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Nearly every structured prediction problem in computer vision requires approximate inference due to large and complex dependencies among output labels. While graphical models provide a clean separation between modeling and inference, learning these models with approximate inference is not well understood. Furthermore, even if a good model is learned, predictions are often inaccurate due to approximations. In this work, instead of performing inference over a graphical model, we instead consider the inference procedure as a composition of predictors. Specifically, we focus on message-passing algorithms, such as Belief Propagation, and show how they can be viewed as procedures that sequentially predict label distributions at each node over a graph. Given labeled graphs, we can then train the sequence of predictors to output the correct labeling s. The result no longer corresponds to a graphical model but simply defines an inference procedure, with strong theoretical properties, that can be used to classify new graphs. We demonstrate the scalability and efficacy of our approach on 3D point cloud classification and 3D surface estimation from single images.