Learning from indirect supervision signals is important in real-world AI applications when, often, gold labels are missing or too costly. In this paper, we develop a unified theoretical framework for multi-class classification when the supervision is provided by a variable that contains nonzero mutual information with the gold label. The nature of this problem is determined by (i) the transition probability from the gold labels to the indirect supervision variables and (ii) the learners prior knowledge about the transition. Our framework relaxes assumptions made in the literature, and supports learning with unknown, non-invertible and instance-dependent transitions. Our theory introduces a novel concept called emph{separation}, which characterizes the learnability and generalization bounds. We also demonstrate the application of our framework via concrete novel results in a variety of learning scenarios such as learning with superset annotations and joint supervision signals.
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