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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