Spark机器学习(12)：神经网络算法

1. 神经网络基础知识

1.1 神经元

神经网络（Neural Net）是由大量的处理单元相互连接形成的网络。神经元是神经网络的最小单元，神经网络由若干个神经元组成。一个神经元的结构如下：

上面的神经元x₁，x₂，x₃和1是输入，h_w,b(x)是输出。

其中f(x)是激活函数，常用的激活函数有sigmoid函数和tanh（双曲正切）函数。

sigmoid函数：

tanh（双曲正切）函数：

1.2 神经网络

神经网络由若干个层次，相邻层次之间的神经元存在输入的关系。第一层称为输入层，最后一层称为输出层，中间的层次称为隐含层。

1.3 信号前向传播和误差反向传播

设神经网络有n层，第1层为L₁，第2层为L₂，第n层为L_n，第p(p=1,2,...n)层的神经元节点数量是m_p。a_j^(k)表示第k层第j个节点的输出值。则对于L_1（也就是输入层），有

第(k+1)层第j个神经元的输出

设一个训练样本的误差为

整体误差函数

 

为了防止过拟合，增加了第二项L2正则化。

目标是求(w,b)，使得J(w,b)最小。为此使用梯度下降法，每次迭代按照下面的公式对w和b进行更新

第n层（也就是输出层）的输出神经元j，其残差为

第k层第i个节点的残差为

求解(w,b)的过程如下：

1) 对于所有的k，令w^(k):=0，b^(k):=0；

2) 信号前向传播，根据每个样本的输入值和w^(k)、w^(k)，逐层计算出h_w,b(x)；

3) 误差反向传播，逐层计算出每一层每个神经元的残差；

4) 对w和b的值进行更新。

反复进行步骤(2)~(4)，直到完成指定的迭代次数为止。

2. MLlib神经网络的实现

MLlib的神经网络类是NerualNet。主要参数包括：

Size：Array[Int]，神经网络每一层的节点数量；

Layer：神经网络的层数；

Activation_function：激活函数，可以是sigm或tanh

Ouput_function：输出函数，可以是sigm、softmax或linear。

代码：

import org.apache.log4j.{ Level, Logger }
import org.apache.spark.{ SparkConf, SparkContext }
import breeze.linalg.{
DenseMatrix => BDM,
max => Bmax,
min => Bmin
}
import scala.collection.mutable.ArrayBuffer

/**
  * Created by Administrator on 2017/7/27.
  */
object NNTest {

  def main(args: Array[String]) = {
    // 设置运行环境
    val conf = new SparkConf().setAppName("Neural Net")
      .setMaster("spark://master:7077").setJars(Seq("E:\Intellij\Projects\MachineLearning\MachineLearning.jar"))
    val sc = new SparkContext(conf)
    Logger.getRootLogger.setLevel(Level.WARN)

    // 随机生成样本数据
    Logger.getRootLogger.setLevel(Level.WARN)
    val sampleRow = 1000
    val sampleColumn = 5
    val randSamp_01 = RandSampleData.RandM(sampleRow, sampleColumn, -10, 10, "sphere")
    // 归一化
    val norMax = Bmax(randSamp_01(::, breeze.linalg.*))
    val norMin = Bmin(randSamp_01(::, breeze.linalg.*))
    val nor1 = randSamp_01 - (BDM.ones[Double](randSamp_01.rows, 1)) * norMin
    val nor2 = nor1 :/ ((BDM.ones[Double](nor1.rows, 1)) * (norMax - norMin))
    // 转换样本
    val randSamp_02 = ArrayBuffer[BDM[Double]]()
    for (i <- 0 to sampleRow - 1) {
      val mi = nor2(i, ::)
      val mi1 = mi.inner
      val mi2 = mi1.toArray
      val mi3 = new BDM(1, mi2.length, mi2)
      randSamp_02 += mi3
    }
    val randSamp_03 = sc.parallelize(randSamp_02, 10)
    sc.setCheckpointDir("hdfs://master:9000/ml/data/checkpoint")
    randSamp_03.checkpoint()
    val trainRDD = randSamp_03.map(f => (new BDM(1, 1, f(::, 0).data), f(::, 1 to -1)))
    // 训练，建立模型
    val opts = Array(100.0, 50.0, 0.0)
    trainRDD.cache
    val numExamples = trainRDD.count()
    println(s"Number of Examples: $numExamples")
    val NNModel = new NeuralNet().
      setSize(Array(5, 10, 10, 10, 10, 10, 1)).
      setLayer(7).
      setActivation_function("tanh_opt").
      setLearningRate(2.0).
      setScaling_learningRate(1.0).
      setWeightPenaltyL2(0.0).
      setNonSparsityPenalty(0.0).
      setSparsityTarget(0.05).
      setInputZeroMaskedFraction(0.0).
      setDropoutFraction(0.0).
      setOutput_function("sigm").
      NNtrain(trainRDD, opts)

    // 测试模型
    val NNPrediction = NNModel.predict(trainRDD)
    val NNPredictionError = NNModel.Loss(NNPrediction)
    println(s"NNerror = $NNPredictionError")
    val showPrediction = NNPrediction.map(f => (f.label.data(0), f.predict_label.data(0))).take(100)
    println("Prediction Result")
    println("Value" + "	" + "Prediction" + "	" + "Error")
    for (i <- 0 until showPrediction.length)
      println(showPrediction(i)._1 + "	" + showPrediction(i)._2 + "	" + (showPrediction(i)._2 - showPrediction(i)._1))

    var tmpWeight = NNModel.weights(0)
    for (i <-0 to 5) {
      tmpWeight = NNModel.weights(i)
      println(s"Weight of Layer ${i+1}")
      for (j <- 0 to tmpWeight.rows - 1) {
        for (k <- 0 to tmpWeight.cols - 1) {
          print(tmpWeight(j, k) + "	")
        }
        println()
      }
    }

  }

}

以上代码建立了一个7层的神经网络，各层的节点数量为Array(5, 10, 10, 10, 10, 10, 1)，对Sphere函数进行了测试。

运行结果：

Number of Examples: 1000
epoch: numepochs = 1 , Took = 17 seconds; Full-batch train mse = 0.066738, val mse = 0.000000.
epoch: numepochs = 2 , Took = 12 seconds; Full-batch train mse = 0.069649, val mse = 0.000000.
epoch: numepochs = 3 , Took = 10 seconds; Full-batch train mse = 0.055260, val mse = 0.000000.
epoch: numepochs = 4 , Took = 10 seconds; Full-batch train mse = 0.016346, val mse = 0.000000.
epoch: numepochs = 5 , Took = 9 seconds; Full-batch train mse = 0.013802, val mse = 0.000000.
epoch: numepochs = 6 , Took = 13 seconds; Full-batch train mse = 0.045142, val mse = 0.000000.
epoch: numepochs = 7 , Took = 7 seconds; Full-batch train mse = 0.031211, val mse = 0.000000.
epoch: numepochs = 8 , Took = 7 seconds; Full-batch train mse = 0.016334, val mse = 0.000000.
epoch: numepochs = 9 , Took = 9 seconds; Full-batch train mse = 0.013348, val mse = 0.000000.
epoch: numepochs = 10 , Took = 7 seconds; Full-batch train mse = 0.017879, val mse = 0.000000.
epoch: numepochs = 11 , Took = 7 seconds; Full-batch train mse = 0.012627, val mse = 0.000000.
epoch: numepochs = 12 , Took = 7 seconds; Full-batch train mse = 0.018080, val mse = 0.000000.
epoch: numepochs = 13 , Took = 7 seconds; Full-batch train mse = 0.016755, val mse = 0.000000.
epoch: numepochs = 14 , Took = 7 seconds; Full-batch train mse = 0.012250, val mse = 0.000000.
epoch: numepochs = 15 , Took = 7 seconds; Full-batch train mse = 0.044833, val mse = 0.000000.
epoch: numepochs = 16 , Took = 7 seconds; Full-batch train mse = 0.024345, val mse = 0.000000.
epoch: numepochs = 17 , Took = 7 seconds; Full-batch train mse = 0.039005, val mse = 0.000000.
epoch: numepochs = 18 , Took = 7 seconds; Full-batch train mse = 0.012298, val mse = 0.000000.
epoch: numepochs = 19 , Took = 7 seconds; Full-batch train mse = 0.012371, val mse = 0.000000.
epoch: numepochs = 20 , Took = 6 seconds; Full-batch train mse = 0.014077, val mse = 0.000000.
epoch: numepochs = 21 , Took = 7 seconds; Full-batch train mse = 0.040328, val mse = 0.000000.
epoch: numepochs = 22 , Took = 6 seconds; Full-batch train mse = 0.036575, val mse = 0.000000.
epoch: numepochs = 23 , Took = 6 seconds; Full-batch train mse = 0.033986, val mse = 0.000000.
epoch: numepochs = 24 , Took = 6 seconds; Full-batch train mse = 0.026421, val mse = 0.000000.
epoch: numepochs = 25 , Took = 6 seconds; Full-batch train mse = 0.036776, val mse = 0.000000.
epoch: numepochs = 26 , Took = 6 seconds; Full-batch train mse = 0.011838, val mse = 0.000000.
epoch: numepochs = 27 , Took = 6 seconds; Full-batch train mse = 0.010749, val mse = 0.000000.
epoch: numepochs = 28 , Took = 6 seconds; Full-batch train mse = 0.012717, val mse = 0.000000.
epoch: numepochs = 29 , Took = 6 seconds; Full-batch train mse = 0.011883, val mse = 0.000000.
epoch: numepochs = 30 , Took = 7 seconds; Full-batch train mse = 0.010562, val mse = 0.000000.
epoch: numepochs = 31 , Took = 6 seconds; Full-batch train mse = 0.010591, val mse = 0.000000.
epoch: numepochs = 32 , Took = 6 seconds; Full-batch train mse = 0.010389, val mse = 0.000000.
epoch: numepochs = 33 , Took = 6 seconds; Full-batch train mse = 0.015908, val mse = 0.000000.
epoch: numepochs = 34 , Took = 6 seconds; Full-batch train mse = 0.012413, val mse = 0.000000.
epoch: numepochs = 35 , Took = 6 seconds; Full-batch train mse = 0.010442, val mse = 0.000000.
epoch: numepochs = 36 , Took = 6 seconds; Full-batch train mse = 0.056686, val mse = 0.000000.
epoch: numepochs = 37 , Took = 6 seconds; Full-batch train mse = 0.054850, val mse = 0.000000.
epoch: numepochs = 38 , Took = 6 seconds; Full-batch train mse = 0.019422, val mse = 0.000000.
epoch: numepochs = 39 , Took = 6 seconds; Full-batch train mse = 0.016443, val mse = 0.000000.
epoch: numepochs = 40 , Took = 6 seconds; Full-batch train mse = 0.010289, val mse = 0.000000.
epoch: numepochs = 41 , Took = 7 seconds; Full-batch train mse = 0.022615, val mse = 0.000000.
epoch: numepochs = 42 , Took = 6 seconds; Full-batch train mse = 0.010723, val mse = 0.000000.
epoch: numepochs = 43 , Took = 6 seconds; Full-batch train mse = 0.010289, val mse = 0.000000.
epoch: numepochs = 44 , Took = 6 seconds; Full-batch train mse = 0.033933, val mse = 0.000000.
epoch: numepochs = 45 , Took = 7 seconds; Full-batch train mse = 0.030156, val mse = 0.000000.
epoch: numepochs = 46 , Took = 7 seconds; Full-batch train mse = 0.022068, val mse = 0.000000.
epoch: numepochs = 47 , Took = 7 seconds; Full-batch train mse = 0.029382, val mse = 0.000000.
epoch: numepochs = 48 , Took = 6 seconds; Full-batch train mse = 0.021275, val mse = 0.000000.
epoch: numepochs = 49 , Took = 6 seconds; Full-batch train mse = 0.039427, val mse = 0.000000.
epoch: numepochs = 50 , Took = 7 seconds; Full-batch train mse = 0.016674, val mse = 0.000000.
NNerror = 0.016674267332022572
Prediction Result
Value    Prediction    Error
0.6048934040010798    0.19097551722554007    -0.41391788677553976
0.5917463309959767    0.35726681238891195    -0.23447951860706479
0.5798180746808277    0.19232727566724744    -0.38749079901358024
0.39808885303777447    0.1926440400752866    -0.20544481296248787
0.4140924247674261    0.19529777426853168    -0.2187946504988944
0.08847408598189055    0.19110126347316514    0.10262717749127459
0.3583460134199821    0.21170344602417424    -0.1466425673958079
0.29635258460747904    0.3549086780038481    0.05855609339636908
0.21947238532147648    0.19156569159762857    -0.02790669372384791
0.5357166982629155    0.36018248221537214    -0.17553421604754332
0.5547810234563126    0.1912501730851674    -0.36353085037114524
0.40529948654006304    0.21826323152039923    -0.1870362550196638
0.4765320387665492    0.34409113646061484    -0.13244090230593436
0.05759629179315594    0.1914373047341408    0.13384101294098488
0.25415182638221206    0.29169483353745973    0.037543007155247665
0.2731217394258585    0.19452719525740314    -0.07859454416845535
0.021103715077802527    0.19131792203441428    0.17021420695661174
0.24098254783013137    0.334302879677641    0.09332033184750962
0.6300811731076671    0.3595001582783692    -0.2705810148292979
0.41827613603130404    0.195477735057971    -0.22279840097333303
0.2526404805902617    0.1945578268820965    -0.05808265370816518
0.16619916368077442    0.191265206532793    0.025066042852018577
0.007724491831775392    0.1909446242319318    0.1832201324001564
0.08926696720959378    0.19197139383958237    0.10270442662998859
0.4822857005955674    0.19244393418394434    -0.28984176641162307
0.12166559083216193    0.19242076231047756    0.07075517147831563
0.2883494676971952    0.30939742289582284    0.02104795519862762
0.38817298742061984    0.1909921814285587    -0.19718080599206114
0.34588396966368695    0.1957690915303307    -0.15011487813335625
0.19958641570784796    0.19348928314854685    -0.0060971325593011105
0.31340425691874024    0.19828489007869293    -0.11511936684004731
0.31775749422734    0.19211592601952254    -0.12564156820781747
0.48789392695999645    0.19120722177454247    -0.296686705185454
0.4359840834351843    0.3604340050247724    -0.07555007841041189
0.17359981155470314    0.1914334455263964    0.01783363397169327
0.3629355770221922    0.2004476969345776    -0.1624878800876146
0.4627621372503198    0.2111988404691097    -0.2515632967812101
0.49652077030838826    0.19101585452942166    -0.3055049157789666
0.12618599928245963    0.19939585850613975    0.07320985922368012
0.45276204270081455    0.1924159942977412    -0.26034604840307335
0.2837721853443281    0.2016124468403725    -0.08215973850395558
0.34590164213713354    0.3601210376231753    0.014219395486041786
0.1961497656762427    0.19408639222665872    -0.0020633734495839884
0.22135763175909048    0.27616370537642354    0.054806073617333056
0.43356473411523927    0.19150317510575426    -0.242061559009485
0.09566706862199378    0.19087327269062435    0.09520620406863056
0.29830626566849494    0.19959705355592236    -0.09870921211257258
0.3070532379895792    0.34322116560057725    0.036167927610998074
0.07052673330364767    0.19118739087384276    0.12066065757019509
0.5501181200918814    0.2024015375945202    -0.34771658249736126
0.31894277127298554    0.1917670097886867    -0.12717576148429885
0.08585450906008718    0.20848620726607436    0.12263169820598718
0.20245657700014166    0.19218060734066275    -0.010275969659478912
0.1712767340967007    0.1913375355437103    0.020060801447009613
0.3779192242827297    0.2035011707996587    -0.17441805348307102
0.241909871430447    0.19089783315658176    -0.051012038273865246
0.40578032667620945    0.3561807946562045    -0.049599532020004944
0.20834390560196567    0.19103138812628986    -0.017312517475675804
0.49675932490421343    0.1915234454414188    -0.3052358794627946
0.2342257039800733    0.1920213029433058    -0.04220440103676751
0.18045883957051312    0.20420037376704497    0.023741534196531855
0.2309607430153665    0.1912620988835584    -0.0396986441318081
0.40644947116571745    0.19173032204451546    -0.214719149121202
0.11691561072493983    0.19280159115148832    0.07588598042654848
0.05696889589626215    0.19083593927270395    0.13386704337644179
0.47164532559761124    0.2506614607550888    -0.22098386484252242
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0.1496556423910719    0.19090570957335065    0.04125006718227875
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0.0    0.19143943303505245    0.19143943303505245
0.36555981464056164    0.19189800228180368    -0.17366181235875797
0.3963164889187304    0.19451555510717428    -0.20180093381155612
0.313325868748335    0.19168776655589245    -0.12163810219244256
0.5034713123520999    0.3339326133308013    -0.16953869902129853
0.4224576693623929    0.3539965263782299    -0.06846114298416295
0.08523050351506854    0.19132247714662606    0.10609197363155752
0.26080914691197654    0.19095418139777426    -0.06985496551420228
0.1324640982588358    0.19304336020222349    0.0605792619433877
0.13055674031551295    0.19224589387242375    0.06168915355691079
0.23625412018106318    0.1917614371123628    -0.044492683068700384
0.5019570376831385    0.3081554524341633    -0.19380158524897517
0.030390738837917763    0.19083521852879112    0.16044447969087336
0.34274552561551896    0.19120112478612986    -0.1515444008293891
0.4514974655646171    0.1916124319598441    -0.259885033604773
0.531777023034474    0.3515396924867077    -0.18023733054776636
0.367772718094668    0.3317275143536775    -0.03604520374099052
0.41600472261866916    0.22278029398575255    -0.1932244286329166
0.36506543552315474    0.325628070833062    -0.039437364690092735
0.314008782918081    0.32408907795815034    0.010080295040069354
0.2925887989109779    0.1921158349811155    -0.10047296392986238
0.4658619691058588    0.2146831464338164    -0.2511788226720424
0.2280242270958607    0.19158705334902099    -0.03643717374683972
0.5003581077100195    0.19149431703681175    -0.3088637906732078
0.4448442553362914    0.19086228548828346    -0.2539819698480079
Weight of Layer 1
1.3741710823232989    1.0997962988037757    -2.3077515870713716    2.0946962013291297    2.24588083756021    0.7186952475525394    
-1.1885813301306254    0.25046447165487246    -1.253986920617667    1.535570764339994    0.1440090623878452    1.2110656874633237    
-0.23784821158321864    -0.5133767761738681    0.5355594752965599    -0.9862762256909807    2.234245108441277    -0.5216923380767392    
2.0496153507146033    -0.9000455162282417    1.3406201642695788    2.1185256789014897    1.038387978643167    -0.011886136436036997    
2.4017180810229086    0.5342060426581219    2.188686239727936    -0.604587031465719    0.061697537675081446    -0.48429030459304306    
-1.234468451038262    0.7790398934631602    -0.22067594788975725    -2.0414139797176176    -0.9324514648411226    0.798505045375407    
0.843464180836847    1.8612698144445792    2.290144904438349    1.2291878648431667    2.3639566790099784    -2.1175568466779437    
2.1488480623696975    2.253851655104785    -1.879142801282798    -0.23011616258273254    2.4342506675413413    -2.184430097374211    
1.3446335417651347    0.39411399422872706    -1.4588967794444714    2.6567285233270366    -0.8576819932211762    -1.9914472547514181    
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Weight of Layer 2
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