MLLIB-22. Support negative implicit input in ALS

I'm back with another less trivial suggestion for ALS:

In ALS for implicit feedback, input values are treated as weights on squared-errors in a loss function (or rather, the weight is a simple function of the input r, like c = 1 + alpha*r). The paper on which it's based assumes that the input is positive. Indeed, if the input is negative, it will create a negative weight on squared-errors, which causes things to go haywire. The optimization will try to make the error in a cell as large possible, and the result is silently bogus.

There is a good use case for negative input values though. Implicit feedback is usually collected from signals of positive interaction like a view or like or buy, but equally, can come from "not interested" signals. The natural representation is negative values.

The algorithm can be extended quite simply to provide a sound interpretation of these values: negative values should encourage the factorization to come up with 0 for cells with large negative input values, just as much as positive values encourage it to come up with 1.

The implications for the algorithm are simple:
* the confidence function value must not be negative, and so can become 1 + alpha*|r|
* the matrix P should have a value 1 where the input R is _positive_, not merely where it is non-zero. Actually, that's what the paper already says, it's just that we can't assume P = 1 when a cell in R is specified anymore, since it may be negative

This in turn entails just a few lines of code change in `ALS.scala`:
* `rs(i)` becomes `abs(rs(i))`
* When constructing `userXy(us(i))`, it's implicitly only adding where P is 1. That had been true for any us(i) that is iterated over, before, since these are exactly the ones for which P is 1. But now P is zero where rs(i) <= 0, and should not be added

I think it's a safe change because:
* It doesn't change any existing behavior (unless you're using negative values, in which case results are already borked)
* It's the simplest direct extension of the paper's algorithm
* (I've used it to good effect in production FWIW)

Tests included.

I tweaked minor things en route:
* `ALS.scala` javadoc writes "R = Xt*Y" when the paper and rest of code defines it as "R = X*Yt"
* RMSE in the ALS tests uses a confidence-weighted mean, but the denominator is not actually sum of weights

Excuse my Scala style; I'm sure it needs tweaks.

Author: Sean Owen <sowen@cloudera.com>

Closes #500 from srowen/ALSNegativeImplicitInput and squashes the following commits:

cf902a9 [Sean Owen] Support negative implicit input in ALS
953be1c [Sean Owen] Make weighted RMSE in ALS test actually weighted; adjust comment about R = X*Yt

mllib/src/main/scala/org/apache/spark/mllib/recommendation/ALS.scala

@@ -64,7 +64,7 @@ case class Rating(val user: Int, val product: Int, val rating: Double)
  * Alternating Least Squares matrix factorization.
  *
  * ALS attempts to estimate the ratings matrix `R` as the product of two lower-rank matrices,
- * `X` and `Y`, i.e. `Xt * Y = R`. Typically these approximations are called 'factor' matrices.
+ * `X` and `Y`, i.e. `X * Yt = R`. Typically these approximations are called 'factor' matrices.
  * The general approach is iterative. During each iteration, one of the factor matrices is held
  * constant, while the other is solved for using least squares. The newly-solved factor matrix is
  * then held constant while solving for the other factor matrix.
@@ -384,8 +384,16 @@ class ALS private (var numBlocks: Int, var rank: Int, var iterations: Int, var l
               userXtX(us(i)).addi(tempXtX)
               SimpleBlas.axpy(rs(i), x, userXy(us(i)))
             case true =>
-              userXtX(us(i)).addi(tempXtX.mul(alpha * rs(i)))
-              SimpleBlas.axpy(1 + alpha * rs(i), x, userXy(us(i)))
+              // Extension to the original paper to handle rs(i) < 0. confidence is a function
+              // of |rs(i)| instead so that it is never negative:
+              val confidence = 1 + alpha * abs(rs(i))
+              userXtX(us(i)).addi(tempXtX.mul(confidence - 1))
+              // For rs(i) < 0, the corresponding entry in P is 0 now, not 1 -- negative rs(i)
+              // means we try to reconstruct 0. We add terms only where P = 1, so, term below
+              // is now only added for rs(i) > 0:
+              if (rs(i) > 0) {
+                SimpleBlas.axpy(confidence, x, userXy(us(i)))
+              }
           }
         }
       }

mllib/src/test/java/org/apache/spark/mllib/recommendation/JavaALSSuite.java

@@ -19,7 +19,6 @@
 
 import java.io.Serializable;
 import java.util.List;
-import java.lang.Math;
 
 import org.junit.After;
 import org.junit.Assert;
@@ -46,7 +45,7 @@ public void tearDown() {
     System.clearProperty("spark.driver.port");
   }
 
-  void validatePrediction(MatrixFactorizationModel model, int users, int products, int features, 
+  static void validatePrediction(MatrixFactorizationModel model, int users, int products, int features,
       DoubleMatrix trueRatings, double matchThreshold, boolean implicitPrefs, DoubleMatrix truePrefs) {
     DoubleMatrix predictedU = new DoubleMatrix(users, features);
     List<scala.Tuple2<Object, double[]>> userFeatures = model.userFeatures().toJavaRDD().collect();
@@ -84,15 +83,15 @@ void validatePrediction(MatrixFactorizationModel model, int users, int products,
         for (int p = 0; p < products; ++p) {
           double prediction = predictedRatings.get(u, p);
           double truePref = truePrefs.get(u, p);
-          double confidence = 1.0 + /* alpha = */ 1.0 * trueRatings.get(u, p);
+          double confidence = 1.0 + /* alpha = */ 1.0 * Math.abs(trueRatings.get(u, p));
           double err = confidence * (truePref - prediction) * (truePref - prediction);
           sqErr += err;
-          denom += 1.0;
+          denom += confidence;
         }
       }
       double rmse = Math.sqrt(sqErr / denom);
       Assert.assertTrue(String.format("Confidence-weighted RMSE=%2.4f above threshold of %2.2f",
-              rmse, matchThreshold), Math.abs(rmse) < matchThreshold);
+              rmse, matchThreshold), rmse < matchThreshold);
     }
   }
 
@@ -103,7 +102,7 @@ public void runALSUsingStaticMethods() {
     int users = 50;
     int products = 100;
     scala.Tuple3<List<Rating>, DoubleMatrix, DoubleMatrix> testData = ALSSuite.generateRatingsAsJavaList(
-        users, products, features, 0.7, false);
+        users, products, features, 0.7, false, false);
 
     JavaRDD<Rating> data = sc.parallelize(testData._1());
     MatrixFactorizationModel model = ALS.train(data.rdd(), features, iterations);
@@ -117,7 +116,7 @@ public void runALSUsingConstructor() {
     int users = 100;
     int products = 200;
     scala.Tuple3<List<Rating>, DoubleMatrix, DoubleMatrix> testData = ALSSuite.generateRatingsAsJavaList(
-        users, products, features, 0.7, false);
+        users, products, features, 0.7, false, false);
 
     JavaRDD<Rating> data = sc.parallelize(testData._1());
 
@@ -134,7 +133,7 @@ public void runImplicitALSUsingStaticMethods() {
     int users = 80;
     int products = 160;
     scala.Tuple3<List<Rating>, DoubleMatrix, DoubleMatrix> testData = ALSSuite.generateRatingsAsJavaList(
-      users, products, features, 0.7, true);
+        users, products, features, 0.7, true, false);
 
     JavaRDD<Rating> data = sc.parallelize(testData._1());
     MatrixFactorizationModel model = ALS.trainImplicit(data.rdd(), features, iterations);
@@ -148,7 +147,7 @@ public void runImplicitALSUsingConstructor() {
     int users = 100;
     int products = 200;
     scala.Tuple3<List<Rating>, DoubleMatrix, DoubleMatrix> testData = ALSSuite.generateRatingsAsJavaList(
-      users, products, features, 0.7, true);
+        users, products, features, 0.7, true, false);
 
     JavaRDD<Rating> data = sc.parallelize(testData._1());
 
@@ -158,4 +157,19 @@ public void runImplicitALSUsingConstructor() {
       .run(data.rdd());
     validatePrediction(model, users, products, features, testData._2(), 0.4, true, testData._3());
   }
+
+  @Test
+  public void runImplicitALSWithNegativeWeight() {
+    int features = 2;
+    int iterations = 15;
+    int users = 80;
+    int products = 160;
+    scala.Tuple3<List<Rating>, DoubleMatrix, DoubleMatrix> testData = ALSSuite.generateRatingsAsJavaList(
+        users, products, features, 0.7, true, true);
+
+    JavaRDD<Rating> data = sc.parallelize(testData._1());
+    MatrixFactorizationModel model = ALS.trainImplicit(data.rdd(), features, iterations);
+    validatePrediction(model, users, products, features, testData._2(), 0.4, true, testData._3());
+  }
+
 }

mllib/src/test/scala/org/apache/spark/mllib/recommendation/ALSSuite.scala

@@ -18,9 +18,9 @@
 package org.apache.spark.mllib.recommendation
 
 import scala.collection.JavaConversions._
+import scala.math.abs
 import scala.util.Random
 
-import org.scalatest.BeforeAndAfterAll
 import org.scalatest.FunSuite
 
 import org.jblas._
@@ -34,7 +34,8 @@ object ALSSuite {
       products: Int,
       features: Int,
       samplingRate: Double,
-      implicitPrefs: Boolean): (java.util.List[Rating], DoubleMatrix, DoubleMatrix) = {
+      implicitPrefs: Boolean,
+      negativeWeights: Boolean): (java.util.List[Rating], DoubleMatrix, DoubleMatrix) = {
     val (sampledRatings, trueRatings, truePrefs) =
       generateRatings(users, products, features, samplingRate, implicitPrefs)
     (seqAsJavaList(sampledRatings), trueRatings, truePrefs)
@@ -45,7 +46,8 @@ object ALSSuite {
       products: Int,
       features: Int,
       samplingRate: Double,
-      implicitPrefs: Boolean = false): (Seq[Rating], DoubleMatrix, DoubleMatrix) = {
+      implicitPrefs: Boolean = false,
+      negativeWeights: Boolean = false): (Seq[Rating], DoubleMatrix, DoubleMatrix) = {
     val rand = new Random(42)
 
     // Create a random matrix with uniform values from -1 to 1
@@ -56,7 +58,9 @@ object ALSSuite {
     val productMatrix = randomMatrix(features, products)
     val (trueRatings, truePrefs) = implicitPrefs match {
       case true =>
-        val raw = new DoubleMatrix(users, products, Array.fill(users * products)(rand.nextInt(10).toDouble): _*)
+        // Generate raw values from [0,9], or if negativeWeights, from [-2,7]
+        val raw = new DoubleMatrix(users, products,
+          Array.fill(users * products)((if (negativeWeights) -2 else 0) + rand.nextInt(10).toDouble): _*)
         val prefs = new DoubleMatrix(users, products, raw.data.map(v => if (v > 0) 1.0 else 0.0): _*)
         (raw, prefs)
       case false => (userMatrix.mmul(productMatrix), null)
@@ -107,6 +111,10 @@ class ALSSuite extends FunSuite with LocalSparkContext {
     testALS(100, 200, 2, 15, 0.7, 0.4, true, true)
   }
 
+  test("rank-2 matrices implicit negative") {
+    testALS(100, 200, 2, 15, 0.7, 0.4, true, false, true)
+  }
+
   /**
    * Test if we can correctly factorize R = U * P where U and P are of known rank.
    *
@@ -118,13 +126,14 @@ class ALSSuite extends FunSuite with LocalSparkContext {
    * @param matchThreshold max difference allowed to consider a predicted rating correct
    * @param implicitPrefs  flag to test implicit feedback
    * @param bulkPredict    flag to test bulk prediciton
+   * @param negativeWeights whether the generated data can contain negative values
    */
   def testALS(users: Int, products: Int, features: Int, iterations: Int,
     samplingRate: Double, matchThreshold: Double, implicitPrefs: Boolean = false,
-    bulkPredict: Boolean = false)
+    bulkPredict: Boolean = false, negativeWeights: Boolean = false)
   {
     val (sampledRatings, trueRatings, truePrefs) = ALSSuite.generateRatings(users, products,
-      features, samplingRate, implicitPrefs)
+      features, samplingRate, implicitPrefs, negativeWeights)
     val model = implicitPrefs match {
       case false => ALS.train(sc.parallelize(sampledRatings), features, iterations)
       case true => ALS.trainImplicit(sc.parallelize(sampledRatings), features, iterations)
@@ -166,13 +175,13 @@ class ALSSuite extends FunSuite with LocalSparkContext {
       for (u <- 0 until users; p <- 0 until products) {
         val prediction = predictedRatings.get(u, p)
         val truePref = truePrefs.get(u, p)
-        val confidence = 1 + 1.0 * trueRatings.get(u, p)
+        val confidence = 1 + 1.0 * abs(trueRatings.get(u, p))
         val err = confidence * (truePref - prediction) * (truePref - prediction)
         sqErr += err
-        denom += 1
+        denom += confidence
       }
       val rmse = math.sqrt(sqErr / denom)
-      if (math.abs(rmse) > matchThreshold) {
+      if (rmse > matchThreshold) {
         fail("Model failed to predict RMSE: %f\ncorr: %s\npred: %s\nU: %s\n P: %s".format(
           rmse, truePrefs, predictedRatings, predictedU, predictedP))
       }