Playing games with approximation algorithms

Sham M. Kakade*, Adam Tauman Kalai, Katrina Ligett

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

24 Scopus citations


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.

Original languageAmerican English
Title of host publicationSTOC'07
Subtitle of host publicationProceedings of the 39th Annual ACM Symposium on Theory of Computing
Number of pages10
StatePublished - 2007
Externally publishedYes
EventSTOC'07: 39th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: 11 Jun 200713 Jun 2007

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


ConferenceSTOC'07: 39th Annual ACM Symposium on Theory of Computing
Country/TerritoryUnited States
CitySan Diego, CA


  • Approximation algorithms
  • Online linear optimization
  • Regret minimization


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