There is such a problem: we have to process multi-label classification (assignmet of tags) of text articles, using some pre-labeled training set. But for many texts in the training set, should be assigned more tags, that it was done by their authors.
For example, there is a text about war medicine, and there is assigned only the text "war", and tag "medicine" is omitted.
Are there any common, conventional methods to deal with such data?
Based on the question and your comments, you can consider every 1-vs-all classification problem as a so-called PU learning problem (*P*ositive and *U*nlabeled). PU learning is a branch of semi-supervised learning in which you only have labels of the positive class and (typically a lot of) unlabeled instances.
Every tag is a class in your case. Instances associated with said tag are positive for the associated 1-vs-all classifier, all others are unlabeled. In your example: war medicine is a known positive for the war class and an unlabeled instance for the medicine class.
Here is a simple but comprehensive reference on the subject: http://cseweb.ucsd.edu/~elkan/posonly.pdf
war medicineis unlabeled from themedicineclass' perspective. One way to go would be to treat every binary 1-vs-all classification problem as semi-supervised and use whichever technique you like most. $\endgroup$