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We study simultaneous learning and covering problems: submodular set cover problems that depend on the solution to an active (query) learning problem. The goal is to jointly minimize the cost of both learning and covering. We extend recent work in this setting to allow for a limited amount of adversarial noise. Certain noisy query learning problems are a special case of our problem. Crucial to our analysis is a lemma showing the logical OR of two submodular cover constraints can be reduced to a single submodular set cover constraint. Combined with known results, this new lemma allows for arbitrary monotone circuits of submodular cover constraints to be reduced to a single constraint. As an example practical application, we present a movie recommendation website that minimizes the total cost of learning what the user wants to watch and recommending a set of movies.
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