Fast and Optimal Prediction on a Labeled Tree

We characterize, up to constant factors, the number of mistakes necessary and sufficient for sequentially predicting a given tree with binary labeled nodes. We provide an efficient algorithm achieving this number of mistakes on any tree. Tree prediction algorithms can solve the general graph prediction problem by representing the graph via one of its spanning trees. In order to cope with adversarial assignments of labels over a general graph, we advocate the use of random spanning trees, which have the additional advantage of retaining relevant spectral information of the original graph.