Github user actuaryzhang commented on a diff in the pull request:
https://github.com/apache/spark/pull/16131#discussion_r90784498
--- Diff:
mllib/src/main/scala/org/apache/spark/ml/regression/GeneralizedLinearRegression.scala
---
@@ -505,7 +505,7 @@ object GeneralizedLinearRegression extends
DefaultParamsReadable[GeneralizedLine
override def initialize(y: Double, weight: Double): Double = {
require(y >= 0.0, "The response variable of Poisson family " +
s"should be non-negative, but got $y")
- y
+ y + 0.1
--- End diff --
The issue is initializing mu = y for the case of y = 0 can lead to almost
zero weight in subsequent IWLS. I'll explain in detail below.
The following initialization method from `FamilyAndLink` shows that the
mean `mu` is initialized using `family.initialize`. So `mu` could be zero if `y
= 0`, which is incorrect since the mean of the Poisson can never be zero.
```
val newInstances = instances.map { instance =>
val mu = family.initialize(instance.label, instance.weight)
val eta = predict(mu)
Instance(eta, instance.weight, instance.features)
}
```
In each iteration of the reweighted least squares, the weight fed to WLS is
defined as follows (`reweightFunc`):
```
val eta = model.predict(instance.features)
val mu = fitted(eta)
val offset = eta + (instance.label - mu) * link.deriv(mu)
val weight = instance.weight / (math.pow(this.link.deriv(mu), 2.0)
* family.variance(mu))
```
Let's use one observation as an example. Suppose mu = y = 0. Then running
the above prints:
```
offset: -32.43970761868737 weight: 2.2177289643212133E-14
```
The weight is almost zero, which is the cause of the issue. To see why, in
the poisson case with log link, we have `(math.pow(this.link.deriv(mu), 2.0) *
family.variance(mu))` = 1/mu where `link.deriv(mu)` = 1/mu^2 and
`family.variance(mu)` is `mu`. So the weight is basically mu.
That also explains why using small epsilon as initialization may not be
helpful. I added 0.1 because that is how R does it:
```
> poisson()$initialize
expression({
if (any(y < 0))
stop("negative values not allowed for the 'Poisson' family")
n <- rep.int(1, nobs)
mustart <- y + 0.1
})
```
@srowen
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