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I recently read a paper quoting:

General unsupervised learning is a long-standing conceptual problem in machine learning. Supervised learning is successful because it can be solved by the minimization of the training error cost function. Unsupervised learning is not as successful, because the unsupervised objective may be unrelated to the supervised task of interest.

I am wondering, is that the only challenge deep unsupervised learning is facing ? Shouldn't deep neural network work well with lots of data?

Semi-supervised learning in contrast seems to have a clear objective due to an available training error cost function. Why is it also considered to be hard ? Are the challenges the same as in unsupervised learning or does it have some unique challenges ?

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The last sentence of your quotation answers your question. Adding more data won't solve a poorly-defined problem. In the semi-supervised case, adding more unlabeled data won't inherently make the problem easier because you still need a way to score how correct some method is, and in the absence of labels, that's a challenge.

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