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I've been learning a little bit about generalization theory, and in particular, the PAC (and PAC-Bayes) approach to thinking about this problem.

So, I started to wonder if there is an analogous version of "generalization" in Unsupervised Learning? I.e., is there a general framework that encapsulates how "good" an unsupervised learning method is? There's reconstruction error for learning lower dimensional representations, but what about unsupervised clustering?

Any ideas?

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In the paper Generalization in Unsupervised Learning (2015), Abou-Moustafa and Schuurmans develop an approach to assess the generalization of an unsupervised learning algorithm $A$ on a given dataset $S$ and how to compare the generalization ability of two unsupervised learning algorithms $A_1$ and $A_2$, for the same learning task.

They first provide a more abstract and general definition of an unsupervised learning algorithm and loss function. Then they define the expected risk, empirical risk and generalization gap in a similar way to the case of supervised learning. Finally, they derive an upper bound on $A$'s expected loss.

Of course, you should read the paper for more details. Specifically, section 2 (page 3) describes their setting in detail.

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