  Contrastive learning has shown outstanding performances in both supervised
and unsupervised learning, and has recently been introduced to solve weakly
supervised learning problems such as semi-supervised learning and noisy label
learning. Despite the empirical evidence showing that semi-supervised labels
improve the representations of contrastive learning, it remains unknown if
noisy supervised information can be directly used in training instead of after
manual denoising. Therefore, to explore the mechanical differences between
semi-supervised and noisy-labeled information in helping contrastive learning,
we establish a unified theoretical framework of contrastive learning under weak
supervision. Specifically, we investigate the most intuitive paradigm of
jointly training supervised and unsupervised contrastive losses. By translating
the weakly supervised information into a similarity graph under the framework
of spectral clustering based on the posterior probability of weak labels, we
establish the downstream classification error bound. We prove that
semi-supervised labels improve the downstream error bound whereas noisy labels
have limited effects under such a paradigm. Our theoretical findings here
provide new insights for the community to rethink the role of weak supervision
in helping contrastive learning.
