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...). Recent work has extended Bayesian optimization to multiple tasks and objectives (Krause and Ong, 2011; Swersky et al., 2013; Zuluaga et al., 2013) and high dimensional problems (Wang et al., 2013; =-=Djolonga et al., 2013-=-). Strong theoretical results have also been developed (Srinivas et al., 2010; Bull, 2011; de Freitas et al., 2012). Bayesian optimization has been shown to be a powerful method for the meta-optimizat...

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...trarily over time. In this framework, the aforementioned transformation does not preserve the sparsity assumption. Indeed, if (`1, . . . , `d) is a s-sparse loss vector, the corresponding gain vector (1− `1, . . . , 1− `d) may even have full support. Consequently, results for loss vectors do not apply directly to sparse gains, and vice versa. It turns out that both setups are fundamentally different. The sparsity assumption is actually quite natural in learning and have also received some attention in online learning (Gerchinovitz, 2013; Carpentier and Munos, 2012; AbbasiYadkori et al., 2012; Djolonga et al., 2013). In the case of gains, it reflects the fact that the problem has some hidden structure and that many options are irrelevant. For instance, in the canonical click-through-rate example, a website displays an ad and gets rewarded if the user clicks on it; we can safely assume that there are only a small number of ads on which a user would click. The sparse scenario can also be seen through the scope of prediction with experts. Given a finite set of expert, we call the winner of a stage the expert with the highest revenue (or the smallest loss); ties are broken arbitrarily. And the objective woul...

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.... , |A|}. For each type i action there are two choices vi ∈ {0, 1}, where vi = 0 implies that action inA corresponding to type i action in the graphical bandit problem is not chosen, and vi = 1 implies that is is chosen. Thus, the action constraint for the graphical bandit problem becomes ∑ i∈TA vi = 1. The methods introduced in [12] cannot be used for this problem due to the additional constraint and the unknown relation. Moreover, [12] only consider finite context spaces. Large dimensional action spaces, where the rewards depend on a subset of the types of actions are considered in [13] and [14]. [13] considers the problem when the reward is Holder continuous in an unknown low-dimensional tuple of types, and uses a special discretization of the action space to achieve dimension independent bounds on the regret. This discretization can be effectively used since the learner can select the actions, as opposed to our case where the learner does not have any control over contexts. [14] considers the problem of optimizing high dimensional functions that have an unknown low dimensional structure from noisy observations. 5 Conclusion In this paper we formalized the problem of learning the b...

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...and analytic properties (Jones, 2001; Osborne et al., 2009). Recent work has sought to improve the performance of the GP approach through accommodating higher dimensional problems (Wang et al., 2013; =-=Djolonga et al., 2013-=-), input nonstationarities (Snoek et al., 2014) and initialization through meta-learning (Feurer et al., 2015). Random forests, which scale linearly with the data, have also been used successfully for...

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...s within). In particular the authors in [17, 18] considered the problem of approximating functions of the form f(x) = g(Ax) from point queries. Very recently and independently a work parallel to ours =-=[19]-=- considered the same bandit problem as ours i.e. they also assume the d-variate reward functions to depend on k ≪ d unknown linear parameters. Although they consider the mean reward function to reside...

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...). Recent work has extended Bayesian optimization to multiple tasks and objectives (Krause and Ong, 2011; Swersky et al., 2013; Zuluaga et al., 2013) and high dimensional problems (Wang et al., 2013; =-=Djolonga et al., 2013-=-). Strong theoretical results have also been developed (Srinivas et al., 2010; Bull, 2011; de Freitas et al., 2012). Bayesian optimization has been shown to be a powerful method for the meta-optimizat...

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... & Ong, 2011; Seldin et al., 2011; Yue et al., 2012; Proceedings of the 31 st International Conference on Machine Learning, Beijing, China, 2014. JMLR:W&CP volume 32. Copyright 2014 by the author(s). =-=Djolonga et al., 2013-=-), and references therein) have rapidly become a reference algorithmic technique for implementing adaptive recommender systems. Within the above scenarios, the widespread adoption of online social net...

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...orse dependence on the geometry of the set of uj , as compared to E[SD(uit)]. The analysis goes along the very same lines as the one of Theorem 1. 6. Related Work The most closely related papers are (=-=Djolonga et al., 2013-=-; Azar et al., 2013; Brunskill & Li, 2013; Maillard & Mannor, 2014). In (Azar et al., 2013), the authors define a transfer learning problem within a stochastic multiarmed bandit setting, where a prior...