  Semisupervised methods are techniques for using labeled data
$(X_1,Y_1),\ldots,(X_n,Y_n)$ together with unlabeled data $X_{n+1},\ldots,X_N$
to make predictions. These methods invoke some assumptions that link the
marginal distribution $P_X$ of X to the regression function f(x). For example,
it is common to assume that f is very smooth over high density regions of
$P_X$. Many of the methods are ad-hoc and have been shown to work in specific
examples but are lacking a theoretical foundation. We provide a minimax
framework for analyzing semisupervised methods. In particular, we study methods
based on metrics that are sensitive to the distribution $P_X$. Our model
includes a parameter $\alpha$ that controls the strength of the semisupervised
assumption. We then use the data to adapt to $\alpha$.
