What is the meaning of "CD(x, candidateGroup;model)-CD(x,selectedGroup; model)"?

[LinkToPast1990]

Hi, @csinva

Referring to the CD paper, we have p=Softmax(Wβ+Wγ), and Wβ provides a quantitative score for the given phrase.
But for different phrases, the Wβ terms may have very different values while the softmax results remain similar, is it reasonable to do subtraction between two Wβ terms?

Thanks!

[csinva]

Hello!

A good question! Indeed for the final layer of a classification network, we have p=Softmax(Wβ+Wγ). A reasonable way to think of the CD-score for a group of features (i.e. the group corresponding to β) is like the input * a logistic regression coefficient, in a 2-term logistic regression. You can take the CD score either with or without the softmax.

What you suggest, with the subtraction does, in fact, make sense: it is like taking the importance to be one coefficient - another coefficient. In some cases, this may make sense, but in general just taking the original CD score for a group is probably more intuitive.

[LinkToPast1990]

Hi, @csinva
Thanks!
The logistic regression is

Could you kindly further explain how could we "think of the CD-score for a group of features (i.e. the group corresponding to β) is like the input * a logistic regression coefficient"?

CD score is Wβ, right? Which one is the coefficient, W or β?

[csinva]

Hey again - yes CD score is Wβ. Thus Wβ would be the coefficient for the group of features we are considering. Wγ would be the coefficient corresponding to the rest of the features.

[LinkToPast1990]

Thanks again.
In CD, we could find that f(x) = SoftMax(g(x)) = SoftMax(Wβ+Wγ).
I thought the β vector is representing the group of features we are considering previously.
But based on your comments, it seems that the logistic regression should be:
SoftMax(logits) = SoftMax(bias + Wβ * group_considering + Wγ * group_rest), right?
In where we model this regression?

[csinva]

Yes sorry your insights are correct and my description is a little confusing.

CD computes Wβ, and this represents the coefficient * the group - both are already multiplied and we can't disentangle them and say one is the coefficient and one is the input.

When we write, SoftMax(logits) = SoftMax(Wβ * group_considering + Wγ * group_rest), I like to think of group_considering and group_rest as binary 0/1 variables. Alternatively, you can just think of SoftMax(Wβ+Wγ) without the need for the binary terms.

Hope that helps!