Abstract: We address the problem of extractive question answering using document-level distant super-vision, pairing questions and relevant documents with answer strings. We compare previously used probability space and distant supervision assumptions (assumptions on the correspondence between the weak answer string labels and possible answer mention spans). We show that these assumptions interact, and that different configurations provide complementary benefits. We demonstrate that a multi-objective model can efficiently combine the advantages of multiple assumptions and outperform the best individual formulation. Our approach outperforms previous state-of-the-art models by 4.3 points in F1 on TriviaQA-Wiki and 1.7 points in Rouge-L on NarrativeQA summaries.