Loss factorization, weakly supervised learning and label noise robustness

We prove that the empirical risk of most well-known loss functions factors into a linear term aggregating all labels with a term that is label free, and can further be expressed by sums of the loss. This holds true even for non-smooth, non-convex losses and in any RKHS. The first term is a (kernel) mean operator --the focal quantity of this work-- which we characterize as the sufficient statistic for the labels. The result tightens known generalization bounds and sheds new light on their interpretation. Factorization has a direct application on weakly supervised learning. In particular, we demonstrate that algorithms like SGD and proximal methods can be adapted with minimal effort to handle weak supervision, once the mean operator has been estimated. We apply this idea to learning with asymmetric noisy labels, connecting and extending prior work. Furthermore, we show that most losses enjoy a data-dependent (by the mean operator) form of noise robustness, in contrast with known negative results.