Playing games with approximation algorithms

In an online linear optimization problem, on each period t, an online algorithm chooses s _t S from a fixed (possibly infinite) set S of feasible decisions. Nature (who may be adversarial) chooses a weight vector w _t R, and the algorithm incurs cost c(s _t,w _t), where c is a fixed cost function that is linear in the weight vector. In the full-information setting, the vector w _t is then revealed to the algorithm, and in the bandit setting, only the cost experienced, c(s _t,w _t), is revealed. The goal of the online algorithm is to perform nearly as well as the best fixed s S in hindsight. Many repeated decision-making problems with weights fit naturally into this framework, such as online shortest-path, online TSP, online clustering, and online weighted set cover. Previously, it was shown how to convert any efficient exact offline optimization algorithm for such a problem into an efficient online bandit algorithm in both the full-information and the bandit settings, with average cost nearly as good as that of the best fixed s S in hindsight. However, in the case where the offline algorithm is an approximation algorithm with ratio α > 1, the previous approach only worked for special types of approximation algorithms. We show how to convert any offline approximation algorithm for a linear optimization problem into a corresponding online approximation algorithm, with a polynomial blowup in runtime. If the offline algorithm has an α-approximation guarantee, then the expected cost of the online algorithm on any sequence is not much larger than α times that of the best s S, where the best is chosen with the benefit of hindsight. Our main innovation is combining Zinkevich's algorithm for convex optimization with a geometric transformation that can be applied to any approximation algorithm. Standard techniques generalize the above result to the bandit setting, except that a "Barycentric Spanner" for the problem is also (provably) necessary as input.Our algorithm can also be viewed as a method for playing largerepeated games, where one can only compute approximate best-responses, rather than best-responses.