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In order to improve the performance of Bayesian optimisation, we develop amodified Gaussian process upper confidence bound (GP-UCB) acquisition function.This is done by sampling the exploration-exploitation trade-off parameter froma distribution. We prove that this allows the expected trade-off parameter tobe altered to better suit the problem without compromising a bound on thefunction's Bayesian regret. We also provide results showing that our methodachieves better performance than GP-UCB in a range of real-world and syntheticproblems.

Randomised Gaussian Process Upper Confidence Bound for Bayesian Optimisation