Efficient Bregman Projections onto the Permutahedron and Related Polytopes
; Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51:1205-1213, 2016.
The problem of projecting onto the permutahedron \mathbfPH(c) – the convex hull of all permutations of a fixed vector c – under a uniformly separable Bregman divergence is shown to be reducible to the Isotonic Optimization problem. This allows us to employ known fast algorithms to improve on several recent results on Bregman projections onto permutahedra. In addition, we present a new algorithm \rm mergepool that have better complexity when the number of distinct entries d in the vector c is small, the simplex being one such example, with c=(1,0,0,\dotsc,0)^T and d=2. \rm mergepool runs in O(n \log d) for certain popular Bregman divergence measures and requires O((n \log d) \log \fracU ε) to find ε-close solutions for general uniformly separable Bregman divergences, where U is a bound on the width of the interval containing the dual solution components. These estimates matches or improves best known bounds for all Bregman projection problems onto various permutahedra, including recent results for projection onto the simplex. The same complexity bounds apply to signed permutahedra, a class that includes the \ell_1-ball as a special case. In summary, this work describes a fast unified approach to this well-known class of problems.