批量梯度下降的邏輯回歸可以參考這篇文章：http://blog.csdn.net/pakko/article/details/37878837
看了一些Scala語法后，打算看看MlLib的機器學習算法的并行化，那就是邏輯回歸，找到package org.apache.spark.mllib.classification下的LogisticRegressionWithSGD這個類，直接搜train()函數。
def train(input: RDD[LabeledPoint],numIterations: Int,stepSize: Double,miniBatchFraction: Double,initialWeights: Vector): LogisticRegressionModel = {new LogisticRegressionWithSGD(stepSize, numIterations, 0.0, miniBatchFraction).run(input, initialWeights)}
發現它調用了GeneralizedLinearAlgorithm下的一個run函數，這個類GeneralizedLinearAlgorithm是個抽象類，并且在GeneralizedLinearAlgorithm.scala文件下，并且類LogisticRegressionWithSGD是繼承了GeneralizedLinearAlgorithm
def run(input: RDD[LabeledPoint], initialWeights: Vector): M = {if (numFeatures < 0) {numFeatures = input.map(_.features.size).first()}if (input.getStorageLevel == StorageLevel.NONE) {logWarning("The input data is not directly cached, which may hurt performance if its"+ " parent RDDs are also uncached.")}// Check the data properties before running the optimizerif (validateData && !validators.forall(func => func(input))) {throw new SparkException("Input validation failed.")}/*** Scaling columns to unit variance as a heuristic to reduce the condition number:** During the optimization process, the convergence (rate) depends on the condition number of* the training dataset. Scaling the variables often reduces this condition number* heuristically, thus improving the convergence rate. Without reducing the condition number,* some training datasets mixing the columns with different scales may not be able to converge.** GLMNET and LIBSVM packages perform the scaling to reduce the condition number, and return* the weights in the original scale.* See page 9 in http://cran.r-project.org/web/packages/glmnet/glmnet.pdf** Here, if useFeatureScaling is enabled, we will standardize the training features by dividing* the variance of each column (without subtracting the mean), and train the model in the* scaled space. Then we transform the coefficients from the scaled space to the original scale* as GLMNET and LIBSVM do.** Currently, it's only enabled in LogisticRegressionWithLBFGS*/val scaler = if (useFeatureScaling) {new StandardScaler(withStd = true, withMean = false).fit(input.map(_.features))} else {null}// Prepend an extra variable consisting of all 1.0's for the intercept.// TODO: Apply feature scaling to the weight vector instead of input data.val data =if (addIntercept) {if (useFeatureScaling) {input.map(lp => (lp.label, appendBias(scaler.transform(lp.features)))).cache()} else {input.map(lp => (lp.label, appendBias(lp.features))).cache()}} else {if (useFeatureScaling) {input.map(lp => (lp.label, scaler.transform(lp.features))).cache()} else {input.map(lp => (lp.label, lp.features))}}/*** TODO: For better convergence, in logistic regression, the intercepts should be computed* from the prior probability distribution of the outcomes; for linear regression,* the intercept should be set as the average of response.*/val initialWeightsWithIntercept = if (addIntercept && numOfLinearPredictor == 1) {appendBias(initialWeights)} else {/** If `numOfLinearPredictor > 1`, initialWeights already contains intercepts. */initialWeights}val weightsWithIntercept = optimizer.optimize(data, initialWeightsWithIntercept) //這里進入優化val intercept = if (addIntercept && numOfLinearPredictor == 1) {weightsWithIntercept(weightsWithIntercept.size - 1)} else {0.0}var weights = if (addIntercept && numOfLinearPredictor == 1) {Vectors.dense(weightsWithIntercept.toArray.slice(0, weightsWithIntercept.size - 1))} else {weightsWithIntercept}/*** The weights and intercept are trained in the scaled space; we're converting them back to* the original scale.** Math shows that if we only perform standardization without subtracting means, the intercept* will not be changed. w_i = w_i' / v_i where w_i' is the coefficient in the scaled space, w_i* is the coefficient in the original space, and v_i is the variance of the column i.*/if (useFeatureScaling) {if (numOfLinearPredictor == 1) {weights = scaler.transform(weights)} else {/*** For `numOfLinearPredictor > 1`, we have to transform the weights back to the original* scale for each set of linear predictor. Note that the intercepts have to be explicitly* excluded when `addIntercept == true` since the intercepts are part of weights now.*/var i = 0val n = weights.size / numOfLinearPredictorval weightsArray = weights.toArraywhile (i < numOfLinearPredictor) {val start = i * nval end = (i + 1) * n - { if (addIntercept) 1 else 0 }val partialWeightsArray = scaler.transform(Vectors.dense(weightsArray.slice(start, end))).toArraySystem.arraycopy(partialWeightsArray, 0, weightsArray, start, partialWeightsArray.size)i += 1}weights = Vectors.dense(weightsArray)}}// Warn at the end of the run as well, for increased visibility.if (input.getStorageLevel == StorageLevel.NONE) {logWarning("The input data was not directly cached, which may hurt performance if its"+ " parent RDDs are also uncached.")}// Unpersist cached dataif (data.getStorageLevel != StorageLevel.NONE) {data.unpersist(false)}createModel(weights, intercept)}
?在上面代碼中的optimizer.optimize，傳入了數據data和初始化的theta，然后optimizer在LogisticRegressionWithSGD中被初始化為：
class LogisticRegressionWithSGD private[mllib] (private var stepSize: Double,private var numIterations: Int,private var regParam: Double,private var miniBatchFraction: Double)extends GeneralizedLinearAlgorithm[LogisticRegressionModel] with Serializable {private val gradient = new LogisticGradient()private val updater = new SquaredL2Updater()@Since("0.8.0")override val optimizer = new GradientDescent(gradient, updater).setStepSize(stepSize).setNumIterations(numIterations).setRegParam(regParam).setMiniBatchFraction(miniBatchFraction)override protected val validators = List(DataValidators.binaryLabelValidator)/*** Construct a LogisticRegression object with default parameters: {stepSize: 1.0,* numIterations: 100, regParm: 0.01, miniBatchFraction: 1.0}.*/@Since("0.8.0")def this() = this(1.0, 100, 0.01, 1.0)override protected[mllib] def createModel(weights: Vector, intercept: Double) = {new LogisticRegressionModel(weights, intercept)}
}
?optimizer被賦值為GradientDescent(gradient, updater)，然后又看GradientDescent這個類：
class GradientDescent private[spark] (private var gradient: Gradient, private var updater: Updater)extends Optimizer with Logging {private var stepSize: Double = 1.0private var numIterations: Int = 100private var regParam: Double = 0.0private var miniBatchFraction: Double = 1.0private var convergenceTol: Double = 0.001...@DeveloperApidef optimize(data: RDD[(Double, Vector)], initialWeights: Vector): Vector = {val (weights, _) = GradientDescent.runMiniBatchSGD(data,gradient,updater,stepSize,numIterations,regParam,miniBatchFraction,initialWeights,convergenceTol)weights}
}
?發現調用的是隨機梯度下降的miniBatch方法，runMiniBatchSGD：
def runMiniBatchSGD(data: RDD[(Double, Vector)],gradient: Gradient,updater: Updater,stepSize: Double,numIterations: Int,regParam: Double,miniBatchFraction: Double,initialWeights: Vector,convergenceTol: Double): (Vector, Array[Double]) = {// convergenceTol should be set with non minibatch settingsif (miniBatchFraction < 1.0 && convergenceTol > 0.0) {logWarning("Testing against a convergenceTol when using miniBatchFraction " +"< 1.0 can be unstable because of the stochasticity in sampling.")}val stochasticLossHistory = new ArrayBuffer[Double](numIterations)// Record previous weight and current one to calculate solution vector differencevar previousWeights: Option[Vector] = Nonevar currentWeights: Option[Vector] = Noneval numExamples = data.count()// if no data, return initial weights to avoid NaNsif (numExamples == 0) {logWarning("GradientDescent.runMiniBatchSGD returning initial weights, no data found")return (initialWeights, stochasticLossHistory.toArray)}if (numExamples * miniBatchFraction < 1) {logWarning("The miniBatchFraction is too small")}// Initialize weights as a column vectorvar weights = Vectors.dense(initialWeights.toArray)val n = weights.size/*** For the first iteration, the regVal will be initialized as sum of weight squares* if it's L2 updater; for L1 updater, the same logic is followed.*/var regVal = updater.compute(weights, Vectors.zeros(weights.size), 0, 1, regParam)._2 //計算正則化的值var converged = false // indicates whether converged based on convergenceTolvar i = 1while (!converged && i <= numIterations) { //迭代開始，在小于最大迭代數的時候不斷運行val bcWeights = data.context.broadcast(weights)// 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, miniBatchSize) = data.sample(false, miniBatchFraction, 42 + i).treeAggregate((BDV.zeros[Double](n), 0.0, 0L))(seqOp = (c, v) => {// c: (grad, loss, count), v: (label, features)val l = gradient.compute(v._2, v._1, bcWeights.value, Vectors.fromBreeze(c._1)) //計算一個batch中每條數據的梯度(c._1, c._2 + l, c._3 + 1)},combOp = (c1, c2) => {// c: (grad, loss, count)(c1._1 += c2._1, c1._2 + c2._2, c1._3 + c2._3) //將batch中所有數據的梯度相加，損失函數值相加，記錄batch的size})if (miniBatchSize > 0) {/*** 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) //原來損失函數是這樣計算batch的總損失值除以batchSize再加上正則化值val update = updater.compute(weights, Vectors.fromBreeze(gradientSum / miniBatchSize.toDouble), //更新權重和下次的正則化值stepSize, i, regParam)weights = update._1regVal = update._2previousWeights = currentWeightscurrentWeights = Some(weights)if (previousWeights != None && currentWeights != None) {converged = isConverged(previousWeights.get,currentWeights.get, convergenceTol)}} else {logWarning(s"Iteration ($i/$numIterations). The size of sampled batch is zero")}i += 1}logInfo("GradientDescent.runMiniBatchSGD finished. Last 10 stochastic losses %s".format(stochasticLossHistory.takeRight(10).mkString(", ")))(weights, stochasticLossHistory.toArray)}
?發現要對Batch中每一條數據計算梯度，調用的是gradient.compute函數，對于二值分類：
override def compute(data: Vector,label: Double,weights: Vector,cumGradient: Vector): Double = {val dataSize = data.size// (weights.size / dataSize + 1) is number of classesrequire(weights.size % dataSize == 0 && numClasses == weights.size / dataSize + 1)numClasses match {case 2 =>/*** For Binary Logistic Regression.** Although the loss and gradient calculation for multinomial one is more generalized,* and multinomial one can also be used in binary case, we still implement a specialized* binary version for performance reason.*/val margin = -1.0 * dot(data, weights)val multiplier = (1.0 / (1.0 + math.exp(margin))) - labelaxpy(multiplier, data, cumGradient) //梯度的計算就是multiplier * data即，(h(x) - y)*xif (label > 0) {// The following is equivalent to log(1 + exp(margin)) but more numerically stable.MLUtils.log1pExp(margin) //返回損失函數值} else {MLUtils.log1pExp(margin) - margin}... //下面有多分類，還沒看
}
?利用treeAggregate并行化batch所有數據后，得到gradientSum要除以miniBatchSize，然后進入updater.compute進行權重theta和正則化值的更新，為了下一次迭代：
@DeveloperApi
class SquaredL2Updater extends Updater {override def compute(weightsOld: Vector,gradient: Vector,stepSize: Double,iter: Int,regParam: Double): (Vector, Double) = {// add up both updates from the gradient of the loss (= step) as well as// the gradient of the regularizer (= regParam * weightsOld)// w' = w - thisIterStepSize * (gradient + regParam * w)// w' = (1 - thisIterStepSize * regParam) * w - thisIterStepSize * gradient //這個就是權重更新的迭代式子，這個是L2正則化后的更新，神奇的是(1 - thisIterStepSize * regParam)val thisIterStepSize = stepSize / math.sqrt(iter) //記得更新式子不是w‘ = w - alpha*gradient alpha就是學習率也就是thisIterStepSizeval brzWeights: BV[Double] = weightsOld.toBreeze.toDenseVector //你會發現alpha = thisIterStepSize = 1/sqrt(iter)也就是隨著迭代次數越多學習率越低，邁出的步伐越小brzWeights :*= (1.0 - thisIterStepSize * regParam)brzAxpy(-thisIterStepSize, gradient.toBreeze, brzWeights)val norm = brzNorm(brzWeights, 2.0)(Vectors.fromBreeze(brzWeights), 0.5 * regParam * norm * norm) //正則化值就是w'的二范數的平方乘以正則化參數regParam乘以0.5}
}
?
