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Assume I have a couple of thousand hens that I want to classify into those that never lay an egg and those that will at some point in their life lay an egg. Assume that already works perfectly.

Now there are a few hens who do lay eggs, but at some point won't do so for a couple of years. Those hens are a really small minority - let's say a hundred.

Now, i want my network to classify a hen that at some point will lay an egg but won't do so for a couple of years as a third class.

My intuition tells me, if I oversample the minority category, my model will simply memorize those hundred examples and fail at generalizing.

However when using weights, my intuition would tell me that my model can't memorize those samples because it doesn't encounter them frequently enough - kind of like a higher learning rate leads to better generalisation, but worse fitting due to the coarse steps.

However all the posts on CrossValidated actually say that oversampling works better - but why? Is that also the case for really small classes like in my case?

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This depends at least a little on the model being used. Most often, simple oversampling is asymptotically equivalent to using class weights: an integer weight $w$ on a datapoint has an equivalent effect on loss calculations as duplicating the datapoint $w$ times. Oversampling then is just a discrete version of class-weighting, so asymptotically they should be equivalent, but also for small samples sizes it doesn't seem clear that the discrete version should lead to consistently more or less overfitting.

If your model does any bagging though, things change: by oversampling, you are likely to include a subset of the duplicates of one point, whereas when weighting the subsetting happens before the weights come into play. However, it's still not clear to me that the final effect will be positive or negative in the sense of overfitting. (Unless you're also planning on using out-of-bag scores, in which case this would be quite bad, being very similar to the resampling-before-splitting in cross-validation.)

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