Is Supervised Learning Really That Different from Unsupervised?

We demonstrate how supervised learning can be decomposed into a two-stage procedure, where (1) all model parameters are selected in an unsupervised manner, and (2) the outputs y are added to the model, without changing the parameter values. This is achieved by a new model selection criterion that - in contrast to cross-validation - can be used also without access to y. For linear ridge regression, we bound the asymptotic out-of-sample risk of our method in terms of the optimal asymptotic risk. We also demonstrate on real and synthetic data that versions of linear and kernel ridge regression, smoothing splines, and neural networks, which are trained without access to y, perform similarly to their standard y-based counterparts. Hence, our results suggest that the difference between supervised and unsupervised learning is less fundamental than it may appear.