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## Playing games with approximation algorithms (2007)

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### Other Repositories/Bibliography

Venue: | In Proceedings of the 39 th annual ACM Symposium on Theory of Computing |

Citations: | 27 - 2 self |

### Citations

1211 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,”
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...input, the solution they find differs from the optimal solution by a factor of at most α in every coordinate. They observe that a number of algorithms, such as the GoemansWilliamson max-cut algorithm =-=[11]-=-, have this property. Balcan and Blum [3] observe that the previous approach applies to another type of approximation algorithm: one that uses an optimal decision for another linear optimization probl... |

475 | Some Aspects of the Sequential Design of Experiments.
- Robbins
- 1952
(Show Context)
Citation Context ...version, the player is then informed of wt, while in the bandit version she is only informed of the value c(st,wt). (The name bandit refers to the similarity to the classic multi-armed bandit problem =-=[15]-=-). The player’s goal is to achieve low average cost. In particular, we compare her cost with that of the best fixed decision: she would like her average cost to approach that of the best single point ... |

298 | Online convex programming and generalized infinitesimal gradient ascent.
- Zinkevich
- 2003
(Show Context)
Citation Context ...e average performance of the best static decision in hindsight. Our new approach is inspired by Zinkevich’s algorithm for the problem of minimizing convex functions over a convex feasible set S ⊆ R n =-=[16]-=-. However, the application is not direct and requires a geometric transformation that can be applied to any approximation algorithm. Example 1 (Online metric TSP). Every day, a delivery company serves... |

192 | Efficient algorithms for the online decision problem.
- Kalai, Vempala
- 2003
(Show Context)
Citation Context ..., even if this action could be chosen after observing the opponent’s play. Kalai and Vempala showed that Hannan’s approach can be used to efficiently solve online linear optimization problems as well =-=[13]-=-. Hannan’s algorithm relied on the ability to find best responses to an opponent’s play history. Informally speaking, Kalai and Vempala replaced this best-reply computation with an efficient black-box... |

150 |
Approximation to Bayes risk in repeated play,” in Contributions to the Theory of Games, Volume III, ser.
- Hannan
- 1957
(Show Context)
Citation Context ...e can only compute approximate best-responses, rather than best-responses. 1. Introduction. In the 1950’s, Hannan gave an algorithm for playing repeated two-player games against an arbitrary opponent =-=[12]-=-. His was one of the earliest algorithms with the no-regret property: against any opponent, his algorithm achieved expected performance asymptotically near that of the best single action, where the be... |

123 |
Online convex optimization in the bandit setting: gradient descent without a gradient.
- Flaxman, Kalai, et al.
- 2005
(Show Context)
Citation Context ...4(α + 2) 2 T. 4. Bandit algorithm. We now describe how to extend Algorithm 3.1 to the partial-information model, where the only feedback we receive is the cost we incur at each period. Flaxman et al. =-=[10]-=- also use a gradient descent style algorithm for online optimization in the bandit setting, but the details of their approach differ significantly from ours. The algorithm we describe here requires ac... |

95 |
Adaptive routing with end-to-end feedback: distributed learning and geometric approaches,” in
- Awerbuch, Kleinberg
- 2004
(Show Context)
Citation Context ...t a priori bounds on W and R by a simple change of basis so that RW = O(n). It is possible to do this from the set W alone. In particular, one can compute a 2-barycentric spanner (BS) e1,...,en for W =-=[2]-=- and perform a change of basis so that Φ(e1),...,Φ(en) is the standard basis (as we describe in greater detail in §4). By the definition of a 2-BS, this implies that W ⊆ [−2,2] n and hence W = 2 √ n i... |

80 | Competing in the Dark: An Efficient Algorithm for Bandit Linear Optimization.
- Abernethy, Hazan, et al.
- 2008
(Show Context)
Citation Context ...nline weighted set cover, the vendors are fixed sets P1,...,Pn ⊆ [m]. Each period, we choose a legal cover st ⊆ [n], that is, ⋃ i∈st Pi = [m]. There is an unknown sequence of cost vectors w1,w2,... ∈ =-=[0,1]-=- n , indicating the quarterly vendor costs. Each quarter, our total cost c(st,wt) is the sum of the costs of the vendors we chose for that quarter. In the full-information setting, at the end of the q... |

76 | Approximation algorithms and online mechanisms for item pricing,”
- Balcan, Blum
- 2006
(Show Context)
Citation Context ...problem nearly as efficiently as one can solve the offline problem. (They used the offline optimizer as a black box.) However, in many cases of interest, such as online combinatorial auction problems =-=[3]-=-, even the offline problem is NP-hard. Hannan’s “follow-the-perturbed-leader” approach can also be applied to some special types of approximation algorithms, but fails to work directly in general. Fin... |

71 | Online geometric optimization in the bandit setting against an adaptive adversary.
- McMahan, Blum
- 2004
(Show Context)
Citation Context ...tly achieve this property using the previous approach. 1.2.2. Bandit results. Previous work in the bandit setting constructs an “exploration basis” to allow the algorithm to discover better decisions =-=[2, 14, 7]-=-. In particular, Awerbuch and Kleinberg [2] introduce a so-called Barycentric Spanner (BS) as their exploration basis and show how to construct one from an optimization oracle A : R n → S. However, in... |

58 |
The price of bandit information for online optimization,
- Dani, Hayes, et al.
- 2008
(Show Context)
Citation Context ...ar optimization problem (without additional input) to a bandit algorithm guaranteeing low α-regret. We note that the above regret is sub-optimal in terms of the T dependence. Furthermore, recent work =-=[8, 4, 1]-=- presents algorithms for online linear optimization that achieve the optimal √ T regret even in the bandit setting (these results either do not explicitly consider the computational issues or assume a... |

54 | Robbing the bandit: less regret in online geometric optimization against an adaptive adversary
- Dani, Hayes
- 2006
(Show Context)
Citation Context ...] that using Hannan’s approach [12], one can guarantee O(T −1/2 ) regret for any linear optimization problem, in the full-information version, as the number of periods T increases. It was later shown =-=[2, 14, 7]-=- how to convert exact algorithms to achieve O(T −1/3 ) regret in the more difficult bandit setting. This prior work was actually a reduction showing that one can solve the online problem nearly as eff... |

50 | A simple polynomial-time rescaling algorithm for solving linear programs.
- Dunagan, Vempala
- 2006
(Show Context)
Citation Context ... δ λ and ‖x‖ ≤ 1 2 √ δ λ . Then ApproxProj(z,s,x) terminates after at most ‖x−z‖2 δλ iterations. Proof. The analysis is reminiscent of that of the perceptron algorithm (see, e.g., Dunagan and Vempala =-=[9]-=-). Let H = 1 √ δ 2 λ . To bound the number of recursivePLAYING GAMES WITH APPROXIMATION ALGORITHMS 13 calls to Approx-Proj, it suffices to show that the non-negative quantity ‖x − z‖ 2 decreases by a... |

13 | Randomized metarounding. Random Structures Algorithms - Carr, Vempala - 2002 |

8 |
Randomized metarounding. Random Struct.
- Carr, Vempala
- 2002
(Show Context)
Citation Context ... can accept input w. By the definition of α-approximation, we have w · 6 Note that representing a given feasible point as a convex combination of feasible points is similar to randomized metarounding =-=[5]-=-. It would be interesting to extend their approach, based on the ellipsoid algorithm, to our problem and potentially achieve a more efficient algorithm. Related but simpler issues arise in [6].PLAYIN... |

8 | Design is as easy as optimization
- Chakrabarty, Mehta, et al.
(Show Context)
Citation Context ...ounding [5]. It would be interesting to extend their approach, based on the ellipsoid algorithm, to our problem and potentially achieve a more efficient algorithm. Related but simpler issues arise in =-=[6]-=-.PLAYING GAMES WITH APPROXIMATION ALGORITHMS 11 Fig. 3.2. An approximation algorithm run on vector w ∈ W always returns a point s ∈ S such that the set αK is contained in the halfspace tangent to Φ(s... |

2 |
High probability regret bounds for bandit online optimization
- Bartlett, Dani, et al.
- 2008
(Show Context)
Citation Context ...ar optimization problem (without additional input) to a bandit algorithm guaranteeing low α-regret. We note that the above regret is sub-optimal in terms of the T dependence. Furthermore, recent work =-=[8, 4, 1]-=- presents algorithms for online linear optimization that achieve the optimal √ T regret even in the bandit setting (these results either do not explicitly consider the computational issues or assume a... |