Predicting the Labelling of a Graph via Minimum p-Seminorm Interpolation
Mark Herbster and Guy Lever
In: The 22nd Annual Conference on Learning Theory (COLT 2009), 18-21 Jun 2009, Montreal, Canada.

## Abstract

We study the problem of predicting the labelling of a graph. The graph is given and a trial sequence of (vertex,label) pairs is then incrementally revealed to the learner. On each trial a vertex is queried and the learner predicts a boolean label. The true label is then returned. The learner's goal is to minimise mistaken predictions. We propose {\em minimum $p$-seminorm interpolation} to solve this problem. To this end we give a $p$-seminorm on the space of graph labellings. Thus on every trial we predict using the labelling which {\em minimises} the $p$-seminorm and is also {\em consistent} with the revealed (vertex, label) pairs. When $p=2$ this is the {\em harmonic energy minimisation} procedure of~\cite{\ZhuHarmonicFunctions}, also called (Laplacian) {\em interpolated regularisation} in~\cite{Belkin2}. In the limit as $p\rightarrow 1$ this is equivalent to predicting with a label-consistent mincut. We give mistake bounds relative to a label-consistent mincut and a resistive cover of the graph. We say an edge is {\em cut} with respect to a labelling if the connected vertices have disagreeing labels. We find that minimising the $p$-seminorm with $p=1+\epsilon$ where $\epsilon\into 0$ as the graph diameter $D\into\infty$ gives a bound of $O(\cut^2 \log D)$ versus a bound of $O(\cut D)$ when $p=2$ where $\cut$ is the number of cut edges.