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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