spark mllib源码分析之逻辑回归弹性网络ElasticNet（一）_逻辑回归 elasticnetparam

相关文章
spark mllib源码分析之逻辑回归弹性网络ElasticNet（二）
spark源码分析之L-BFGS
spark mllib源码分析之OWLQN
spark中的online均值/方差统计
spark源码分析之二分类逻辑回归evaluation
spark正则化

spark在ml包中将逻辑回归封装了下，同时在算法中引入了L1和L2正则化，通过elasticNetParam来调节两种正则化的系数，同时根据选择的正则化，决定使用L-BFGS还是OWLQN优化，是谓Elastic Net。

1. 辅助类

我们首先介绍下模型训练和预测，评价中使用到的一些类。

1.1. MultiClassSummarizer

主要用在样本的训练过程中，统计数据中各种label出现的次数及其weight，这里引入了样本weight，可以用在unbalance的数据中，通过惩罚数量大的class达到样本均衡，默认为1

class MultiClassSummarizer extends Serializable {
   
  private val distinctMap = new mutable.HashMap[Int, (Long, Double)]
  private var totalInvalidCnt: Long = 0L
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distinctMap的key是label，类型为Long，value是个tuple，第一个元素是label出现的次数，第二维是weight的和，

$$\sum w_il=l*\sum w_i$$
$l$是label，weight为1的时候，这里相当于label的数量。
因为这个累积器主要用在treeAggregator中，重要的是两个函数，add用于累积样本，merge用于两个MultiClassSummarizer的合并

/**
 * Add a new label into this MultilabelSummarizer, and update the distinct map.
 *
 * @param label The label for this data point.
 * @param weight The weight of this instances.
 * @return This MultilabelSummarizer
 */
def add(label: Double, weight: Double = 1.0): this.type = { 
  require(weight >= 0.0, s"instance weight, $weight has to be >= 0.0")

  if (weight == 0.0) return this
  //这里要求label必须为整数，否则认为非法
  if (label - label.toInt != 0.0 || label < 0) {
    totalInvalidCnt += 1
    this
  }else {
    val (counts: Long, weightSum: Double) = distinctMap.getOrElse(label.toInt, (0L, 0.0))
    //累加样本次数及weight
    distinctMap.put(label.toInt, (counts + 1L, weightSum + weight))
    this
  }
}

/**
 * Merge another MultilabelSummarizer, and update the distinct map.
 * (Note that it will merge the smaller distinct map into the larger one using in-place
 * merging, so either `this` or `other` object will be modified and returned.)
 *
 * @param other The other MultilabelSummarizer to be merged.
 * @return Merged MultilabelSummarizer object.
 */
def merge(other: MultiClassSummarizer): MultiClassSummarizer = { 
//将size小的并入大的，性能
  val (largeMap, smallMap) = if (this.distinctMap.size > other.distinctMap.size) {
    (this, other)
  } else {
    (other, this)
  }
  smallMap.distinctMap.foreach {
    case (key, value) =>
      val (counts: Long, weightSum: Double) = largeMap.distinctMap.getOrElse(key, (0L, 0.0))
      //直接累加
      largeMap.distinctMap.put(key, (counts + value._1, weightSum + value._2))
  }
  largeMap.totalInvalidCnt += smallMap.totalInvalidCnt
  largeMap
}
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返回统计到的class数，默认从0开始，所以是最大label+1

def numClasses: Int = if (distinctMap.isEmpty) 0 else distinctMap.keySet.max + 1
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返回weight累积和

def histogram: Array[Double] = { 
  val result = Array.ofDim[Double](numClasses)
  var i = 0 
  //应该是val len = numClasses
  val len = result.length
  //这里要求class从0到k-1
  while (i < len) { 
    result(i) = distinctMap.getOrElse(i, (0L, 0.0))._2
    i += 1 
  } 
  result
}
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对比numClasses，可以看到这里result实现是有点问题的，必须要求class从0到k-1全部出现了，否则会丢失部分的class的统计。

1.2. MultivariateOnlineSummarizer

在spark中的online均值/方差统计中已有介绍，计算样本集的方差，用于归一化。

1.3 LogisticRegressionModel

逻辑回归model，放着训练得到的系数矩阵，矩阵，class数，是否多分类等参数。

1.3.1. 预测

override protected def predict(features: Vector): Double = if (isMultinomial) {
  super.predict(features)
} else {
  // Note: We should use getThreshold instead of $(threshold) since getThreshold is overridden.
  if (score(features) > getThreshold) 1 else 0
}
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可以看到二分类与多分类是分开处理的，其原理是不同的

1.3.1.1. 二分类

从上面可以看到二分类的预测是通过计算特征得分，与threshold比较，大于为1，否则0，score函数代码

private val score: Vector => Double = (features) => { 
  val m = margin(features)
  1.0 / (1.0 + math.exp(-m))
}
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从score函数可以看到，这里是将margin带入了sigmoid函数，我们看margin函数

private val margin: Vector => Double = (features) => { 
  BLAS.dot(features, _coefficients) + _intercept
}
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就是将特征与系数相乘，再加上截距。
二分类中还实现了一些低级API，用在evaluate model，分别计算margin，预测值，预测label

//计算二分类的margin，返回DenseVector
override protected def predictRaw(features: Vector): Vector = { 
  val m = margin(features)
  Vectors.dense(-m, m)
} 
//由margin计算原始的预测值，也就是经过sigmoid函数的值
override protected def raw2probabilityInPlace(rawPrediction: Vector): Vector = { 
  rawPrediction match { 
    case dv: DenseVector =>
      var i = 0 
      val size = dv.size
      while (i < size) { 
        dv.values(i) = 1.0 / (1.0 + math.exp(-dv.values(i)))
        i += 1 
      } 
      dv
    case sv: SparseVector =>
      throw new RuntimeException("Unexpected error in LogisticRegressionModel:" + 
        " raw2probabilitiesInPlace encountered SparseVector")
  } 
}
//由原始的预测值，预测label，从上面可知vector(1)为实际的预测值，用来预测label
override protected def raw2prediction(rawPrediction: Vector): Double = { 
  // Note: We should use getThreshold instead of $(threshold) since getThreshold is overridden.
  val t = getThreshold
  val rawThreshold = if (t == 0.0) { 
    Double.NegativeInfinity
  } else if (t == 1.0) { 
    Double.PositiveInfinity
  } else { 
    math.log(t / (1.0 - t))
  } 
  if (rawPrediction(1) > rawThreshold) 1 else 0 
}
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1.3.1.2. 多分类

多分类时，其调用了父类的predict函数

override protected def predict(features: FeaturesType): Double = { 
  raw2prediction(predictRaw(features))
}
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调用了raw2prediction函数

override protected def raw2prediction(rawPrediction: Vector): Double = { 
  if (!isDefined(thresholds)) { 
    rawPrediction.argmax
  } else { 
    probability2prediction(raw2probability(rawPrediction))
  } 
}
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可以看到，如果没有设置thresholds数组（一般不会设置），直接返回了入参rawPrediction向量中最大元素所在的位置（index），举例来说rawPrediction如果是[2.3, 1.2, 5.1, 3.4]，则返回2(最大值5.1)。rawPrediction来自于predictRaw函数

override protected def predictRaw(features: Vector): Vector = { 
  if (isMultinomial) { 
    margins(features)
  } else { 
    val m = margin(features)
    Vectors.dense(-m, m)
  } 
}
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直接调用了margins函数

private val margins: Vector => Vector = (features) => { 
  val m = interceptVector.toDense.copy
  //m = alpha * coefficientMatrix * features + beta * m
  BLAS.gemv(1.0, coefficientMatrix, features, 1.0, m)
  m 
}
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代码比较简单，系数矩阵分别于特征向量相乘，再与截距向量相加。

1.3.2. save model

使用LogisticRegressionModelWriter将训练的参数和得到的系数矩阵写入hdfs

class LogisticRegressionModelWriter(instance: LogisticRegressionModel)
  extends MLWriter with Logging { 

  private case class Data(
      numClasses: Int,
      numFeatures: Int,
      interceptVector: Vector,
      coefficientMatrix: Matrix,
      isMultinomial: Boolean)

  override protected def saveImpl(path: String): Unit = { 
    //训练时的参数
    DefaultParamsWriter.saveMetadata(instance, path, sc)
    //保存训练结果
    val data = Data(instance.numClasses, instance.numFeatures, instance.interceptVector,
      instance.coefficientMatrix, instance.isMultinomial)
    val dataPath = new Path(path, "data").toString
    sparkSession.createDataFrame(Seq(data)).repartition(1).write.parquet(dataPath)
  } 
} 
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metadata中除了训练参数，还保存了训练时的环境，官方demo的训练参数保存结果

{
    "class": "org.apache.spark.ml.classification.LogisticRegressionModel",
    "timestamp": 1500886361787,
    "sparkVersion": "2.0.2",
    "uid": "logreg_ea57ce7dcde4",
    "paramMap": {
        "fitIntercept": true,
        "rawPredictionCol": "rawPrediction",
        "predictionCol": "prediction",
        "tol": 0.000001,
        "labelCol": "label",
        "standardization": true,
        "regParam": 0.3,
        "probabilityCol": "probability",
        "featuresCol": "features",
        "maxIter": 10,
        "elasticNetParam": 0.8,
        "threshold": 0.5
    }
}
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1.3.3. load model

使用LogisticRegressionModelReader将save保存的模型读取回来，metadata使用json解析回来，解析parquet获取系数矩阵，截距等，比较简单。

1.4. LogisticAggregator

LogisticAggregator用于训练过程中，计算每轮迭代的梯度和loss，需要分布式计算，类似于上面的summarizer，也是用在treeAggregator中。

1.4.1. 算法

用于训练过程中计算梯度与loss，在前面介绍L-BFGS时说过其训练结果返回的系数向量只有k-1维，预测时则默认class 0的margin是0，这种是带pivot class，二分类属于这种；这里的多分类不使用这种方法，而是训练得到k个class分别对应的系数。

1.4.1. 1. 二分类

如前文所述，二分类是有pivot，一般二分类的梯度

$$\frac{\delta}{\delta_{\theta _j}}J(\theta)=-\frac{1}{m}\sum_{i=1}^m (h_\theta(x_i)-y_i)x_{i,j}\\ J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}[y^i\log h_\theta(x^i)+(1-y^i)\log(1-h_\theta(x^i))]$$
这里的m是样本数，对于单个样本m=1，h是sigmoid函数，y是label，整理可得