large margins and the growth function

[ilya239]

I'm trying to understand why the large-margin requirement affects the growth function. For any size margin, we can find three points far enough from each other that they are shattered by perceptrons with at least that margin. How, then, is the growth function at n=3 less than 8?

Or, put differently: the growth function is a property of the hypothesis set. The large-margin requirement does not remove any hypotheses from the hypothesis set; it just prevents us from using particular hypotheses for particular training sets. This limitation is a property of the learning algorithm, but the VC analysis was independent of learning algorithm. If the hypothesis set has not changed, how can the growth function change?

[yaser]

Originally Posted by ilya239

I'm trying to understand why the large-margin requirement affects the growth function. For any size margin, we can find three points far enough from each other that they are shattered by perceptrons with at least that margin.

The generalization result that relates to the margin assumes all the points lie within a limited-size region (so the value of the margin is meaningful relative to that).

the growth function is a property of the hypothesis set. The large-margin requirement does not remove any hypotheses from the hypothesis set; it just prevents us from using particular hypotheses for particular training sets. This limitation is a property of the learning algorithm, but the VC analysis was independent of learning algorithm.

You are right. In this result, some liberty is taken in distinguishing the hypothesis set from the learning algorithm. The same liberty is also taken in the case of nearest-neighbor classifiers (a simple model that will be mentioned briefly in Lecture 16).

I would take the margin-based arguments for generalization as just motivational, and rely on the generalization results that relate to the number of support vectors.

[ilya239]

Originally Posted by yaser

The generalization result that relates to the margin assumes all the points lie within a limited-size region (so the value of the margin is meaningful relative to that).

Ah, thanks, makes sense. In real problems, parameters often do have only limited ranges.

Originally Posted by yaser

some liberty is taken in distinguishing the hypothesis set from the learning algorithm. The same liberty is also taken in the case of nearest-neighbor classifiers (a simple model that will be mentioned briefly in Lecture 16).

I see now -- it's like with heavy regularization, if the cost of complex hypotheses becomes so exorbitant that they're never chosen in practice, you could just as well say they've been removed from the hypothesis set. If they're rarely chosen, it's as if each hypothesis in the set has a "weight". I wonder if there exists a "fuzzy" version of VC analysis that formalizes this "middle ground" created by margins or regularizers.

Thanks for help!