On Learning Latent Models with Multi-Instance Weak Supervision

We consider a weakly supervised learning scenario where the supervision signal is generated by a transition function $\sigma$ of labels associated with multiple input instances. We formulate this problem as *multi-instance Partial Label Learning (multi-instance PLL)*, which is an extension to the standard PLL problem. Our problem is met in different fields, including latent structural learning and neuro-symbolic integration. Despite the existence of many learning techniques, limited theoretical analysis has been dedicated to this problem. In this paper, we provide the first theoretical study of multi-instance PLL with possibly an unknown transition $\sigma$. Our main contributions are as follows: First, we proposed a necessary and sufficient condition for the learnability of the problem. This condition nontrivially generalizes and relaxes the existing *small ambiguity degree* in PLL literature since we allow the transition to be deterministic. Second, we derived Rademacher-style error bounds based on the top-$k$ surrogate loss that is widely used in the neuro-symbolic literature. Furthermore, we conclude with empirical experiments for learning with an unknown transition. The empirical results align with our theoretical findings; however, they also expose the issue of scalability in the weak supervision literature.