Online Learning of Convex Sets on Graphs

Abstract

We study online learning of general convex sets and halfspaces on graphs. While online learning of halfspaces in Euclidean space is a classical learning problem, the corresponding problem on graphs is understudied. In this context, a set of vertices is convex if it contains all connecting shortest paths and a halfspace is a convex set whose complement is also convex. We discuss mistake bounds based on the Halving algorithm and shortest path covers. Halving achieves near-optimal bounds but is inefficient in general. The shortest path cover based algorithm is efficient but provides optimal bounds only for restricted graph families such as trees. To mitigate the weaknesses of both approaches, we propose a novel polynomial time algorithm which achieves near-optimal bounds on graphs that are $K_{2,k}$ minor-free for some constant $k\in \mathbb {N}$. In contrast to previous mistake bounds on graphs, which typically depend on the induced cut of the labelling, our bounds only depend on the graph itself. Finally, we discuss the agnostic version of this problem and introduce an adaptive variant of Halving for k-intersections of halfspaces.