We present a novel approach for structure prediction that addresses the difficulty of obtaining labeled structures for training. We observe that structured output problems often have a companion learning problem of determining whether a given input possesses a good structure. For example, the companion problem for the part-of-speech (POS) tagging task asks whether a given sequence of words has a corresponding sequence of POS tags that is "legitimate". While obtaining direct supervision for structures is difficult and expensive, it is often very easy to obtain indirect supervision from the companion binary decision problem. In this paper, we develop a large margin framework that jointly learns from both direct and indirect forms of supervision. Our experiments ex-hibit the significant contribution of the easy-to-get indirect binary supervision on three important NLP structure learning problems.