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The problem of sequentially maximizing the expectation of a function seeks tomaximize the expected value of a function of interest without having directcontrol on its features. Instead, the distribution of such features depends ona given context and an action taken by an agent. In contrast to Bayesianoptimization, the arguments of the function are not under agent's control, butare indirectly determined by the agent's action based on a given context. Ifthe information of the features is to be included in the maximization problem,the full conditional distribution of such features, rather than its expectationonly, needs to be accounted for. Furthermore, the function is itself unknown,only counting with noisy observations of such function, and potentiallyrequiring the use of unmatched data sets. We propose a novel algorithm for theaforementioned problem which takes into consideration the uncertainty derivedfrom the estimation of both the conditional distribution of the features andthe unknown function, by modeling the former as a Bayesian conditional meanembedding and the latter as a Gaussian process. Our algorithm empiricallyoutperforms the current state-of-the-art algorithm in the experimentsconducted.

Sequential Decision Making on Unmatched Data using Bayesian Kernel Embeddings