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Multi-cloud computing has become increasingly popular with enterprises looking to avoid vendor lock-in. While most cloud providers offer similar functionality, they may differ significantly in terms of performance and/or cost. A customer looking to benefit from such differences will naturally want to solve the multi-cloud configuration problem: given a workload, which cloud provider should be chosen and how should its nodes be configured in order to minimize runtime or cost? In this work, we consider solutions to this optimization problem. We develop and evaluate possible adaptations of state-of-the-art cloud configuration solutions to the multi-cloud domain. Furthermore, we identify an analogy between multi-cloud configuration and the selection-configuration problems commonly studied in the automated machine learning (AutoML) field. Inspired by this connection, we utilize popular optimizers from AutoML to solve multi-cloud configuration. Finally, we propose a new algorithm for solving multi-cloud configuration, CloudBandit (CB). It treats the outer problem of cloud provider selection as a best-arm identification problem, in which each arm pull corresponds to running an arbitrary black-box optimizer on the inner problem of node configuration. Our experiments indicate that (a) many state-of-the-art cloud configuration solutions can be adapted to multi-cloud, with best results obtained for adaptations which utilize the hierarchical structure of the multi-cloud configuration domain, (b) hierarchical methods from AutoML can be used for the multi-cloud configuration task and can outperform state-of-the-art cloud configuration solutions and (c) CB achieves competitive or lower regret relative to other tested algorithms, whilst also identifying configurations that have 65% lower median cost and 20% lower median time in production, compared to choosing a random provider and configuration.