前向人工神经网络敏感性研究

1、1前向人工神经网络敏感性研究曾晓勤曾晓勤河海大学计算机及信息工程学院2003年10月 2一一. . 引言引言 1. 前向神经网络前向神经网络( (FNN)FNN)介绍介绍 神经元 离散型：自适应线性元（Adaline） 连续型：感知机（Perceptron） 神经网络 离散型：多层自适应线性网（Madaline） 连续型：多层感知机（BP网或MLP）3问题问题 硬件精度对权的影响 环境噪音对输入的影响 动机动机 参数的扰动对网络会产生怎样影响？ 如何衡量网络输出偏差的大小？2. 研究提出研究提出4建立网络输出与网络参数扰动之间的关系分析该关系，揭示网络的行为规律量化网络输出偏差3. 研究内容研

2、究内容),(WXSY),(WXSY),(WXSY5指导网络设计，增强网络抗干扰能力度量网络性能，如容错和泛化能力研究其它网络课题的基础，如网络结构的 裁剪和参数的挑选等4. 研究意义研究意义61. Madaline的敏感性n维几何模型（超球面） M. Stevenson, R. Winter, and B. Widrow, “Sensitivity of Feedforward Neural Networks to Weight Errors,” IEEE Trans. on Neural, Networks, vol. 1, no. 1, 1990. 统计模型（方差） S. W. Pich,

3、 “The Selection of Weight Accuracies for Madalines,” IEEE Trans. on Neural Networks, vol. 6, no. 2, 1995.（典型（典型方法和方法和）7分析方法（偏微分） S. Hashem, “Sensitivity Analysis for Feed- Forward Artificial Neural Networks with Differentiable Activation Functions”, Proc. of IJCNN, vol. 1, 1992. 统计方法（标准差） J. Y. Choi

4、 & C. H. Choi, “Sensitivity Ana- lysis of Multilayer Perceptron with Differ- entiable Activation Functions,” IEEE Trans. on Neural Networks, vol. 3, no. 1, 1992.2. MLP的敏感性8输入属性筛选 J. M. Zurada, A. Malinowski, S. Usui, “Perturbation Method for Deleting Redundant Inputs of Perceptron Networks”, Neu

5、rocomputing, vol. 14, 1997. 网络结构裁减 A. P. Engelbrecht, “A New Pruning Heuristic Based on Variance Analysis of Sensitivity Information”, IEEE Trans. on Neural Networks, vol. 12, no. 6, 2001.3. 敏感性的应用9 J.L. Bernier et al, “A Quantitive Study of Fault Tolerance, Noise Immunity and Generalization Ability

6、 of MLPs,” Neural Computation, vol. 12, 2000. 容错和泛化问题10三三. . 研究方法研究方法1. 自底向上自底向上方法方法单个神经元整个网络2. 概率统计方法概率统计方法概率率（离散型）均值（连续型）3. n-维几何模型维几何模型超矩形的顶点（离散型）超矩形体（连续型）11四四. .已获成果（代表性论文）已获成果（代表性论文） 敏感性分析： “Sensitivity Analysis of Multilayer Percep- tron to Input and Weight Perturbations,” IEEE Trans. on Neura

7、l Networks, vol. 12, no.6, pp. 1358-1366, Nov. 2001. 12 敏感性量化： “A Quantified Sensitivity Measure for Multi- layer Perceptron to Input Perturbation,” Neural Computation, vol. 15, no. 1, pp. 183-212, Jan. 2003.13隐层节点的裁剪（敏感性应用）： “Hidden Neuron Pruning for Multilayer Perceptrons Using Sensitivity Measur

8、e,” Proc. of IEEE ICMLC2002, pp. 1751-1757, Nov. 2002. 输入属性重要性的判定（敏感性应用）： “Determining the Relevance of Input Features for Multilayer Perceptrons,” Proc. of IEEE SMC2003, Oct. 2003.14五五. . 未来工作未来工作 进一步完善已有的结果进一步完善已有的结果, ,使之更加实用使之更加实用 放松限制条件 扩大分析范围 精确量化计算 进一步应用所得的结果进一步应用所得的结果, ,解决实际问题解决实际问题 探索新方法探索新方

9、法, ,研究新类型的网络研究新类型的网络15结束谢谢各位！谢谢各位！ 16 Inputs Weights Neurons action Output 1x 1w 2x 2w . 1 0jjwx . jjwx )(jjwxf y . 1 0jjwx 1nw 1nx nw nx 1, 1,1, 1yRWXnn17 Inputs Weights Neurons action Output 1x 1w 2x 2w . . jjwx )(jjwxf y= 1/(1+jjwxe) . 1nw 1nx nw nx 1 , 0,1 , 0yRWXnn18 Input X1 Layer 1 . Layer (L

10、-1) Layer L Output YL 2n 2Ln 11x 2n 2Ln Ly1 12x 110nx 10nx 2n 2Ln LnLy 2n 2Ln 0ninputs 1n neurons 1Lnneurons Lnneurons19 WW X FNN YY 20 W XX FNN YY 21Effects of input & weight deviations on neurons sensitivitySensitivity increases with input and weigh deviations, but the increase has an upper bo

11、und.22Effects of input dimension on neurons sensitivityThere exists an optimal value for the dimension of input, which yields the highest sensitivity value.23Effects of input & weight deviations on MLPs sensitivitySensitivity of an MLP increases with the input and weight deviations. 24Effects of

12、 the number of neurons in a layer Sensitivity of MLPs: n-2-2-1 | 1n 10 to the dimension of input. 25 Sensitivity of MLPs: 2-n-2-1 | 1n 10 to the number of neurons in the 1st layer. 26 Sensitivity of MLPs: 2-2-n-1 | 1n 10 to the number of neurons in the 2nd layer .There exists an optimal value for th

13、e number of neurons in a layer, which yields the highest sensitivity value. The nearer a layer to the output layer is, The more effect the number of neurons in the layer has.27Effects of the number of layers Sensitivity of MLPs:2-1,2-2-1,.,2-2-2-2-2-2-2-2-2-2-1 to the number of layers.Sensitivity de

14、creases with the number increasing, and the decrease almost levels off when the number becomes large.28Sensitivity of the neurons with 2-dimensional input 29Sensitivity of the neurons with 3-dimensional input 30Sensitivity of the neurons with 4-dimensional input 31Sensitivity of the neurons with 5-d

15、imensional input 32Sensitivity of the MLPs: 2-2-1, 2-3-1,2-2-2-1 33 Simulation 1 (Function Approximation) Implement an MLP to approximate the function: where Implementation considerations The MLP architecture is restricted to 2-n-1. The convergence condition is MES-goal=0.01&Epoch105. The lowest

16、 trainable number of hidden neurons is n=5. The pruning processes start with MLPs of 2-5-1 and stop at an architecture of 2-4-1. The relevant data used by and resulted from the pruning process are listed in Table 1 and Table 2.21121215 .0),(xxexxxxxF 1 , 0 1 , 0),(21xx34TABLE 1. Data for 3 MLPs with

17、 5 hidden neurons to realize the functionMLP2-5-1EpochMSE (training)MSE (testing)Trained weights and biasMSE-(goal=0.01 & epoch=100000)Sensitivity Relevance 1 30586 0.000999816 0.0117005-12.9212 -0.2999 33.7943 -34.6057 31.4768 -31.0169-0.5607 -0.8140 1.1737 -1.1026-5.4507 12.7341 -13.0816 -12.0

18、171 8.7152 bias=00.0317940.0022720.0014060.0270660.0018150.17330.02890.01840.32530.0158 2 65209 0.000999959 0.0124573 32.6223 -33.3731-0.7361 0.7202-31.8412 31.2399-15.1872 -0.0937-0.3989 -1.0028 11.9959 -15.4905 12.2103 -6.0877 -12.5057 bias=00.0021760.0004630.0018210.0310170.0270680.02610.00720.02

19、220.18880.3385 3 26094 0.000999944 0.0120354-15.0940 17.6184-19.9163 21.4109-14.0535 -0.8460 1.0263 -0.1258 26.7757 -26.1259 8.8172 -18.6532 -6.8307 16.8506 -10.4671 bias=00.0135470.0066610.0262200.0283520.0023240.11940.12420.17910.47770.024335TABLE 2. Data for the 3 pruned MLPs with 4 hidden neuron

20、s to realize the functionMLP2-4-1EpochMSE (training)MSE (testing)Retrained weights and bias(goal=0.01 & epoch=100000)SensitivityRelevance1(Obtained by removing the 5th neuron from the MLP of 2-5-1) 2251 0.000999998 0.0114834-14.4387 -0.7003 34.8366 -35.6080 33.1285 -32.6271-1.5065 0.0184-5.7036

21、13.0579 -13.2457 -12.1803 bias=4.23490.0270140.0021000.0014600.0313430.15410.02740.01930.38182(Obtained by removing the 2nd neuron from the MLP of 2-5-1) 1945 0.000999921 0.0119645 33.5805 -34.2727-32.9313 32.3172-15.8016 -0.5610-1.3318 0.010312.6267 12.7961 -6.1782 -13.3652 bias=-7.94680.0019540.00

22、18000.0269020.0292830.02470.02300.16620.39143(Obtained by removing the 5th neuron from the MLP of 2-5-1) 13253 0.000999971 0.011926-34.3974 33.8148-34.3250 34.7990-1.2909 0.0198 11.8097 0.8879 15.7984 -15.6503 -12.9606 6.0722 bias=-1.41940.0016370.0013160.0288340.0281220.02590.02060.37370.170836 Sim

23、ulation 2 (Classification) Implement an MLP to solve the XOR problem: 0 1Implementation considerations The MLP architecture is restricted to 2-n-1. The convergence condition is MES-goal=0.1&Epoch105. The pruning processes start with MLPs of 2-5-1 and stop at an architecture of 2-4-1. The relevan

24、t data used by and resulted from the pruning process are listed in Table 3 and Table 4.),(21xxF15 . 0&15 . 05 . 00&5 . 002121xxorxx5 . 00&15 . 015 . 0&5 . 002121xxorxx37TABLE 3. Data for 3 MLPs with 5 hidden neurons to realize the functionMLP2-5-1EpochMSE (training)MSE (testing)Train

25、ed weights and bias(goal=0.1 & epoch=100000)SensitivityRelevance 1 44518 0.0999997 0.109217 2.8188 -8.1143 2.4420 -0.5450 2.5766 3.7037 1.4955 -2.9245-2.5714 -3.7124 14.0153 -43.9907 28.0636 19.5486 -68.6432 bias=00.0475990.0357470.0315180.0273550.0315130.66711.57250.88450.53482.1632 2 51098 0.0

26、999998 0.113006 1.4852 -3.8902 1.0692 0.1466-1.0723 -0.1455-7.0301 2.5695-3.1382 -2.8094 23.9314 -19.1824 27.1565 14.9694 -91.6363 bias=00.0375930.0201700.0201780.0455040.0325500.89970.38690.54800.68122.9828 3 33631 0.0999994 0.11369 3.2920 2.9094-1.0067 3.4724-7.0578 2.4377-3.2921 -2.9096 1.5303 -0.0606 45.7579 -30.0598 16.5386 -52.2874 -29.7040 bias=00.0314980.0391660.0462100.0314970.0317151.44131.17730.76421.64690.942138TABLE 4. Data for the 3 pruned MLPs with 4 hidde