Spark0.9.0机器学习包MLlib-Optimization代码阅读

       基于Spark的一个生态产品--MLlib，实现了经典的机器学算法，源码分8个文件夹，classification文件夹下面包含NB、LR、SVM的实现，clustering文件夹下面包含K均值的实现，linalg文件夹下面包含SVD的实现(稀疏矩阵的表示)，recommendation文件夹下面包含als，矩阵分解实现，regression文件夹下面实现了线性回归，L2的线性回归，L1的线性回归，Util文件夹下面包含了可以为各个算法生成toy-data的文件，另外还有一个DataValidators.scala文件，api文件夹下面是PythonMLLibAPI.scala 文件，最后一个也是本文将要讲的optimization--优化算法模块包含Gradient. scala，GradientDescent.scala，Optimizer.scala，Updater.scala4个文件，作为一个scala语言的新手，如文章标题写的一样，只是对四个文件源码进行了粗读，力求搞清楚MLlib的优化算法模块的代码架构是什么样的，实现了哪些算法以及采用了什么并行策略等，关于源码中用到的scala语言特性，等熟悉这门语言后，还需要反复阅读代码。走过、路过的朋友发现文中的错误，也烦请指正，谢谢，下面是阅读过程中的一些理解(注：由于源代码有非常多的注释，为节省空间，本文有选择性的删除了，详细注释请参考源码，另外貌似博客园没有scala语言的插入模板)。

 

Gradient.scala文件

第一部分，定义了Gradient 的抽象类

 

 1 package org.apache.spark.mllib.optimization
 2 
 3 import org.jblas.DoubleMatrix
 4 
 5 /**
 6 
 7  * Class used to compute the gradient for a loss function, given a single data point.
 8 
 9  */
10 
11   abstract class Gradient extends Serializable {
12 
13   /**
14 
15    * Compute the gradient and loss given the features of a single data point.
16 
17    * @param data - Feature values for one data point. Column matrix of size dx1
18 
19    * where d is the number of features.
20 
21    * @param label - Label for this data item.
22 
23    * @param weights - Column matrix containing weights for every feature.
24 
25    * @return A tuple of 2 elements. The first element is a column matrix containing the computed
26 
27    * gradient and the second element is the loss computed at this data point.
28 
29    */
30 
31   def compute(data: DoubleMatrix, label: Double, weights: DoubleMatrix): 
32 
33       (DoubleMatrix, Double)
34 
35 }

       可以从上面的注释上看出compute的参数data是一个样本的特征(d*1维度)，label就是一个double型变量，该数据点(a single data point)的标签，weights就是特征变量的回归系数也是d*1维度，该函数返回2个东西，第1个是该样本点下计算的梯度，第2个该样本点下的损失loss

 

第二部分， Gradient 对 三种不同损失函数(Log-Loss， LeastSquares -Loss，Hinge-Loss)的派生类

 

针对log-loss损失函数，重写抽象类的compute函数

 1 /**
 2 
 3  * Compute gradient and loss for a logistic loss function, as used in binary classification.
 4 
 5  * See also the documentation for the precise formulation.
 6 
 7  */
 8 
 9 class LogisticGradient extends Gradient {
10 
11   override def compute(data: DoubleMatrix, label: Double, weights: DoubleMatrix): 
12 
13       (DoubleMatrix, Double) = {
14 
15     val margin: Double = -1.0 * data.dot(weights)
16 
17     val gradientMultiplier = (1.0 / (1.0 + math.exp(margin))) - label
18 
19     val gradient = data.mul(gradientMultiplier)
20 
21     val loss =
22 
23       if (label > 0) {
24 
25         math.log(1 + math.exp(margin))
26 
27       } else {
28 
29         math.log(1 + math.exp(margin)) - margin
30 
31       }
32 
33     (gradient, loss)
34 
35   }
36 
37 }

       我们知道对于log-loss的表达式loss=-[y*log(g(wx))+(1-y)*log(1-g(wx))]， 其中g(wx)=1/(1+exp(-wx))，二分类(0,1)，对这个loss进行求w偏导，d(loss)/d(w)=[g(wx)-y] * x  (为书写方便，用d代表偏导的符号了)，具体的表达式推导请移步http://www.cnblogs.com/kobedeshow/p/3340240.html

       结合上面代码，margin得到-wx（不明白为什么取margin的名字，函数间隔？但是函数间隔也是y*g(wx)呀），接着 gradientMultiplier是求上面梯度公式的左边， gradient  就是该点的梯度，最后求loss，当label=1的时候，上面的log-loss表达式=-[1*log(g(wx))]=-log[1/(1+exp(-wx)]=log(1+exp(margin))，当 label=0 的时候， 上面的log-loss表达式=-[log(1-g(wx))]=-[log(1-1/(1+exp(-wx)))]=-log[exp(-wc)/(1+exp(-wx))]=log(1+exp(-wx))-wc= log(1+exp(margin)) -margin

 

针对l eastsquares -loss损失函数，重写抽象类的compute函数

 1 /**
 2 
 3  * Compute gradient and loss for a Least-squared loss function, as used in linear regression.
 4 
 5  * This is correct for the averaged least squares loss function (mean squared error)
 6 
 7  * L = 1/n ||A weights-y||^2
 8 
 9  * See also the documentation for the precise formulation.
10 
11  */
12 
13 class LeastSquaresGradient extends Gradient {
14 
15   override def compute(data: DoubleMatrix, label: Double, weights: DoubleMatrix): 
16 
17       (DoubleMatrix, Double) = {
18 
19     val diff: Double = data.dot(weights) - label
20 
21     val loss = diff * diff
22 
23     val gradient = data.mul(2.0 * diff)
24 
25     (gradient, loss)
27   }
28 
29 }

 

         l eastsquares -loss的表达式 如注释所示： L = 1/n ||A weights-y||^2，这里n=1，文中代码的变量diff，就是f(wx)-y的值，损失loss=diff*diff，梯度就是data.mul(2.0 * diff)，注意.mul是 DoubleMatrix( jblas )的一个方法，是元素跟矩阵的点乘，.mull是矩阵跟矩阵的乘法，.dot是向量的内积

 

针对hinge -loss损失函数，重写抽象类的compute函数

 1 /**
 2 
 3  * Compute gradient and loss for a Hinge loss function, as used in SVM binary classification.
 4 
 5  * See also the documentation for the precise formulation.
 6 
 7  * NOTE: This assumes that the labels are {0,1}
 8 
 9  */
10 
11 class HingeGradient extends Gradient {
12 
13   override def compute(data: DoubleMatrix, label: Double, weights: DoubleMatrix):
14 
15       (DoubleMatrix, Double) = {
16 
17     val dotProduct = data.dot(weights)
18 
19     // Our loss function with {0, 1} labels is max(0, 1 - (2y – 1) (f_w(x)))
20 
21     // Therefore the gradient is -(2y - 1)*x
22 
23     val labelScaled = 2 * label - 1.0
24 
25     if (1.0 > labelScaled * dotProduct) {
26 
27       (data.mul(-labelScaled), 1.0 - labelScaled * dotProduct)
28 
29     } else {
30 
31       (DoubleMatrix.zeros(1, weights.length), 0.0)
32 
33     }
35   }
37 }

       hinge-loss的二分类(-1,1)的表达式是max(0，1- y * f(x))，代码中映射到(0,1)，变成max(0, 1 - (2y – 1) (f(x)))，这时候当样本错分的时候(也就是labelScaled * dotProduct<1)，梯度是data.mul(-labelScaled)，损失是1-labelScaled * dotProduct

 

Updater.scala文件

第一部分，定义了 Updater  的抽象类

 1 /**
 2 
 3  * Class used to perform steps (weight update) using Gradient Descent methods.
 4 
 5  * For general minimization problems, or for regularized problems of the form
 6 
 7  * min L(w) + regParam * R(w),
 8 
 9  * the compute function performs the actual update step, when given some
10 
11  * (e.g. stochastic) gradient direction for the loss L(w),
12 
13  * and a desired step-size (learning rate).
14 
15  *
16 
17  * The updater is responsible to also perform the update coming from the
18 
19  * regularization term R(w) (if any regularization is used).
20 
21  */
22 
23 abstract class Updater extends Serializable {
24 
25   /**
26 
27    * Compute an updated value for weights given the gradient, stepSize, iteration number and
28 
29    * regularization parameter. Also returns the regularization value regParam * R(w)
30 
31    * computed using the *updated* weights.
32 
33    * @param weightsOld - Column matrix of size dx1 where d is the number of features.
34 
35    * @param gradient - Column matrix of size dx1 where d is the number of features.
36 
37    * @param stepSize - step size across iterations
38 
39    * @param iter - Iteration number
40 
41    * @param regParam - Regularization parameter
42 
43    *
44 
45    * @return A tuple of 2 elements. The first element is a column matrix containing updated weights,
46 
47    * and the second element is the regularization value computed using updated weights.
48 
49    */
50 
51   def compute(weightsOld: DoubleMatrix, gradient: DoubleMatrix, stepSize: Double, iter: Int,
52 
53       regParam: Double): (DoubleMatrix, Double)
54 
55 }

      compute的参数weightsOld是更新前的变量回归系数(d*1维)，gradient是根据指定的损失函数计算出的当前梯度，stepSize 是步长也就是学习速率，iter 迭代次数，regParam 是正则参数值，该函数返回2个东西，第1个是更新后的回归系数，第2个是更新后的regParam * R(w) 值。

 

第二部分， Updater  对 三种不同正则方式(无正则，L1，L2)的派生类

 

针对 无正则  ，重写抽象类的compute函数

 1 /**
 2 
 3  * A simple updater for gradient descent *without* any regularization.
 4 
 5  * Uses a step-size decreasing with the square root of the number of iterations.
 6 
 7  */
 8 
 9 class SimpleUpdater extends Updater {
10 
11   override def compute(weightsOld: DoubleMatrix, gradient: DoubleMatrix,
12 
13       stepSize: Double, iter: Int, regParam: Double): (DoubleMatrix, Double) = {
14 
15     val thisIterStepSize = stepSize / math.sqrt(iter)
16 
17     val step = gradient.mul(thisIterStepSize)
18 
19     (weightsOld.sub(step), 0)
20 
21   }
22 
23 }

      对于梯度下降算法，w -= a*gradient，a是学习率对应代码里面的thisIterStepSize(相当于一开始步长很大，随迭代次数，增加而减小)，式子中的a*gradient对应着step，最后，weightsNew=weightsOld.sub(step)

 

针对 L1正则  ，重写抽象类的compute函数

 1 /**
 2 
 3  * Updater for L1 regularized problems.
 4 
 5  * R(w) = ||w||_1
 6 
 7  * Uses a step-size decreasing with the square root of the number of iterations.
 8 
 9  * Instead of subgradient of the regularizer, the proximal operator for the
10 
11  * L1 regularization is applied after the gradient step. This is known to
12 
13  * result in better sparsity of the intermediate solution.
14 
15  * The corresponding proximal operator for the L1 norm is the soft-thresholding
16 
17  * function. That is, each weight component is shrunk towards 0 by shrinkageVal.
18 
19  * If w > shrinkageVal, set weight component to w-shrinkageVal.
20 
21  * If w < -shrinkageVal, set weight component to w+shrinkageVal.
22 
23  * If -shrinkageVal < w < shrinkageVal, set weight component to 0.
24 
25  * Equivalently, set weight component to signum(w) * max(0.0, abs(w) - shrinkageVal)
26 
27  */
28 
29 class L1Updater extends Updater {
30 
31   override def compute(weightsOld: DoubleMatrix, gradient: DoubleMatrix,
32 
33       stepSize: Double, iter: Int, regParam: Double): (DoubleMatrix, Double) = {
34 
35     val thisIterStepSize = stepSize / math.sqrt(iter)
36 
37     val step = gradient.mul(thisIterStepSize)
38 
39     // Take gradient step
40 
41     val newWeights = weightsOld.sub(step)
42 
43     // Apply proximal operator (soft thresholding)
44 
45     val shrinkageVal = regParam * thisIterStepSize
46 
47     (0 until newWeights.length).foreach { i =>
48 
49       val wi = newWeights.get(i)
50 
51       newWeights.put(i, signum(wi) * max(0.0, abs(wi) - shrinkageVal))
52 
53     }
54 
55     (newWeights, newWeights.norm1 * regParam)
56 
57   }
58 
59 }

       加了正则项之后，前几步都一样，然后关键是对后面的处理（后面的理论暂时还不太理解，可以参考http://freemind.pluskid.org/machine-learning/sparsity-and-some-basics-of-l1-regularization/），还是说代码步骤吧，变量shrinkageVal =regParam * thisIterStepSize(注意：要*thisIterStepSize，因为w -= a*gradient  里面的gradient包括L（w）还包括正则的R(w))，然后对加正则前更新的newWeights，上遍历每一个元素，直接对该元素赋值newWeights.put(i, signum(wi) * max(0.0, abs(wi) - shrinkageVal))，对应着代码注释的红体部分。

 

针对 L2正则  ，重写抽象类的compute函数

 1 /**
 2 
 3  * Updater for L2 regularized problems.
 4 
 5  * R(w) = 1/2 ||w||^2
 6 
 7  * Uses a step-size decreasing with the square root of the number of iterations.
 8 
 9  */
10 
11 class SquaredL2Updater extends Updater {
12 
13   override def compute(weightsOld: DoubleMatrix, gradient: DoubleMatrix,
14 
15       stepSize: Double, iter: Int, regParam: Double): (DoubleMatrix, Double) = {
16 
17     val thisIterStepSize = stepSize / math.sqrt(iter)
18 
19     val step = gradient.mul(thisIterStepSize)
20 
21     // add up both updates from the gradient of the loss (= step) as well as
22 
23     // the gradient of the regularizer (= regParam * weightsOld)
24 
25     val newWeights = weightsOld.mul(1.0 - thisIterStepSize * regParam).sub(step)
26 
27     (newWeights, 0.5 * pow(newWeights.norm2, 2.0) * regParam)
28 
29   }
30 
31 }

       L2正则项加入后，损失函数变为loss1=loss+1/2 *regParam* ||w||^2，按梯度下降的更新公式:w=w-学习速率 * (d(loss1)/d(w));后面的d(loss1)=d(loss1)/d(w) + d(1/2*regParam*||w||^2) / d(w)了，那么更新公式变成了w=w-学习速率*d(loss)/d(w)-学习速率*d(1/2*regParam*||w|| ^2)/d(w)=(1-学习速率*regParam)*w-学习速率*d(loss)/d(w)，这个也就对应了第25行代码的意思

 

GradientDescent.scala文件
第一部分，定义了GradientDescent 类

  1 package org.apache.spark.mllib.optimization
  2 
  3 import org.apache.spark.Logging
  4 
  5 import org.apache.spark.rdd.RDD
  6 
  7 import org.jblas.DoubleMatrix
  8 
  9 import scala.collection.mutable.ArrayBuffer
 10 
 11 /**
 12 
 13  * Class used to solve an optimization problem using Gradient Descent.
 14 
 15  * @param gradient Gradient function to be used.
 16 
 17  * @param updater Updater to be used to update weights after every iteration.
 18 
 19  */
 20 
 21 class GradientDescent(var gradient: Gradient, var updater: Updater)
 22 
 23   extends Optimizer with Logging
 24 
 25 {
 26 
 27   private var stepSize: Double = 1.0
 28 
 29   private var numIterations: Int = 100
 30 
 31   private var regParam: Double = 0.0
 32 
 33   private var miniBatchFraction: Double = 1.0
 34 
 35   /**
 36 
 37    * Set the initial step size of SGD for the first step. Default 1.0.
 38 
 39    * In subsequent steps, the step size will decrease with stepSize/sqrt(t)
 40 
 41    */
 42 
 43   def setStepSize(step: Double): this.type = {
 44 
 45     this.stepSize = step
 46 
 47     this
 48 
 49   }
 50 
 51   /**
 52 
 53    * Set fraction of data to be used for each SGD iteration.
 54 
 55    * Default 1.0 (corresponding to deterministic/classical gradient descent)
 56 
 57    */
 58 
 59   def setMiniBatchFraction(fraction: Double): this.type = {
 60 
 61     this.miniBatchFraction = fraction
 62 
 63     this
 64 
 65   }

 66 
 67   /**
 68 
 69    * Set the number of iterations for SGD. Default 100.
 70 
 71    */
 72 
 73   def setNumIterations(iters: Int): this.type = {
 74 
 75     this.numIterations = iters
 76 
 77     this
 78 
 79   }
 80 
 81   /**
 82 
 83    * Set the regularization parameter. Default 0.0.
 84 
 85    */
 86 
 87   def setRegParam(regParam: Double): this.type = {
 88 
 89     this.regParam = regParam
 90 
 91     this
 92 
 93   }
 94 
 95   /**
 96 
 97    * Set the gradient function (of the loss function of one single data example)
 98 
 99    * to be used for SGD.
100 
101    */
102 
103   def setGradient(gradient: Gradient): this.type = {
104 
105     this.gradient = gradient
106 
107     this
108 
109   }
110 
111   /**
112 
113    * Set the updater function to actually perform a gradient step in a given direction.
114 
115    * The updater is responsible to perform the update from the regularization term as well,
116 
117    * and therefore determines what kind or regularization is used, if any.
118 
119    */
120 
121   def setUpdater(updater: Updater): this.type = {
122 
123     this.updater = updater
124 
125     this
126 
127   }
128 
129   def optimize(data: RDD[(Double, Array[Double])], initialWeights: Array[Double])
130 
131     : Array[Double] = {
132 
133     val (weights, stochasticLossHistory) = GradientDescent.runMiniBatchSGD(
134 
135         data,
136 
137         gradient,
138 
139         updater,
140 
141         stepSize,
142 
143         numIterations,
144 
145         regParam,
146 
147         miniBatchFraction,
148 
149         initialWeights)
150 
151     weights
152 
153   }
154 
155 }

       该类的输入有2个参数，第一个是前面都是gradient对应了用户需要选哪个损失函数计算梯度，第二个updater 对应了用户选择哪一种正则方式，程序开头都设置了stepSize，numIterations，regParam，miniBatchFraction的默认值最后一个函数optimize，输入RDD数据，跟初始的回归系数weight,返回weights权重

 

第二部分，定义了object GradientDescent 

  1 // Top-level method to run gradient descent.
  2 
  3 object GradientDescent extends Logging {
  4 
  5   /**
  6 
  7    * Run stochastic gradient descent (SGD) in parallel using mini batches.
  8 
  9    * In each iteration, we sample a subset (fraction miniBatchFraction) of the total data
 10 
 11    * in order to compute a gradient estimate.
 12 
 13    * Sampling, and averaging the subgradients over this subset is performed using one standard
 14 
 15    * spark map-reduce in each iteration.
 16 
 17    *
 18 
 19    * @param data - Input data for SGD. RDD of the set of data examples, each of
 20 
 21    * the form (label, [feature values]).
 22 
 23    * @param gradient - Gradient object (used to compute the gradient of the loss function of
 24 
 25    * one single data example)
 26 
 27    * @param updater - Updater function to actually perform a gradient step in a given direction.
 28 
 29    * @param stepSize - initial step size for the first step
 30 
 31    * @param numIterations - number of iterations that SGD should be run.
 32 
 33    * @param regParam - regularization parameter
 34 
 35    * @param miniBatchFraction - fraction of the input data set that should be used for
 36 
 37    * one iteration of SGD. Default value 1.0.
 38 
 39    *
 40 
 41    * @return A tuple containing two elements. The first element is a column matrix containing
 42 
 43    * weights for every feature, and the second element is an array containing the
 44 
 45    * stochastic loss computed for every iteration.
 46 
 47    */
 48 
 49   def runMiniBatchSGD(
 50 
 51     data: RDD[(Double, Array[Double])],
 52 
 53     gradient: Gradient,
 54 
 55     updater: Updater,
 56 
 57     stepSize: Double,
 58 
 59     numIterations: Int,
 60 
 61     regParam: Double,
 62 
 63     miniBatchFraction: Double,
 64 
 65     initialWeights: Array[Double]) : (Array[Double], Array[Double]) = {
 66 
 67     val stochasticLossHistory = new ArrayBuffer[Double](numIterations)
 68 
 69     val nexamples: Long = data.count()
 70 
 71     val miniBatchSize = nexamples * miniBatchFraction
 72 
 73     // Initialize weights as a column vector
 74 
 75     var weights = new DoubleMatrix(initialWeights.length, 1, initialWeights:_*)
 76 
 77     var regVal = 0.0
 78 
 79     for (i <- 1 to numIterations) {
 80 
 81       // Sample a subset (fraction miniBatchFraction) of the total data
 82 
 83       // compute and sum up the subgradients on this subset (this is one map-reduce)
 84 
 85       val (gradientSum, lossSum) = data.sample(false, miniBatchFraction, 42 + i).map {
 86 
 87         case (y, features) =>
 88 
 89           val featuresCol = new DoubleMatrix(features.length, 1, features:_*)
 90 
 91           val (grad, loss) = gradient.compute(featuresCol, y, weights)
 92 
 93           (grad, loss)
 94 
 95       }.reduce((a, b) => (a._1.addi(b._1), a._2 + b._2))
 96 
 97       /**
 98 
 99        * NOTE(Xinghao): lossSum is computed using the weights from the previous iteration
100 
101        * and regVal is the regularization value computed in the previous iteration as well.
102 
103        */
104 
105       stochasticLossHistory.append(lossSum / miniBatchSize + regVal)
106 
107       val update = updater.compute(
108 
109         weights, gradientSum.div(miniBatchSize), stepSize, i, regParam)
110 
111       weights = update._1
112 
113       regVal = update._2
114 
115     }
116 
117     logInfo("GradientDescent.runMiniBatchSGD finished. Last 10 stochastic losses %s".format(
118 
119       stochasticLossHistory.takeRight(10).mkString(", ")))
120 
121     (weights.toArray, stochasticLossHistory.toArray)
122 
123   }
124 
125 }

       该object进行了整个的优化过程，输出是回归系数跟每次迭代的loss，这里实现的是minibatch-sgd的并行，前面的var weights = new DoubleMatrix(initialWeights.length, 1, initialWeights:_*)，这个操作是把array型的搞成矩阵型的d*1维矩阵。关键代码for (i <- 1 to numIterations) 里面的，首先data是spark的RDD数据类型，data.sample方法第一个参数指是否又放回的抽样，第二个是抽样比例，第三个是随机种子，data.sample返回抽样后的RDD，然后RDD.map,RDD.reduce操作就是一个完整的map-reduce操作。接着，把得到的gradientSum除以miniBatchSize，扔到updater里面去更新梯度，关于minibatch-sgd的并行策略可以参考我之前的文章《常见数据挖掘算法的Map-Reduce策略(2)》里面的algorithm3。