Spark MLlib SVM算法

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1.1 SVM支持向量机算法

支持向量机理论知识参照以下文档：

支持向量机SVM（一）

支持向量机SVM（二）

支持向量机（三）核函数

支持向量机（四）

支持向量机（五）SMO算法

SVM的目标函数及梯度下降更新公式如下：





MLlib 中 SVM的代码结构如下：

1.2 Spark Mllib SVM源码分析

1.2.1 SVMWithSGD

SVM算法的train方法，由SVMWithSGD类的object定义了train函数，在train函数中新建了SVMWithSGD对象。

package org.apache.spark.mllib.classification

// 1 类：SVMWithSGD

class SVMWithSGD private (

privatevar stepSize: Double,

privatevar numIterations: Int,

privatevar regParam: Double,

privatevar miniBatchFraction: Double)

extends GeneralizedLinearAlgorithm[SVMModel] with Serializable {

privateval gradient = new HingeGradient()

privateval updater = new SquaredL2Updater()

overrideval optimizer = new GradientDescent(gradient, updater)

.setStepSize(stepSize)

.setNumIterations(numIterations)

.setRegParam(regParam)

.setMiniBatchFraction(miniBatchFraction)

overrideprotectedval validators = List(DataValidators.binaryLabelValidator)

/**

* Construct a SVM object with default parameters: {stepSize: 1.0, numIterations: 100,

* regParm: 0.01, miniBatchFraction: 1.0}.

*/

defthis() = this(1.0, 100, 0.01, 1.0)

overrideprotecteddef createModel(weights: Vector, intercept: Double) = {

new SVMModel(weights, intercept)

}

}

SVMWithSGD类中参数说明：

stepSize: 迭代步长，默认为1.0

numIterations: 迭代次数，默认为100

regParam: 正则化参数，默认值为0.0

miniBatchFraction: 每次迭代参与计算的样本比例，默认为1.0

gradient：HingeGradient ()，梯度下降；

updater：SquaredL2Updater ()，正则化，L2范数；

optimizer：GradientDescent (gradient, updater)，梯度下降最优化计算。

// 2 train方法

object SVMWithSGD {

/**

* Train a SVM model given an RDD of (label, features) pairs. We run a fixed number

* of iterations of gradient descent using the specified step size. Each iteration uses

* `miniBatchFraction` fraction of the data to calculate the gradient. The weights used in

* gradient descent are initialized using the initial weights provided.

*

* NOTE: Labels used in SVM should be {0, 1}.

*

* @param input RDD of (label, array of features) pairs.

* @param numIterations Number of iterations of gradient descent to run.

* @param stepSize Step size to be used for each iteration of gradient descent.

* @param regParam Regularization parameter.

* @param miniBatchFraction Fraction of data to be used per iteration.

* @param initialWeights Initial set of weights to be used. Array should be equal in size to

*        the number of features in the data.

*/

def train(

input: RDD[LabeledPoint],

numIterations: Int,

stepSize: Double,

regParam: Double,

miniBatchFraction: Double,

initialWeights: Vector): SVMModel = {

new SVMWithSGD(stepSize, numIterations, regParam, miniBatchFraction)

.run(input, initialWeights)

}

train参数说明：

input：样本数据，分类标签lable只能是1.0和0.0两种,feature为double类型

numIterations: 迭代次数，默认为100

stepSize: 迭代步长，默认为1.0

miniBatchFraction: 每次迭代参与计算的样本比例，默认为1.0

initialWeights:初始权重，默认为0向量

run方法来自于继承父类GeneralizedLinearAlgorithm，实现方法如下。

1.2.2 GeneralizedLinearAlgorithm
LogisticRegressionWithSGD中run方法的实现。

package org.apache.spark.mllib.regression

/**

* Run the algorithm with the configured parameters on an input RDD

* of LabeledPoint entries starting from the initial weights provided.

*/

def run(input: RDD[LabeledPoint], initialWeights: Vector): M = {

// 特征维度赋值。

if (numFeatures < 0) {

numFeatures = input.map(_.features.size).first()

}

// 输入样本数据检测。

if (input.getStorageLevel == StorageLevel.NONE) {

logWarning("The input data is not directly cached, which may hurt performance if its"

+ " parent RDDs are also uncached.")

}

// 输入样本数据检测。

// Check the data properties before running the optimizer

if (validateData && !validators.forall(func => func(input))) {

thrownew SparkException("Input validation failed.")

}

val scaler = if (useFeatureScaling) {

new StandardScaler(withStd = true, withMean = false).fit(input.map(_.features))

} else {

null

}

// 输入样本数据处理，输出data(label, features)格式。

// addIntercept：是否增加θ0常数项，若增加，则增加x0=1项。

// 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))

}

}

//初始化权重。

// addIntercept：是否增加θ0常数项，若增加，则权重增加θ0。

/**

* 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

}

//权重优化，进行梯度下降学习，返回最优权重。

val weightsWithIntercept = optimizer.optimize(data, initialWeightsWithIntercept)

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

}

createModel(weights, intercept)

}

其中optimizer.optimize(data, initialWeightsWithIntercept)是实现的核心。

oprimizer的类型为GradientDescent，optimize方法中主要调用GradientDescent伴生对象的runMiniBatchSGD方法，返回当前迭代产生的最优特征权重向量。

GradientDescentd对象中optimize实现方法如下。

1.2.3 GradientDescent
optimize实现方法如下。

package org.apache.spark.mllib.optimization

/**

* :: DeveloperApi ::

* Runs gradient descent on the given training data.

* @param data training data

* @param initialWeights initial weights

* @return solution vector

*/

@DeveloperApi

def optimize(data: RDD[(Double, Vector)], initialWeights: Vector): Vector = {

val (weights, _) = GradientDescent.runMiniBatchSGD(

data,

gradient,

updater,

stepSize,

numIterations,

regParam,

miniBatchFraction,

initialWeights)

weights

}

}

在optimize方法中，调用了GradientDescent.runMiniBatchSGD方法，其runMiniBatchSGD实现方法如下：

/**

* 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, Vector)],

gradient: Gradient,

updater: Updater,

stepSize: Double,

numIterations: Int,

regParam: Double,

miniBatchFraction: Double,

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

//历史迭代误差数组

val stochasticLossHistory = new ArrayBuffer[Double](numIterations)

//样本数据检测，若为空，返回初始值。

val numExamples = data.count()

// if no data, return initial weights to avoid NaNs

if (numExamples == 0) {

logWarning("GradientDescent.runMiniBatchSGD returning initial weights, no data found")

return (initialWeights, stochasticLossHistory.toArray)

}

// miniBatchFraction值检测。

if (numExamples * miniBatchFraction < 1) {

logWarning("The miniBatchFraction is too small")

}

// weights权重初始化。

// Initialize weights as a column vector

var 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.dense(new Array[Double](weights.size)), 0, 1, regParam)._2

// weights权重迭代计算。

for (i <- 1 to numIterations) {

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)

// 采用treeAggregate的RDD方法，进行聚合计算，计算每个样本的权重向量、误差值，然后对所有样本权重向量及误差值进行累加。

// sample是根据miniBatchFraction指定的比例随机采样相应数量的样本 。

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))

(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)

})

// 保存本次迭代误差值，以及更新weights权重向量。

if (miniBatchSize > 0) {

/**

* 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.

*/

// updater.compute更新weights矩阵和regVal（正则化项）。根据本轮迭代中的gradient和loss的变化以及正则化项计算更新之后的weights和regVal。

stochasticLossHistory.append(lossSum / miniBatchSize + regVal)

val update = updater.compute(

weights, Vectors.fromBreeze(gradientSum / miniBatchSize.toDouble), stepSize, i, regParam)

weights = update._1

regVal = update._2

} else {

logWarning(s"Iteration ($i/$numIterations). The size of sampled batch is zero")

}

}

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

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

(weights, stochasticLossHistory.toArray)

}

runMiniBatchSGD的输入、输出参数说明：

data 样本输入数据，格式 (label, [feature values])

gradient 梯度对象，用于对每个样本计算梯度及误差

updater 权重更新对象，用于每次更新权重

stepSize 初始步长

numIterations 迭代次数

regParam 正则化参数

miniBatchFraction 迭代因子，每次迭代参与计算的样本比例

返回结果(Vector, Array[Double])，第一个为权重，每二个为每次迭代的误差值。

在MiniBatchSGD中主要实现对输入数据集进行迭代抽样，通过使用LogisticGradient作为梯度下降算法，使用SquaredL2Updater作为更新算法，不断对抽样数据集进行迭代计算从而找出最优的特征权重向量解。在LinearRegressionWithSGD中定义如下：

privateval gradient = new HingeGradient()

privateval updater = new SquaredL2Updater()

overrideval optimizer = new GradientDescent(gradient, updater)

.setStepSize(stepSize)

.setNumIterations(numIterations)

.setRegParam(regParam)

.setMiniBatchFraction(miniBatchFraction)

runMiniBatchSGD方法中调用了gradient.compute、updater.compute两个方法，其实现方法如下。

1.2.4 gradient & updater
1）gradient

//计算当前计算对象的类标签：(2 * label - 1.0)

//计算当前梯度：-(2y - 1)*x

//计算当前误差：(0, 1 - (2y - 1) (f_w(x)))

//计算权重的更新值

//返回当前训练对象的特征权重向量和误差

/**

* :: DeveloperApi ::

* 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}

*/

@DeveloperApi

class HingeGradient extends Gradient {

overridedef compute(data: Vector, label: Double, weights: Vector): (Vector, Double) = {

val dotProduct = dot(data, 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) {

val gradient = data.copy

scal(-labelScaled, gradient)

(gradient, 1.0 - labelScaled * dotProduct)

} else {

(Vectors.sparse(weights.size, Array.empty, Array.empty), 0.0)

}

}

2）updater

//weihtsOld:上一次迭代计算后的特征权重向量

//gradient:本次迭代计算的特征权重向量

//stepSize:迭代步长

//iter:当前迭代次数

//regParam:正则参数

//以当前迭代次数的平方根的倒数作为本次迭代趋近(下降)的因子

//返回本次剃度下降后更新的特征权重向量

//使用了L2 regularization（R(w) = 1/2 ||w||^2），weights更新规则为：

Spark MLlib SVM算法3

/**

* :: DeveloperApi ::

* 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.

*/

@DeveloperApi

class SquaredL2Updater extends Updater {

overridedef 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

val thisIterStepSize = stepSize / math.sqrt(iter)

val brzWeights: BV[Double] = weightsOld.toBreeze.toDenseVector

brzWeights :*= (1.0 - thisIterStepSize * regParam)

brzAxpy(-thisIterStepSize, gradient.toBreeze, brzWeights)

val norm = brzNorm(brzWeights, 2.0)

(Vectors.fromBreeze(brzWeights), 0.5 * regParam * norm * norm)

}

}

1.3 Mllib SVM实例

1、数据

数据格式为：标签, 特征1 特征2 特征3……

0 128:51 129:159 130:253 131:159 132:50 155:48 156:238 157:252 158:252 159:252 160:237 182:54 183:227 184:253 185:252 186:239 187:233 188:252 189:57 190:6 208:10 209:60 210:224 211:252 212:253 213:252 214:202 215:84 216:252 217:253 218:122 236:163 237:252 238:252
239:252 240:253 241:252 242:252 243:96 244:189 245:253 246:167 263:51 264:238 265:253 266:253 267:190 268:114 269:253 270:228 271:47 272:79 273:255 274:168 290:48 291:238 292:252 293:252 294:179 295:12 296:75 297:121 298:21 301:253 302:243 303:50 317:38 318:165
319:253 320:233 321:208 322:84 329:253 330:252 331:165 344:7 345:178 346:252 347:240 348:71 349:19 350:28 357:253 358:252 359:195 372:57 373:252 374:252 375:63 385:253 386:252 387:195 400:198 401:253 402:190 413:255 414:253 415:196 427:76 428:246 429:252 430:112
441:253 442:252 443:148 455:85 456:252 457:230 458:25 467:7 468:135 469:253 470:186 471:12 483:85 484:252 485:223 494:7 495:131 496:252 497:225 498:71 511:85 512:252 513:145 521:48 522:165 523:252 524:173 539:86 540:253 541:225 548:114 549:238 550:253 551:162
567:85 568:252 569:249 570:146 571:48 572:29 573:85 574:178 575:225 576:253 577:223 578:167 579:56 595:85 596:252 597:252 598:252 599:229 600:215 601:252 602:252 603:252 604:196 605:130 623:28 624:199 625:252 626:252 627:253 628:252 629:252 630:233 631:145
652:25 653:128 654:252 655:253 656:252 657:141 658:37

1 159:124 160:253 161:255 162:63 186:96 187:244 188:251 189:253 190:62 214:127 215:251 216:251 217:253 218:62 241:68 242:236 243:251 244:211 245:31 246:8 268:60 269:228 270:251 271:251 272:94 296:155 297:253 298:253 299:189 323:20 324:253 325:251 326:235 327:66
350:32 351:205 352:253 353:251 354:126 378:104 379:251 380:253 381:184 382:15 405:80 406:240 407:251 408:193 409:23 432:32 433:253 434:253 435:253 436:159 460:151 461:251 462:251 463:251 464:39 487:48 488:221 489:251 490:251 491:172 515:234 516:251 517:251
518:196 519:12 543:253 544:251 545:251 546:89 570:159 571:255 572:253 573:253 574:31 597:48 598:228 599:253 600:247 601:140 602:8 625:64 626:251 627:253 628:220 653:64 654:251 655:253 656:220 681:24 682:193 683:253 684:220

……

2、代码

//1 读取样本数据

valdata_path = "/user/tmp/sample_libsvm_data.txt"

valexamples = MLUtils.loadLibSVMFile(sc, data_path).cache()

//2 样本数据划分训练样本与测试样本

valsplits = examples.randomSplit(Array(0.6, 0.4), seed = 11L)

valtraining = splits(0).cache()

valtest = splits(1)

valnumTraining = training.count()

valnumTest = test.count()

println(s"Training: $numTraining, test: $numTest.")

//3 新建SVM模型，并设置训练参数

valnumIterations = 1000

valstepSize = 1

valminiBatchFraction = 1.0

valmodel = SVMWithSGD.train(training, numIterations, stepSize, miniBatchFraction)
//4 对测试样本进行测试

valprediction = model.predict(test.map(_.features))

valpredictionAndLabel = prediction.zip(test.map(_.label))

//5 计算测试误差

valmetrics = new MulticlassMetrics(predictionAndLabel)

valprecision = metrics.precision

println("Precision = " + precision)