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Many applications require optimizing an unknown, noisy function that is expensive to evaluate. We formalize this task as a multi-armed bandit problem, where the payoff function is either sampled from a Gaussian process (GP) or has low RKHS norm. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze GP-UCB, an intuitive upper-confidence based algorithm, and bound its cumulative regret in terms of maximal information gain, establishing a novel connection between GP optimization and experimental design. Moreover, by bounding the latter in terms of operator spectra, we obtain explicit sublinear regret bounds for many commonly used covariance functions. In some important cases, our bounds have surprisingly weak dependence on the dimensionality. In our experiments on real sensor data, GP-UCB compares favorably with other heuristical GP optimization approaches.
Information-Theoretic Regret Bounds for Gaussian Process Optimization in the Bandit Setting N. Srinivas, A. Krause, S. Kakade, M. SeegerIn IEEE Transactions on Information Theory, volume 58, 2012
Bibtex Entry:
	author = {Niranjan Srinivas and Andreas Krause and Sham Kakade and Matthias Seeger},
	doi = {10.1109/TIT.2011.2182033},
	journal = {IEEE Transactions on Information Theory},
	month = {May},
	number = {5},
	pages = {3250-3265},
	title = {Information-Theoretic Regret Bounds for Gaussian Process Optimization in the Bandit Setting},
	volume = {58},
	year = {2012}}