Global optimization of univariate Lipschitz functions I: survey and properties
Mathematical Programming: Series A and B
Lipschitzian optimization without the Lipschitz constant
Journal of Optimization Theory and Applications
Randomized Distributed Edge Coloring via an Extension of the Chernoff--Hoeffding Bounds
SIAM Journal on Computing
Introduction to Reinforcement Learning
Introduction to Reinforcement Learning
Bayesian Algorithms for One-Dimensional GlobalOptimization
Journal of Global Optimization
Efficient Global Optimization of Expensive Black-Box Functions
Journal of Global Optimization
On Bayesian Methods for Seeking the Extremum
Proceedings of the IFIP Technical Conference
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Pure exploration in multi-armed bandits problems
ALT'09 Proceedings of the 20th international conference on Algorithmic learning theory
Sharp dichotomies for regret minimization in metric spaces
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Bandit problems and the exploration/exploitation tradeoff
IEEE Transactions on Evolutionary Computation
Journal of Approximation Theory
Variable risk control via stochastic optimization
International Journal of Robotics Research
Examples of inconsistency in optimization by expected improvement
Journal of Global Optimization
Bayesian optimization in high dimensions via random embeddings
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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In the efficient global optimization problem, we minimize an unknown function f, using as few observations f(x) as possible. It can be considered a continuum-armed-bandit problem, with noiseless data, and simple regret. Expected-improvement algorithms are perhaps the most popular methods for solving the problem; in this paper, we provide theoretical results on their asymptotic behaviour. Implementing these algorithms requires a choice of Gaussian-process prior, which determines an associated space of functions, its reproducing-kernel Hilbert space (RKHS). When the prior is fixed, expected improvement is known to converge on the minimum of any function in its RKHS. We provide convergence rates for this procedure, optimal for functions of low smoothness, and describe a modified algorithm attaining optimal rates for smoother functions. In practice, however, priors are typically estimated sequentially from the data. For standard estimators, we show this procedure may never find the minimum of f. We then propose alternative estimators, chosen to minimize the constants in the rate of convergence, and show these estimators retain the convergence rates of a fixed prior.