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 \emph{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. Firstly, we propose a necessary and sufficient condition for the learnability of the problem. This condition non-trivially generalizes and relaxes the existing small ambiguity degree in the PLL literature, since we allow the transition to be deterministic. Secondly, we derive Rademacher-style error bounds based on a top-$k$ surrogate loss that is widely used in the neuro-symbolic literature. Furthermore, we conclude with empirical experiments for learning under unknown transitions. The empirical results align with our theoretical findings; however, they also expose the issue of scalability in the weak supervision literature.

翻译：我们考虑一种弱监督学习场景，其中监督信号由与多个输入实例相关的标签的转移函数$\sigma$生成。我们将此问题形式化为*多实例部分标签学习（多实例PLL）*，这是对标准PLL问题的扩展。我们的问题出现在不同领域，包括潜在结构学习和神经符号融合。尽管存在许多学习技术，但对此问题的理论分析仍然有限。在本文中，我们首次对可能具有未知转移$\sigma$的多实例PLL进行了理论研究。我们的主要贡献如下。首先，我们提出了问题可学习性的充分必要条件。该条件非平凡地推广并放松了PLL文献中现有的小模糊度条件，因为我们允许转移是确定性的。其次，我们基于神经符号文献中广泛使用的top-$k$替代损失推导了Rademacher风格误差界。此外，我们通过实证实验来学习未知转移下的模型。实验结果与我们的理论发现一致；然而，它们也暴露了弱监督文献中的可扩展性问题。