12 research outputs found
Bandit Online Optimization Over the Permutahedron
The permutahedron is the convex polytope with vertex set consisting of the
vectors for all permutations (bijections) over
. We study a bandit game in which, at each step , an
adversary chooses a hidden weight weight vector , a player chooses a
vertex of the permutahedron and suffers an observed loss of
.
A previous algorithm CombBand of Cesa-Bianchi et al (2009) guarantees a
regret of for a time horizon of . Unfortunately,
CombBand requires at each step an -by- matrix permanent approximation to
within improved accuracy as grows, resulting in a total running time that
is super linear in , making it impractical for large time horizons.
We provide an algorithm of regret with total time
complexity . The ideas are a combination of CombBand and a recent
algorithm by Ailon (2013) for online optimization over the permutahedron in the
full information setting. The technical core is a bound on the variance of the
Plackett-Luce noisy sorting process's "pseudo loss". The bound is obtained by
establishing positive semi-definiteness of a family of 3-by-3 matrices
generated from rational functions of exponentials of 3 parameters
Decomposition algorithm for the single machine scheduling polytope
Given an -vector of processing times of jobs, the single machine scheduling polytope arises as the convex hull of completion times of jobs when these are scheduled without idle time on a single machine. Given a point (x\ud
in C), Carathéodory's theorem implies that can be written as convex combination of at most vertices of . We show that this convex combination can be computed from and in time O(n2), which is linear in the naive encoding of the output. We obtain this result using essentially two ingredients. First, we build on the fact that the scheduling polytope is a zonotope. Therefore, all of its faces are centrally symmetric. Second, instead of , we consider the polytope of half times and its barycentric subdivision. We show that the subpolytopes of this barycentric subdivison of have a simple, linear description. The final decomposition algorithm is in fact an implementation of an algorithm proposed by Grötschel, Lovász, and Schrijver applied to one of these subpolytopes
Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes
Optimization algorithms such as projected Newton's method, FISTA, mirror
descent and its variants enjoy near-optimal regret bounds and convergence
rates, but suffer from a computational bottleneck of computing "projections''
in potentially each iteration (e.g., regret of online mirror
descent). On the other hand, conditional gradient variants solve a linear
optimization in each iteration, but result in suboptimal rates (e.g.,
regret of online Frank-Wolfe). Motivated by this trade-off in
runtime v/s convergence rates, we consider iterative projections of close-by
points over widely-prevalent submodular base polytopes . We develop a
toolkit to speed up the computation of projections using both discrete and
continuous perspectives. We subsequently adapt the away-step Frank-Wolfe
algorithm to use this information and enable early termination. For the special
case of cardinality based submodular polytopes, we improve the runtime of
computing certain Bregman projections by a factor of . Our
theoretical results show orders of magnitude reduction in runtime in
preliminary computational experiments