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I have a collection of images with labels (such as "goat") that I'd like to train a classifier with, using a convolutional neural network or similar technique.

However, while the presence of a label implies presence of the feature, absence of a label does not imply absence of the feature. For example, an image labeled "goat" should have a goat somewhere in the image, but an image without the "goat" label might still have a goat. Perhaps it was a human-curated data set where the second person didn't think to label the goat.

This makes the error difficult to calculate: If my model outputs a probability that an image contains a goat, then I can calculate the error in the positive case (expected = 100%), but not in the negative case (expected = 0% would be incorrect).

This seems to be a ''Supervised Learning'' and ''Classification'' problem, but the techniques I've read about assume that absence of a label implies absence of the feature. Is there a name for this problem, or a reference for how to handle it?

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I'd say this is a clear problem of semi-supervised learning. From the wiki page:

Semi-supervised learning is a class of supervised learning tasks and techniques that also make use of unlabeled data for training – typically a small amount of labeled data with a large amount of unlabeled data.

Here is a great tutorial.

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