Disambiguation of weak supervision with exponential convergence rates

In many applications of machine learning, such as recommender systems, where an input characterizing a user should be matched with a target representing an ordering of a large number of items, accessing fully supervised data (,) is not an option. Instead, one should expect weak information on the target, which could be a list of previously taken (if items are online courses), watched (if items are plays), etc., items by a user characterized by the feature vector. This motivates weakly supervised learning, aiming at learning a mapping from inputs to targets in such a setting where tools from supervised learning can not be applied off-the-shelves. Recent applications of weakly supervised learning showcase impressive results in solving complex tasks such as action retrieval on instructional videos (Miech et al., 2019), image semantic segmentation (Papandreou et al., 2015), salient object detection (Wang et al., 2017), 3D pose estimation (Dabral et al., 2018), text-to-speech synthesis (Jia et al., 2018), to name a few. However, those applications of weakly supervised learning are usually based on clever heuristics, and theoretical foundations of learning from weakly supervised data are scarce, especially when compared to statistical learning literature on supervised learning (Vapnik, 1995; Boucheron et al., 2005; Steinwart and Christmann, 2008). We aim to provide a step in this direction. In this paper, we focus on partial labelling, a popular instance of weak supervision, approached with a structured prediction point of view Ciliberto et al. (2020). We detail this setup in Section 2. Our contributions are organized as follows.