[SPARK-4907][MLlib] Inconsistent loss and gradient in LeastSquaresGra…

…dient compared with R

In most of the academic paper and algorithm implementations,
people use L = 1/2n ||A weights-y||^2 instead of L = 1/n ||A weights-y||^2
for least-squared loss. See Eq. (1) in http://web.stanford.edu/~hastie/Papers/glmnet.pdf

Since MLlib uses different convention, this will result different residuals and
all the stats properties will be different from GLMNET package in R.

The model coefficients will be still the same under this change.

Author: DB Tsai <dbtsai@alpinenow.com>

Closes #3746 from dbtsai/lir and squashes the following commits:

19c2e85 [DB Tsai] make stepsize twice to converge to the same solution
0b2c29c [DB Tsai] first commit

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

@@ -94,16 +94,16 @@ class LogisticGradient extends Gradient {
  * :: DeveloperApi ::
  * Compute gradient and loss for a Least-squared loss function, as used in linear regression.
  * This is correct for the averaged least squares loss function (mean squared error)
- *              L = 1/n ||A weights-y||^2
+ *              L = 1/2n ||A weights-y||^2
  * See also the documentation for the precise formulation.
  */
 @DeveloperApi
 class LeastSquaresGradient extends Gradient {
   override def compute(data: Vector, label: Double, weights: Vector): (Vector, Double) = {
     val diff = dot(data, weights) - label
-    val loss = diff * diff
+    val loss = diff * diff / 2.0
     val gradient = data.copy
-    scal(2.0 * diff, gradient)
+    scal(diff, gradient)
     (gradient, loss)
   }
 
@@ -113,8 +113,8 @@ class LeastSquaresGradient extends Gradient {
       weights: Vector,
       cumGradient: Vector): Double = {
     val diff = dot(data, weights) - label
-    axpy(2.0 * diff, data, cumGradient)
-    diff * diff
+    axpy(diff, data, cumGradient)
+    diff * diff / 2.0
   }
 }
 

mllib/src/test/scala/org/apache/spark/mllib/regression/StreamingLinearRegressionSuite.scala

@@ -52,7 +52,7 @@ class StreamingLinearRegressionSuite extends FunSuite with TestSuiteBase {
     // create model
     val model = new StreamingLinearRegressionWithSGD()
       .setInitialWeights(Vectors.dense(0.0, 0.0))
-      .setStepSize(0.1)
+      .setStepSize(0.2)
       .setNumIterations(25)
 
     // generate sequence of simulated data
@@ -84,7 +84,7 @@ class StreamingLinearRegressionSuite extends FunSuite with TestSuiteBase {
     // create model
     val model = new StreamingLinearRegressionWithSGD()
       .setInitialWeights(Vectors.dense(0.0))
-      .setStepSize(0.1)
+      .setStepSize(0.2)
       .setNumIterations(25)
 
     // generate sequence of simulated data
@@ -118,7 +118,7 @@ class StreamingLinearRegressionSuite extends FunSuite with TestSuiteBase {
     // create model initialized with true weights
     val model = new StreamingLinearRegressionWithSGD()
       .setInitialWeights(Vectors.dense(10.0, 10.0))
-      .setStepSize(0.1)
+      .setStepSize(0.2)
       .setNumIterations(25)
 
     // generate sequence of simulated data for testing