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I know barely anything about semi-supervised learning, but I had the following idea.

I classify documents in two classes, and would like to use the documents having the highest label confidences as new training data (given that my labeled dataset is rather small compared to my unlabeled dataset), and do so recursively.

The documents associated with a very high confidence are likely to be of the same class, but bring a bit of additional information, therefore increasing the training vocabulary, at the cost of increasing the risk of misclassification a bit.

I guess I could increase the confidence threshold necessary to use a document as training data for the next iteration, so as to reduce the risk of diverging badly.

I could also continuously reduce the weight associated with unlabeled data that I use as training data as I keep iterating.

What ML paradigm(s) would be appropriate for such a task? Is there any reference you recommend?

I don't necessarily want to label all the elements, I need a high precision, but do not care as much about recall.

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What you are describing is a part of the expectation maximization (EM) algorithm, which is sometimes used in semi-supervised learning. It is an iterative optimization algorithm where you start giving labeling to unlabeled instances and in a later phase, adjust these labelings to the new distribution of classes you have reached (very simple explained). I would start searching there maybe.

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