[jira] [Created] (SPARK-22555) Possibly incorrect scaling of L2 regularization strength in LinearRegression

["Andrew Crosby (JIRA)" <ji...@apache.org>]

Andrew Crosby created SPARK-22555:
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             Summary: Possibly incorrect scaling of L2 regularization strength in LinearRegression
                 Key: SPARK-22555
                 URL: https://issues.apache.org/jira/browse/SPARK-22555
             Project: Spark
          Issue Type: Bug
          Components: ML
    Affects Versions: 2.2.0
            Reporter: Andrew Crosby
            Priority: Minor


According to the Spark documentation, the linear regression estimator minimizes the regularized sum of squares:

1/N Sum(y - w x)^2^ + λ( (1-α) |w|~2~ + α |w|~1~ )

Under the hood, in order to improve convergence, the optimization algorithms actually work in scaled space using the variables y' = y / σ ~y~, x' = x / σ ~x~ and w' = w / (σ ~x~ / σ ~y~). In terms of these scaled variables, the above expression becomes:

σ ~y~^2^ ( 1/N  Sum(y' - w' x')^2^ + λ( (1-α) / σ ~x~^2^ |w'|~2~ + α / (σ ~x~ σ ~y~) |w'|~1~ ) )

The solution in scaled space is equivalent to the original problem, provided that the regularization strengths are suitably adjusted. The effective L1 regularization strength should be λ α / (σ ~x~ σ ~y~) and the effective L2 regularization strength should be λ (1-α) / σ ~x~^2^.

However, this doesn't quite match the regularization strengths that are actually used. While the factors of σ ~x~ are correctly included (or correctly ommitted if the standardization parameter is set), it appears that the 1 / σ ~y~ scaling is applied to both the L1 and L2 regularization parameters instead of just to the L1 regularization parameter. Both LinearRegression.scala and WeightedLeastSquares.scala contain code along the following lines:

{code}
val effectiveRegParam = $(regParam) / yStd
val effectiveL1RegParam = $(elasticNetParam) * effectiveRegParam
val effectiveL2RegParam = (1.0 - $(elasticNetParam)) * effectiveRegParam
{code}

Admittedly, the unit tests confirm that the current behaviour matches that of R's glmnet, it just doesn't seem to match the behaviour claimed in the documentation.



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