Github user iyounus commented on the pull request: https://github.com/apache/spark/pull/10702#issuecomment-177375091 I've completed this PR. I think all the tests are there. Here, I'm going to document a couple of minor issues just for future reference. __Issue 1__ For the case when `yStd = 0` and `fitIntercept = false`, we've four possibilities (`reParam: zero/non-zero` and `standardization: true/false`). Using `WeightedLeastSquares` (`normal` solver), I _can_ get the following results: ``` # data used for the following results val df = sc.parallelize(Seq( (17.0, Vectors.dense(0.0, 5.0)), (17.0, Vectors.dense(1.0, 7.0)), (17.0, Vectors.dense(2.0, 11.0)), (17.0, Vectors.dense(3.0, 13.0)) ), 2).toDF("label", "features") ``` ``` # coefficients obtained from WeightedLeastSquares (1) reg: 0.0, standardization: false --------> 0.0 [-9.508474576271158,3.457627118644062] (2) reg: 0.0, standardization: true --------> 0.0 [-9.508474576271158,3.457627118644062] (3) reg: 0.1, standardization: false --------> 0.0 [-7.134240246406588,3.010780287474336] (4) reg: 0.1, standardization: true --------> 0.0 [-5.730337078651679,2.7219101123595495] ``` This is with `L2` regularization, and ignoring standardization of the label for the case (4). For the case (4), we throw an error because this is ill-defined, so the user never sees these results. For case (3), even though the standardization is `false`, the label is still standardized because the `standardizeLable` is hardwired to be `true` when calling `WeightedLeastSquares` within `LinearRegression` class. Therefore, an error is thrown in this case too. Which, in my opinion, is not right thing to do because the analytical solution does exist for this case. __Issue 2__ Again, for the case when `yStd = 0` and `fitIntercept = false`, I can get the following results using `l-bfgs` solver: ``` (1) reg: 0.0, standardization: false --------> 0.0 [-9.508474576271176,3.4576271186440652] (2) reg: 0.0, standardization: true --------> 0.0 [-9.508474576271176,3.4576271186440652] (3) reg: 0.1, standardization: false --------> 0.0 [-9.327614273741196,3.423618722197146] (4) reg: 0.1, standardization: true --------> 0.0 [-9.08129403505256,3.374915377479131] ``` Here, results (1) and (2) are identical to what we get from `WeightedLeastSquares` as expected. Case (4) is ill-defined and we throw an error. Now, for case (3), the numerical values are different as compared to `WeightedLeastSquares`. This is because we standardize label using `yMean`. Otherwise, the values obtained from `l-bfgs` are identical to `WeightedLeastSquares`. Note that the user will not see these values because an error is thrown for this case instead. __Issue 3__ The normal equation with regression (Ridge Regression), gives significantly different results as compared to case (3) above. Here is my R code with results: ``` ridge_regression <- function(A, b, lambda, intercept=TRUE){ if (intercept) { A = cbind(rep(1.0, length(b)), A) I = diag(ncol(A)) I[1,1] = 0.0 } else { I = diag(ncol(A)) } R = chol( t(A) %*% A + lambda*I ) z = solve(t(R), t(A) %*% b) w = solve(R, z) return(w) } A <- matrix(c(0, 1, 2, 3, 5, 7, 11, 13), 4, 2) b <- c(17, 17, 17, 17) df <- as.data.frame(cbind(A, b)) ridge_regression(A, b, 0.1, intercept = FALSE) [1,] -8.783272 [2,] 3.321237 ``` In my opinion, when `standardization=flase`, the results from `normal` solver must match this. Even though the user doesn't see this case, it gives me less confidence in the implementation of normal equation, because it doesn't match this simple case. I also wrote about this at https://github.com/apache/spark/pull/10274.
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