raining principles for unsupervised learning are often derived from motivations that appear to be independent of supervised learning, causing a proliferation of semisupervised training methods. In this paper we present a simple unification of several supervised and unsupervised training principles through the concept of optimal reverse prediction: predict the inputs from the target labels, optimizing both over model parameters and any missing labels. In particular, we show how supervised least squares, principal components analysis, k-means clustering and normalized graph-cut clustering can all be expressed as instances of the same training principle, differing only in constraints made on the target labels. Natural forms of semi-supervised regression and classification are then automatically derived, yielding semi-supervised learning algorithms for regression and classification that, surprisingly, are novel and refine the state of the art. These algorithms can all be combined with standard regularizers and made non-linear via kernels.