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）
http://www.cnblogs.com/jerrylead/archive/2011/03/13/1982639.html
支持向量机SVM（二）
http://www.cnblogs.com/jerrylead/archive/2011/03/13/1982684.html
http://www.cnblogs.com/jerrylead/archive/2011/03/18/1988406.html
http://www.cnblogs.com/jerrylead/archive/2011/03/18/1988415.html
支持向量机（五）SMO算法
http://www.cnblogs.com/jerrylead/archive/2011/03/18/1988419.html
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：
/**
* :: 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)