Initialized the regVal for first iteration in SGD optimizer

Ported from https://github.com/apache/incubator-spark/pull/633

In runMiniBatchSGD, the regVal (for 1st iter) should be initialized
as sum of sqrt of weights if it's L2 update; for L1 update, the same logic is followed.

It maybe not be important here for SGD since the updater doesn't take the loss
as parameter to find the new weights. But it will give us the correct history of loss.
However, for LBFGS optimizer we implemented, the correct loss with regVal is crucial to
find the new weights.

Author: DB Tsai <dbtsai@alpinenow.com>

Closes #40 from dbtsai/dbtsai-smallRegValFix and squashes the following commits:

77d47da [DB Tsai] In runMiniBatchSGD, the regVal (for 1st iter) should be initialized as sum of sqrt of weights if it's L2 update; for L1 update, the same logic is followed.

mllib/src/main/scala/org/apache/spark/mllib/optimization/GradientDescent.scala

@@ -149,7 +149,13 @@ object GradientDescent extends Logging {
 
     // Initialize weights as a column vector
     var weights = new DoubleMatrix(initialWeights.length, 1, initialWeights:_*)
-    var regVal = 0.0
+
+    /**
+     * For the first iteration, the regVal will be initialized as sum of sqrt of
+     * weights if it's L2 update; for L1 update; the same logic is followed.
+     */
+    var regVal = updater.compute(
+      weights, new DoubleMatrix(initialWeights.length, 1), 0, 1, regParam)._2
 
     for (i <- 1 to numIterations) {
       // Sample a subset (fraction miniBatchFraction) of the total data

mllib/src/main/scala/org/apache/spark/mllib/optimization/Updater.scala

@@ -111,6 +111,8 @@ class SquaredL2Updater extends Updater {
     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)
+    // w' = w - thisIterStepSize * (gradient + regParam * w)
+    // w' = (1 - thisIterStepSize * regParam) * w - thisIterStepSize * gradient
     val newWeights = weightsOld.mul(1.0 - thisIterStepSize * regParam).sub(step)
     (newWeights, 0.5 * pow(newWeights.norm2, 2.0) * regParam)
   }

mllib/src/test/scala/org/apache/spark/mllib/optimization/GradientDescentSuite.scala

@@ -104,4 +104,45 @@ class GradientDescentSuite extends FunSuite with LocalSparkContext with ShouldMa
     val lossDiff = loss.init.zip(loss.tail).map { case (lhs, rhs) => lhs - rhs }
     assert(lossDiff.count(_ > 0).toDouble / lossDiff.size > 0.8)
   }
+
+  test("Test the loss and gradient of first iteration with regularization.") {
+
+    val gradient = new LogisticGradient()
+    val updater = new SquaredL2Updater()
+
+    // Add a extra variable consisting of all 1.0's for the intercept.
+    val testData = GradientDescentSuite.generateGDInput(2.0, -1.5, 10000, 42)
+    val data = testData.map { case LabeledPoint(label, features) =>
+      label -> Array(1.0, features: _*)
+    }
+
+    val dataRDD = sc.parallelize(data, 2).cache()
+
+    // Prepare non-zero weights
+    val initialWeightsWithIntercept = Array(1.0, 0.5)
+
+    val regParam0 = 0
+    val (newWeights0, loss0) = GradientDescent.runMiniBatchSGD(
+      dataRDD, gradient, updater, 1, 1, regParam0, 1.0, initialWeightsWithIntercept)
+
+    val regParam1 = 1
+    val (newWeights1, loss1) = GradientDescent.runMiniBatchSGD(
+      dataRDD, gradient, updater, 1, 1, regParam1, 1.0, initialWeightsWithIntercept)
+
+    def compareDouble(x: Double, y: Double, tol: Double = 1E-3): Boolean = {
+      math.abs(x - y) / (math.abs(y) + 1e-15) < tol
+    }
+
+    assert(compareDouble(
+      loss1(0),
+      loss0(0) + (math.pow(initialWeightsWithIntercept(0), 2) +
+        math.pow(initialWeightsWithIntercept(1), 2)) / 2),
+      """For non-zero weights, the regVal should be \frac{1}{2}\sum_i w_i^2.""")
+
+    assert(
+      compareDouble(newWeights1(0) , newWeights0(0) - initialWeightsWithIntercept(0)) &&
+      compareDouble(newWeights1(1) , newWeights0(1) - initialWeightsWithIntercept(1)),
+      "The different between newWeights with/without regularization " +
+        "should be initialWeightsWithIntercept.")
+  }
 }