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

 package org.apache.spark.mllib.optimization

 import org.jblas.DoubleMatrix

 /**

  * Class used to compute the gradient for a loss function, given a single data point.

  */

   abstract class Gradient extends Serializable {

   /**

    * Compute the gradient and loss given the features of a single data point.

    * @param data - Feature values for one data point. Column matrix of size dx1

    * where d is the number of features.

    * @param label - Label for this data item.

    * @param weights - Column matrix containing weights for every feature.

    * @return A tuple of 2 elements. The first element is a column matrix containing the computed

    * gradient and the second element is the loss computed at this data point.

    */

   def compute(data: DoubleMatrix, label: Double, weights: DoubleMatrix): 

       (DoubleMatrix, Double)

 }

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

 /**

  * Compute gradient and loss for a logistic loss function, as used in binary classification.

  * See also the documentation for the precise formulation.

  */

 class LogisticGradient extends Gradient {

   override def compute(data: DoubleMatrix, label: Double, weights: DoubleMatrix): 

       (DoubleMatrix, Double) = {

     val margin: Double = -1.0 * data.dot(weights)

     val gradientMultiplier = (1.0 / (1.0 + math.exp(margin))) - label

     val gradient = data.mul(gradientMultiplier)

     val loss =

       if (label > 0) {

         math.log(1 + math.exp(margin))

       } else {

         math.log(1 + math.exp(margin)) - margin

       }

     (gradient, loss)

   }

 }

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leastsquares-loss损失函数，重写抽象类的compute函数

 /**

  * Compute gradient and loss for a Least-squared loss function, as used in linear regression.

  * This is correct for the averaged least squares loss function (mean squared error)

  * L = 1/n ||A weights-y||^2

  * See also the documentation for the precise formulation.

  */

 class LeastSquaresGradient extends Gradient {

   override def compute(data: DoubleMatrix, label: Double, weights: DoubleMatrix): 

       (DoubleMatrix, Double) = {

     val diff: Double = data.dot(weights) - label

     val loss = diff * diff

     val gradient = data.mul(2.0 * diff)

     (gradient, loss)
   }

 }

         leastsquares-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

 /**

  * Compute gradient and loss for a Hinge loss function, as used in SVM binary classification.

  * See also the documentation for the precise formulation.

  * NOTE: This assumes that the labels are {0,1}

  */

 class HingeGradient extends Gradient {

   override def compute(data: DoubleMatrix, label: Double, weights: DoubleMatrix):

       (DoubleMatrix, Double) = {

     val dotProduct = data.dot(weights)

     // Our loss function with {0, 1} labels is max(0, 1 - (2y – 1) (f_w(x)))

     // Therefore the gradient is -(2y - 1)*x

     val labelScaled = 2 * label - 1.0

     if (1.0 > labelScaled * dotProduct) {

       (data.mul(-labelScaled), 1.0 - labelScaled * dotProduct)

     } else {

       (DoubleMatrix.zeros(1, weights.length), 0.0)

     }
   }
 }

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

 /**

  * Class used to perform steps (weight update) using Gradient Descent methods.

  * For general minimization problems, or for regularized problems of the form

  * min L(w) + regParam * R(w),

  * the compute function performs the actual update step, when given some

  * (e.g. stochastic) gradient direction for the loss L(w),

  * and a desired step-size (learning rate).

  *

  * The updater is responsible to also perform the update coming from the

  * regularization term R(w) (if any regularization is used).

  */

 abstract class Updater extends Serializable {

   /**

    * Compute an updated value for weights given the gradient, stepSize, iteration number and

    * regularization parameter. Also returns the regularization value regParam * R(w)

    * computed using the *updated* weights.

    * @param weightsOld - Column matrix of size dx1 where d is the number of features.

    * @param gradient - Column matrix of size dx1 where d is the number of features.

    * @param stepSize - step size across iterations

    * @param iter - Iteration number

    * @param regParam - Regularization parameter

    *

    * @return A tuple of 2 elements. The first element is a column matrix containing updated weights,

    * and the second element is the regularization value computed using updated weights.

    */

   def compute(weightsOld: DoubleMatrix, gradient: DoubleMatrix, stepSize: Double, iter: Int,

       regParam: Double): (DoubleMatrix, Double)

 }

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

 /**

  * A simple updater for gradient descent *without* any regularization.

  * Uses a step-size decreasing with the square root of the number of iterations.

  */

 class SimpleUpdater extends Updater {

   override def compute(weightsOld: DoubleMatrix, gradient: DoubleMatrix,

       stepSize: Double, iter: Int, regParam: Double): (DoubleMatrix, Double) = {

     val thisIterStepSize = stepSize / math.sqrt(iter)

     val step = gradient.mul(thisIterStepSize)

     (weightsOld.sub(step), 0)

   }

 }

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

 /**

  * Updater for L1 regularized problems.

  * R(w) = ||w||_1

  * Uses a step-size decreasing with the square root of the number of iterations.

  * Instead of subgradient of the regularizer, the proximal operator for the

  * L1 regularization is applied after the gradient step. This is known to

  * result in better sparsity of the intermediate solution.

  * The corresponding proximal operator for the L1 norm is the soft-thresholding

  * function. That is, each weight component is shrunk towards 0 by shrinkageVal.

  * If w > shrinkageVal, set weight component to w-shrinkageVal.

  * If w < -shrinkageVal, set weight component to w+shrinkageVal.

  * If -shrinkageVal < w < shrinkageVal, set weight component to 0.

  * Equivalently, set weight component to signum(w) * max(0.0, abs(w) - shrinkageVal)

  */

 class L1Updater extends Updater {

   override def compute(weightsOld: DoubleMatrix, gradient: DoubleMatrix,

       stepSize: Double, iter: Int, regParam: Double): (DoubleMatrix, Double) = {

     val thisIterStepSize = stepSize / math.sqrt(iter)

     val step = gradient.mul(thisIterStepSize)

     // Take gradient step

     val newWeights = weightsOld.sub(step)

     // Apply proximal operator (soft thresholding)

     val shrinkageVal = regParam * thisIterStepSize

     (0 until newWeights.length).foreach { i =>

       val wi = newWeights.get(i)

       newWeights.put(i, signum(wi) * max(0.0, abs(wi) - shrinkageVal))

     }

     (newWeights, newWeights.norm1 * regParam)

   }

 }

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

 /**

  * Updater for L2 regularized problems.

  * R(w) = 1/2 ||w||^2

  * Uses a step-size decreasing with the square root of the number of iterations.

  */

 class SquaredL2Updater extends Updater {

   override def compute(weightsOld: DoubleMatrix, gradient: DoubleMatrix,

       stepSize: Double, iter: Int, regParam: Double): (DoubleMatrix, Double) = {

     val thisIterStepSize = stepSize / math.sqrt(iter)

     val step = gradient.mul(thisIterStepSize)

     // add up both updates from the gradient of the loss (= step) as well as

     // the gradient of the regularizer (= regParam * weightsOld)

     val newWeights = weightsOld.mul(1.0 - thisIterStepSize * regParam).sub(step)

     (newWeights, 0.5 * pow(newWeights.norm2, 2.0) * regParam)

   }

 }

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

 package org.apache.spark.mllib.optimization

 import org.apache.spark.Logging

 import org.apache.spark.rdd.RDD

 import org.jblas.DoubleMatrix

 import scala.collection.mutable.ArrayBuffer

 /**

  * Class used to solve an optimization problem using Gradient Descent.

  * @param gradient Gradient function to be used.

  * @param updater Updater to be used to update weights after every iteration.

  */

 class GradientDescent(var gradient: Gradient, var updater: Updater)

   extends Optimizer with Logging

 {

   private var stepSize: Double = 1.0

   private var numIterations: Int = 100

   private var regParam: Double = 0.0

   private var miniBatchFraction: Double = 1.0

   /**

    * Set the initial step size of SGD for the first step. Default 1.0.

    * In subsequent steps, the step size will decrease with stepSize/sqrt(t)

    */

   def setStepSize(step: Double): this.type = {

     this.stepSize = step

     this

   }

   /**

    * Set fraction of data to be used for each SGD iteration.

    * Default 1.0 (corresponding to deterministic/classical gradient descent)

    */

   def setMiniBatchFraction(fraction: Double): this.type = {

     this.miniBatchFraction = fraction

     this

   }

   /**

    * Set the number of iterations for SGD. Default 100.

    */

   def setNumIterations(iters: Int): this.type = {

     this.numIterations = iters

     this

   }

   /**

    * Set the regularization parameter. Default 0.0.

    */

   def setRegParam(regParam: Double): this.type = {

     this.regParam = regParam

     this

   }

   /**

    * Set the gradient function (of the loss function of one single data example)

    * to be used for SGD.

    */

   def setGradient(gradient: Gradient): this.type = {

     this.gradient = gradient

     this

   }

   /**

    * Set the updater function to actually perform a gradient step in a given direction.

    * The updater is responsible to perform the update from the regularization term as well,

    * and therefore determines what kind or regularization is used, if any.

    */

   def setUpdater(updater: Updater): this.type = {

     this.updater = updater

     this

   }

   def optimize(data: RDD[(Double, Array[Double])], initialWeights: Array[Double])

     : Array[Double] = {

     val (weights, stochasticLossHistory) = GradientDescent.runMiniBatchSGD(

         data,

         gradient,

         updater,

         stepSize,

         numIterations,

         regParam,

         miniBatchFraction,

         initialWeights)

     weights

   }

 }

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 

 // Top-level method to run gradient descent.

 object GradientDescent extends Logging {

   /**

    * Run stochastic gradient descent (SGD) in parallel using mini batches.

    * In each iteration, we sample a subset (fraction miniBatchFraction) of the total data

    * in order to compute a gradient estimate.

    * Sampling, and averaging the subgradients over this subset is performed using one standard

    * spark map-reduce in each iteration.

    *

    * @param data - Input data for SGD. RDD of the set of data examples, each of

    * the form (label, [feature values]).

    * @param gradient - Gradient object (used to compute the gradient of the loss function of

    * one single data example)

    * @param updater - Updater function to actually perform a gradient step in a given direction.

    * @param stepSize - initial step size for the first step

    * @param numIterations - number of iterations that SGD should be run.

    * @param regParam - regularization parameter

    * @param miniBatchFraction - fraction of the input data set that should be used for

    * one iteration of SGD. Default value 1.0.

    *

    * @return A tuple containing two elements. The first element is a column matrix containing

    * weights for every feature, and the second element is an array containing the

    * stochastic loss computed for every iteration.

    */

   def runMiniBatchSGD(

     data: RDD[(Double, Array[Double])],

     gradient: Gradient,

     updater: Updater,

     stepSize: Double,

     numIterations: Int,

     regParam: Double,

     miniBatchFraction: Double,

     initialWeights: Array[Double]) : (Array[Double], Array[Double]) = {

     val stochasticLossHistory = new ArrayBuffer[Double](numIterations)

     val nexamples: Long = data.count()

     val miniBatchSize = nexamples * miniBatchFraction

     // Initialize weights as a column vector

     var weights = new DoubleMatrix(initialWeights.length, 1, initialWeights:_*)

     var regVal = 0.0

     for (i <- 1 to numIterations) {

       // Sample a subset (fraction miniBatchFraction) of the total data

       // compute and sum up the subgradients on this subset (this is one map-reduce)

       val (gradientSum, lossSum) = data.sample(false, miniBatchFraction, 42 + i).map {

         case (y, features) =>

           val featuresCol = new DoubleMatrix(features.length, 1, features:_*)

           val (grad, loss) = gradient.compute(featuresCol, y, weights)

           (grad, loss)

       }.reduce((a, b) => (a._1.addi(b._1), a._2 + b._2))

       /**

        * NOTE(Xinghao): lossSum is computed using the weights from the previous iteration

        * and regVal is the regularization value computed in the previous iteration as well.

        */

       stochasticLossHistory.append(lossSum / miniBatchSize + regVal)

       val update = updater.compute(

         weights, gradientSum.div(miniBatchSize), stepSize, i, regParam)

       weights = update._1

       regVal = update._2

     }

     logInfo("GradientDescent.runMiniBatchSGD finished. Last 10 stochastic losses %s".format(

       stochasticLossHistory.takeRight(10).mkString(", ")))

     (weights.toArray, stochasticLossHistory.toArray)

   }

 }

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。