## Abstract

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 language | English |
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Title of host publication | STOC'07 |

Subtitle of host publication | Proceedings of the 39th Annual ACM Symposium on Theory of Computing |

Pages | 546-555 |

Number of pages | 10 |

DOIs | |

State | Published - 2007 |

Externally published | Yes |

Event | STOC'07: 39th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: 11 Jun 2007 → 13 Jun 2007 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | STOC'07: 39th Annual ACM Symposium on Theory of Computing |
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Country/Territory | United States |

City | San Diego, CA |

Period | 11/06/07 → 13/06/07 |

## Keywords

- Approximation algorithms
- Online linear optimization
- Regret minimization