Bayesian optimization techniques have been successfully applied to robotics, planning, sensor placement, recommendation, advertising, intelligent user interfaces and automatic algorithm configuration. Despite these successes, the approach is restricted to problems of moderate dimension, and several workshops on Bayesian optimization have identified its scaling to high-dimensions as one of the holy grails of the field. In this paper, we introduce a novel random embedding idea to attack this problem. The resulting Random EMbedding Bayesian Optimization (REMBO) algorithm is very simple, has important invariance properties, and applies to domains with both categorical and continuous variables. We present a thorough theoretical analysis of REMBO. Empirical results confirm that REMBO can effectively solve problems with billions of dimensions, provided the intrinsic dimensionality is low. They also show that REMBO achieves state-of-the-art performance in optimizing the 47 discrete parameters of a popular mixed integer linear programming solver.

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The dueling bandits problem is an online learning framework where learning happens ``on-the-fly'' through preference feedback, i.e., from comparisons between a pair of actions. Unlike conventional online learning settings that require absolute feedback for each action, the dueling bandits framework assumes only the presence of (noisy) binary feedback about the relative quality of each pair of actions. The dueling bandits problem is well-suited for modeling settings that elicit subjective or implicit human feedback, which is typically more reliable in preference form. In this survey, we review recent results in the theories, algorithms, and applications of the dueling bandits problem. As an emerging domain, the theories and algorithms of dueling bandits have been intensively studied during the past few years. We provide an overview of recent advancements, including algorithmic advances and applications. We discuss extensions to standard problem formulation and novel application areas, highlighting key open research questions in our discussion.

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