	0.025553	0.033065	0.113584	0.028327	0.054085	0.177761	0.010123	0.046546
x3	0.083647	0.046366	0.066226	0.044534	0.010054	0.077919	0.071001	0.018548	0.035294	0.071642	0.053888	0.028448	0.102273	0.096744	0.078535	0.114879
x4	0.039123	0.113002	0.161581	0.113866	0.070651	0.085571	0.01836	0.015408	0.032978	0.018375	0.068342	0.059137	0.00701	0.070451	0.032602	0.099853
x5	0.013762	0.055823	0.147481	0.055493	0.093444	0.027264	0.082384	0.058674	0.032903	0.033342	0.045937	0.035937	0.110378	0.004487	0.161653	0.041037
x6	0.038079	0.089649	0.013102	0.021696	0.042833	0.109787	0.001024	0.0673	0.036916	0.134038	0.066291	0.146953	0.009803	0.081372	0.098701	0.042456
x7	0.013865	0.001702	0.057869	0.072264	0.104456	0.029761	0.062669	0.07539	0.05715	0.102282	0.017876	0.012626	0.020022	0.100243	0.153076	0.11875
x8	0.002042	0.006925	0.060008	0.13031	0.136406	0.056203	0.061606	0.028064	0.006926	0.099129	0.055122	0.117276	0.06846	0.014505	0.078184	0.078834
x9	0.044205	0.171724	0.153162	0.13818	0.029189	0.025947	0.049391	0.012338	0.100584	0.028133	0.004946	0.017914	0.008463	0.014741	0.066944	0.134139
x10	0.094619	0.0563	0.040223	0.096283	0.10152	0.145036	0.051991	0.075623	0.075216	0.061209	0.057986	0.066076	0.004787	0.01945	0.042341	0.011339
x11	0.078909	0.005603	0.08149	0.125202	0.069081	0.10143	0.122451	0.027058	0.057647	0.016226	0.03275	0.050667	0.036795	0.11072	0.081767	0.002204
x12	0.046159	0.005688	0.006237	0.004618	0.014815	0.005272	0.04598	0.037005	0.190933	0.065535	0.005131	0.015155	0.065812	0.099804	0.172294	0.21956
x13	0.111226	0.027026	0.027497	0.074868	0.139154	0.084413	0.080342	0.038769	0.088824	0.047083	0.056548	0.002081	0.10549	0.049929	0.020529	0.04622
x14	0.092772	0.03999	0.055938	0.128114	0.036386	0.013061	0.083943	0.051033	0.106374	0.007257	0.063049	0.091929	0.084821	0.020458	0.089496	0.035379
x15	0.131586	0.038161	0.176303	0.041758	0.049173	0.096633	0.0033	0.045529	0.084262	0.050839	0.003322	0.063406	0.029601	0.045323	0.116047	0.024757
x16	0.070289	0.05378	0.092472	0.03372	0.052087	0.110236	0.055639	0.124221	0.029371	0.06142	0.04904	0.043376	0.012261	0.041226	0.109564	0.061299
均值	0.060205	0.054331	0.075087	0.072372	0.064091	0.065266	0.065459	0.050404	0.066666	0.054326	0.043998	0.056204	0.049428	0.067134	0.082113	0.072915

DownLoad: CSV

表 4 μ = 3.99时不同参数的预测准确率

Table 4. The prediction accuracy of different parameters when μ = 3.99.

指标	调整前	以各权重阀值调整后			变化趋势
指标	调整前	0.09	0.1	0.11	变化趋势
L₁	16	15	12	10
准确率	57.10％	59.30％	56.40％	51.40％

DownLoad: CSV

表 5 神经元数及预测准确率变化表

Table 5. Table of neuron numbers and prediction accuracy.

模型	调整前L₁	调整后L₁	调整前准确率	调整后准确率	神经元数调整	准确率变化趋势
Logistic (μ = 3.6)	16	15	82.90％	90.70％	–1
Logistic (μ = 3.7)	16	13	70.70％	71.40％	–3
Logistic (μ = 3.8)	20	16	68.60％	68.60％	–4
Logistic (μ = 3.9)	16	12	60.00％	60.00％	–4
Logistic (μ = 3.99)	16	15	57.10％	59.30％	–1
Henon	22	21	67.10％	70.00％	–1
Rossler	16	14	77.10％	83.60％	–2

DownLoad: CSV

表 6 μ = 3.6 时不同参数的预测准确率

Table 6. The prediction accuracy of different parameters when μ = 3.6.

指标	调整前	以各权重阀值调整后			变化趋势
指标	调整前	0.08	0.09	0.095	变化趋势
L₁	16	15	12	10
准确率	82.90％	90.70％	87.90％	78.60％

DownLoad: CSV

表 7 μ = 3.7 时不同参数的预测准确率

Table 7. The prediction accuracy of different parameters when μ = 3.7.

指标	调整前	以各权重阀值调整后			变化趋势
指标	调整前	0.075	0.09	0.105	变化趋势
L₁	16	13	11	9
准确率	70.70％	71.40％	65.00％	60.70％

DownLoad: CSV

表 8 μ = 3.8时不同参数的预测准确率

Table 8. The prediction accuracy of different parameters when μ = 3.8.

指标	调整前	以各权重阀值调整后			变化趋势
指标	调整前	0.085	0.095	0.105	变化趋势
L₁	20	18	16	12
准确率	68.60％	68.60％	68.60％	65.00％

DownLoad: CSV

表 9 μ = 3.9 时不同参数的预测准确率

Table 9. The prediction accuracy of different parameters when μ = 3.9.

指标	调整前	以各权重阀值调整后			变化趋势
指标	调整前	0.09	0.095	0.1	变化趋势
L₁	16	14	12	10
准确率	60.00％	60.00％	60.00％	55.00％

DownLoad: CSV

表 10 Henon系统取不同参数的预测准确率

Table 10. Prediction accuracy of Henon system for different parameters.

指标	调整前	以各权重阀值调整后				变化趋势
指标	调整前	0.1	0.11	0.12	0.14	变化趋势
L₁	22	21	19	16	14
准确率	67.10％	70.00％	66.40％	65.70％	65.00％

DownLoad: CSV

表 11 Rossler系统取不同参数的预测准确率

Table 11. Prediction accuracy of Rossler system for different parameters.

指标	调整前	以各权重阀值调整后				变化趋势
指标	调整前	0.085	0.095	0.105	0.115	变化趋势
L₁	16	14	12	11	8
准确率	77.10％	83.60％	81.40％	80.70％	71.40％

DownLoad: CSV

[1]	邓帅 2019 计算机应用研究 36 1984 Deng S 2019 Appl. Res. Comput. 36 1984
[2]	邵恩泽, 吴正勇, 王灿 2020 工业控制计算机 33 11 Google Scholar Shao E Z, Wu Z Y, Wang C 2020 Ind. Contrl. Comput. 33 11 Google Scholar
[3]	乔俊飞, 樊瑞元, 韩红桂, 阮晓钢 2010 控制理论与应用 27 111 Qiao J F, Fan R Y, Han H G, Ruan X G 2010 Contl. Theor. Appl. 27 111
[4]	陈国茗, 于腾腾, 刘新为 2021 数值计算与计算机应用 42 215 Google Scholar Chen G M, Yu T T, Liu X W 2021 J. Num. Method. Comp. Appl. 42 215 Google Scholar
[5]	魏德志, 陈福集, 郑小雪 2015 物理学报 64 110503 Google Scholar Wei D Z, Chen F J, Zheng X X 2015 Acta Phys. Sin. 64 110503 Google Scholar
[6]	王新迎, 韩敏 2015 物理学报 64 070504 Google Scholar Wang X Y, Han M 2015 Acta Phys. Sin 64 070504 Google Scholar
[7]	黄伟建, 李永涛, 黄远 2021 物理学报 70 010501 Google Scholar Huang W J, Li Y T, Huang Y 2021 Acta Phys. Sin. 70 010501 Google Scholar
[8]	Yamaguti Y, Tsuda I 2021 Chaos 31 013137 Google Scholar
[9]	Graves A 2013 arXiv: 1308.0850 [cs. NE]
[10]	Johnston D E 1978 Proc 8 th BHRA Int Conf Fluid Sealing Durham, UK, 1978 pC1-1
[11]	Sezer O B, Gudelek M U, Ozbayoglu A M 2020 Appl. Soft Comput. J. 90 106181 Google Scholar
[12]	甘文娟, 陈永红, 韩静, 王亚飞 2020 计算机系统应用 29 212 Google Scholar Gan W J, Chen Y H, Han J, Wang Y F 2020 Comput. Syst. Appl. 29 212 Google Scholar
[13]	Farmelo G 2002 It Must Be Beautiful: Great Equations of Modern Science (London: Granta Publications) pp28–45
[14]	Grassberger P, Procaccia I 1983 Physica D 9 189 Google Scholar
[15]	Nauenberg M 1983 Ann. N. Y. Acad. Sci. 410 317 Google Scholar
[16]	张中华, 丁华福 2009 计算机技术与发展 19 185 Google Scholar Zhang Z H, Ding H F 2009 Comput. Technol. Dev. 19 185 Google Scholar
[17]	Butcher J C 1967 J. ACM 14 84 Google Scholar
[18]	Liu C, Yin S Q, Zhang M, Zeng Y, Liu J Y 2014 Appl. Mech. Mater. 644-650 2216
[19]	Bao Y K, Liu Z T 2006 LNCS 4224 504
[20]	Ou Y Y, Chen G H, Oyang Y J 2006 LNCS 4099 1017

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