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，其中监督信号由多个输入实例相关联的标签通过转移函数$\sigma$生成。我们将该问题形式化为\emph{多实例偏标签学习（multi-instance PLL）}，这是对标准PLL问题的扩展。该问题出现在多个领域，包括隐式结构学习与神经符号集成。尽管已有许多学习技术，但针对该问题的理论分析仍较为有限。本文首次对转移函数$\sigma$可能未知情况下的多实例PLL问题进行了理论研究。我们的主要贡献如下：首先，我们提出了该问题可学习性的充要条件。该条件非平凡地推广并放宽了PLL文献中现有的小模糊度假设，因为我们允许转移函数是确定性的。其次，基于神经符号文献中广泛使用的top-$k$代理损失函数，我们推导了拉德马赫风格的误差界。最后，我们通过实验验证了未知转移函数下的学习性能。实验结果与理论发现一致，但同时也暴露出现有弱监督方法在可扩展性方面存在的问题。