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## Regret bounds for gaussian process bandit problems (2010)

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Venue: | In AISTATS |

Citations: | 24 - 3 self |

### Citations

817 | Finite-time analysis of the multi-armed bandit problem.
- Auer, Cesa-Bianchi, et al.
- 2002
(Show Context)
Citation Context ...NTRODUCTION Bandit problems have become a topic of extensive research in recent years and many extensions of the classical framework (Robbins, 1952; Gittins and Jones, 1979; Berry and Fristedt, 1985; =-=Auer et al., 2002-=-; Cesa-Bianchi and Lugosi, 2006) have been developed (among many others, Abernethy et al., 2008; Slivkins and Upfal, 2008; Wang et al., 2008; Bubeck et al., 2008; Kleinberg et al., 2008b). A particula... |

719 | Gaussian Processes for Machine Learning.
- Rasmussen, Williams
- 2006
(Show Context)
Citation Context ...eward distribution. A stochastic process r(x) is Gaussian if and only if for every finite set of indices x1, . . . , xn the variable (r(x1), . . . , r(xn)) is a multivariate Gaussian random variable (=-=Rasmussen and Williams, 2006-=-). A Gaussian process is completely specified by its mean function µ and its covariance function k(x, y) = Cov[r(x), r(y)]. The covariance function k is symmetric and positive semi-definite. 3.1.1 Met... |

481 |
Efficient global optimization of expensive black-box functions
- Jones, Schonlau, et al.
- 1998
(Show Context)
Citation Context ...e if the GP reward functions are similar to that of the concrete optimisation problem at hand. Due to the flexibility and power of GPs, the approach has sparked considerable interest (Schonlau, 1997; =-=Jones et al., 1998-=-; Jones, 2001; Ginsbourger and Riche, 2009; Osborne et al., 2009). The early works were done in the field of global optimisation. Using stochastic processes for global optimisation has a long traditio... |

475 | Some Aspects of the Sequential Design of Experiments.
- Robbins
- 1952
(Show Context)
Citation Context ... there is at most a logarithmic looseness in our upper bounds. 1 INTRODUCTION Bandit problems have become a topic of extensive research in recent years and many extensions of the classical framework (=-=Robbins, 1952-=-; Gittins and Jones, 1979; Berry and Fristedt, 1985; Auer et al., 2002; Cesa-Bianchi and Lugosi, 2006) have been developed (among many others, Abernethy et al., 2008; Slivkins and Upfal, 2008; Wang et... |

353 | Applied Probability Models with Optimization Applications - Ross - 1970 |

235 | A taxonomy of global optimization methods based on response surfaces.
- Jones
- 2001
(Show Context)
Citation Context ...unctions are similar to that of the concrete optimisation problem at hand. Due to the flexibility and power of GPs, the approach has sparked considerable interest (Schonlau, 1997; Jones et al., 1998; =-=Jones, 2001-=-; Ginsbourger and Riche, 2009; Osborne et al., 2009). The early works were done in the field of global optimisation. Using stochastic processes for global optimisation has a long tradition. The earlie... |

197 | Combinatorial methods in density estimation. - Devroye, Lugosi - 2001 |

118 | Nearly tight bounds for the continuum-armed bandit problem. - Kleinberg - 2004 |

110 |
The sizes of compact subsets of hilbert space and continuity of gaussian processes.
- Dudley
- 1967
(Show Context)
Citation Context ...e draws? The key to dealing with these questions lies in a fundamental work from probability theory. One of the key tools is the celebrated chaining argument with, in particular, the Dudley integral (=-=Dudley, 1967-=-). This argument allows us to bound the expected supremum and, from concentration inequalities, with high probability, the supremum for a given GP sample. The interesting property of the Dudley integr... |

82 | The continuum-armed bandit problem. - Agrawal - 1995 |

80 | Pure Exploration in Multi-Armed Bandit Problems. - Bubeck, Munos, et al. - 2009 |

80 | Competing in the Dark: An Efficient Algorithm for Bandit Linear Optimization. - Abernethy, Hazan, et al. - 2008 |

80 |
Computer Experiments and Global Optimization.
- Schonlau
- 1997
(Show Context)
Citation Context ... well in practice if the GP reward functions are similar to that of the concrete optimisation problem at hand. Due to the flexibility and power of GPs, the approach has sparked considerable interest (=-=Schonlau, 1997-=-; Jones et al., 1998; Jones, 2001; Ginsbourger and Riche, 2009; Osborne et al., 2009). The early works were done in the field of global optimisation. Using stochastic processes for global optimisation... |

74 |
Concentration Inequalities and Model Selection: Ecole d’ete de Probabilites de Saint-Flour 23.
- Massart
- 2003
(Show Context)
Citation Context ...ng number N(ε, X), which is the maximal number of points that are all at least ε-distant from each other (in the d(x, y) metric). The bound on the expected supremum is given in the following theorem (=-=Massart, 2003-=-)[Th. 3.18, p. 74]: Theorem 3.1 (Metric Entropy Bound) Let (f(x))x∈X be some centered Gaussian process. Assume that (X, d) is totally bounded and denote by N(ε, X) the ε-packing number of (X, d), for ... |

65 | Bandit Problems
- Berry, Fristedt
- 1985
(Show Context)
Citation Context ...s in our upper bounds. 1 INTRODUCTION Bandit problems have become a topic of extensive research in recent years and many extensions of the classical framework (Robbins, 1952; Gittins and Jones, 1979; =-=Berry and Fristedt, 1985-=-; Auer et al., 2002; Cesa-Bianchi and Lugosi, 2006) have been developed (among many others, Abernethy et al., 2008; Slivkins and Upfal, 2008; Wang et al., 2008; Bubeck et al., 2008; Kleinberg et al., ... |

49 | Online Optimization in X-Armed Bandits. - Bubeck, Munos, et al. - 2011 |

45 | Regrets bounds for sleeping experts and bandits, - Kleinberg, Niculescu-Mizil, et al. - 2008 |

41 | Gaussian processes for global optimization.
- Osborne, Garnett, et al.
- 2009
(Show Context)
Citation Context ...te optimisation problem at hand. Due to the flexibility and power of GPs, the approach has sparked considerable interest (Schonlau, 1997; Jones et al., 1998; Jones, 2001; Ginsbourger and Riche, 2009; =-=Osborne et al., 2009-=-). The early works were done in the field of global optimisation. Using stochastic processes for global optimisation has a long tradition. The earliest works date back to Kushner (1964). The use of GP... |

40 | Improved Rates for the Stochastic Continuum-Armed Bandit Problem. - Auer, Ortner, et al. - 2007 |

39 |
D.M.: A dynamic allocation index for the discounted multiarmed bandit problem.
- Gittins, Jones
- 1979
(Show Context)
Citation Context ...st a logarithmic looseness in our upper bounds. 1 INTRODUCTION Bandit problems have become a topic of extensive research in recent years and many extensions of the classical framework (Robbins, 1952; =-=Gittins and Jones, 1979-=-; Berry and Fristedt, 1985; Auer et al., 2002; Cesa-Bianchi and Lugosi, 2006) have been developed (among many others, Abernethy et al., 2008; Slivkins and Upfal, 2008; Wang et al., 2008; Bubeck et al.... |

35 | A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise. - Kushner - 1964 |

21 | Adapting to a changing environment: the brownian restless bandits, - Slivkins, Upfal - 2008 |

17 | A Problem in the Sequential Design of Experiments,” Sankhya: - Bellman - 1956 |

16 | Multi-armed Bandit Problems in Metric Spaces. In - Kleinberg, Slivkins, et al. - 2008 |

14 | A multi-points criterion for deterministic parallel global optimization based on Gaussian processes. HAL: hal-00260579, - Ginsbourger, Riche, et al. - 2009 |

7 |
Gaussian process bandits without regret: An experimental design approach
- Srinivas, Krause, et al.
- 2009
(Show Context)
Citation Context ...ound is derived for a specific kernel that fulfills the α-Hölder continuity assumption and matches the upper bound up to a logarithmic factor. During the reviewing process of this work, another work (=-=Srinivas et al., 2009-=-) addressing regret bounds in GP optimization has appeared. It provides a different approach linked to optimal empirical design and information gain and studies the cumulative regret. So, both the bou... |

6 | Towards GP-based optimization with finite time horizon
- Ginsbourger, Riche
- 2010
(Show Context)
Citation Context ...similar to that of the concrete optimisation problem at hand. Due to the flexibility and power of GPs, the approach has sparked considerable interest (Schonlau, 1997; Jones et al., 1998; Jones, 2001; =-=Ginsbourger and Riche, 2009-=-; Osborne et al., 2009). The early works were done in the field of global optimisation. Using stochastic processes for global optimisation has a long tradition. The earliest works date back to Kushner... |

2 | Computer Experiments and Global Optimization - Holden-Day, CA |