Best Approximation with Laplacian p-Norms to Predict the Labeling of a Graph

The aim is to predict the labeling of the vertices of a graph. The graph is given. A trial sequence of (vertex,label) pairs is then incrementally revealed to the learner. On each trial a vertex is given and the learner predicts a label and then the true label is returned. The learner’s goal is to minimize mistaken predictions. We propose to solve the problem by the method of best approximation. To this end we propose a (graph Laplacian) p-semi-norm on the space of graph labelings. Thus on every trial we predict with the labeling which minimizes the p-semi-norm and also is consistent with the revealed (vertex,label) pairs. When p = 1 this is the min-cut method of [2] and if p = 2 this is the harmonic energy minimization procedure of [6] also called (Laplacian) interpolated regularization in [1]. We prove mistake bounds for this methodology. We say an edge is cut with respect to a labeling if the connected vertices have disagreeing labels. We find that performing best approximation with p = 1 + where → 0 as the graph diameter D → ∞ gives a bound of O(k logD) versus a bound of O(kD) for best approximation with p = 2 where k is the number of cut edges.