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Adaptive-treed bandits

We describe a novel algorithm for noisy global optimisation and continuum-armed bandits, with good convergence properties over any continuous reward function having finitely many polynomial maxima. Over such functions, our algorithm achieves square-root regret in bandits, and inverse-square-root error in optimisation, without prior information. Our algorithm works by reducing these problems to tree-armed bandits, and we also provide new results in this setting. We show it is possible to adaptively combine multiple trees so as to minimise the regret, and also give near-matching lower bounds on the regret in terms of the zooming dimension.