Online Learning of Permutations Using Extended Formulation

We introduce a new method for efficiently learning permutations that exploits the technique of extended formulation from the combinatorial optimization community. This extended formulation technique encodes the hard-to-describe polytope of permutations as the projection of an easier to describe polytope in a higher dimensional space. Although the best special-purpose algorithm for permutations has a slightly better regret bound, our methodology yields regret bounds like those of other published algorithms. The new methodology can be generalized to other combinatorial objects. It also has an elegant method of determining the algorithm’s initial weight vector and the use of the extended formulation leads to a faster and more natural way to make appropriate predictions.