SPARK-2748 [MLLIB] [GRAPHX] Loss of precision for small arguments to Math.exp, Math.log

In a few places in MLlib, an expression of the form `log(1.0 + p)` is evaluated. When p is so small that `1.0 + p == 1.0`, the result is 0.0. However the correct answer is very near `p`. This is why `Math.log1p` exists.

Similarly for one instance of `exp(m) - 1` in GraphX; there's a special `Math.expm1` method.

While the errors occur only for very small arguments, given their use in machine learning algorithms, this is entirely possible.

Also note the related PR for Python: https://github.com/apache/spark/pull/1652

Author: Sean Owen <srowen@gmail.com>

Closes #1659 from srowen/SPARK-2748 and squashes the following commits:

c5926d4 [Sean Owen] Use log1p, expm1 for better precision for tiny arguments

graphx/src/main/scala/org/apache/spark/graphx/util/GraphGenerators.scala

@@ -100,8 +100,10 @@ object GraphGenerators {
    */
   private def sampleLogNormal(mu: Double, sigma: Double, maxVal: Int): Int = {
     val rand = new Random()
-    val m = math.exp(mu + (sigma * sigma) / 2.0)
-    val s = math.sqrt((math.exp(sigma*sigma) - 1) * math.exp(2*mu + sigma*sigma))
+    val sigmaSq = sigma * sigma
+    val m = math.exp(mu + sigmaSq / 2.0)
+    // expm1 is exp(m)-1 with better accuracy for tiny m
+    val s = math.sqrt(math.expm1(sigmaSq) * math.exp(2*mu + sigmaSq))
     // Z ~ N(0, 1)
     var X: Double = maxVal
 

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

@@ -68,9 +68,9 @@ class LogisticGradient extends Gradient {
     val gradient = brzData * gradientMultiplier
     val loss =
       if (label > 0) {
-        math.log(1 + math.exp(margin))
+        math.log1p(math.exp(margin)) // log1p is log(1+p) but more accurate for small p
       } else {
-        math.log(1 + math.exp(margin)) - margin
+        math.log1p(math.exp(margin)) - margin
       }
 
     (Vectors.fromBreeze(gradient), loss)
@@ -89,9 +89,9 @@ class LogisticGradient extends Gradient {
     brzAxpy(gradientMultiplier, brzData, cumGradient.toBreeze)
 
     if (label > 0) {
-      math.log(1 + math.exp(margin))
+      math.log1p(math.exp(margin))
     } else {
-      math.log(1 + math.exp(margin)) - margin
+      math.log1p(math.exp(margin)) - margin
     }
   }
 }