Bandit online optimization over the permutahedron

The permutahedron is the convex polytope with vertex set consisting of the vectors (π(1),…, π(n)) for all permutations (bijections) π over {1,…, n}. We study a bandit game in which, at each step t, an adversary chooses a hidden weight weight vector s_t, a player chooses a vertex π_t of the permutahedron and suffers an observed instantaneous loss of ∑_i ⁿ=1 π_t(i)s_t(i).We study the problem in two regimes. In the first regime, s_t is a point in the polytope dual to the permutahedron. Algorithm CombBand of Cesa-Bianchi et al (2009) guarantees a regret of O(n√T log n) after T steps. Unfortunately, CombBand requires at each step an n-by-n matrix permanent computation, a #P-hard problem. Approximating the permanent is possible in the impractical running time of O(n¹⁰), with an additional heavy inverse-polynomial dependence on the sought accuracy. We provide an algorithm of slightly worse regret O(n^3/2√T) but with more realistic time complexity O(n³) per step. The technical contribution is a bound on the variance of the Plackett-Luce noisy sorting process’s ‘pseudo loss’, obtained by establishing positive semi-definiteness of a family of 3-by-3 matrices of rational functions in exponents of 3 parameters.In the second regime, st is in the hypercube. For this case we present and analyze an algorithm based on Bubeck et al.’s (2012) OSMD approach with a novel projection and decomposition technique for the permutahedron. The algorithm is efficient and achieves a regret of O(n√T), but for a more restricted space of possible loss vectors.