  Despite the empirical success and practical significance of (relational)
knowledge distillation that matches (the relations of) features between teacher
and student models, the corresponding theoretical interpretations remain
limited for various knowledge distillation paradigms. In this work, we take an
initial step toward a theoretical understanding of relational knowledge
distillation (RKD), with a focus on semi-supervised classification problems. We
start by casting RKD as spectral clustering on a population-induced graph
unveiled by a teacher model. Via a notion of clustering error that quantifies
the discrepancy between the predicted and ground truth clusterings, we
illustrate that RKD over the population provably leads to low clustering error.
Moreover, we provide a sample complexity bound for RKD with limited unlabeled
samples. For semi-supervised learning, we further demonstrate the label
efficiency of RKD through a general framework of cluster-aware semi-supervised
learning that assumes low clustering errors. Finally, by unifying data
augmentation consistency regularization into this cluster-aware framework, we
show that despite the common effect of learning accurate clusterings, RKD
facilitates a "global" perspective through spectral clustering, whereas
consistency regularization focuses on a "local" perspective via expansion.
