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- W2051259245 abstract "In an online linear optimization problem, on each period t, an online algorithm chooses $s_tinmathcal{S}$ from a fixed (possibly infinite) set $mathcal{S}$ of feasible decisions. Nature (who may be adversarial) chooses a weight vector $w_tinmathbb{R}^n$, 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 $sinmathcal{S}$ in hindsight. Many repeated decision-making problems with weights fit naturally into this framework, such as online shortest-path, online traveling salesman problem (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 algorithm in both the full-information and the bandit settings, with average cost nearly as good as that of the best fixed $sinmathcal{S}$ in hindsight. However, in the case where the offline algorithm is an approximation algorithm with ratio $alpha >1$, the previous approach worked only 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 $alpha$-approximation guarantee, then the expected cost of the online algorithm on any sequence is not much larger than $alpha$ times that of the best $sinmathcal{S}$, 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 large repeated games, where one can compute only approximate best responses, rather than best responses." @default.
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- W2051259245 date "2009-01-01" @default.
- W2051259245 modified "2023-09-30" @default.
- W2051259245 title "Playing Games with Approximation Algorithms" @default.
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- W2051259245 doi "https://doi.org/10.1137/070701704" @default.
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