  We study the problem of estimation and testing in logistic regression with
class-conditional noise in the observed labels, which has an important
implication in the Positive-Unlabeled (PU) learning setting. With the key
observation that the label noise problem belongs to a special sub-class of
generalized linear models (GLM), we discuss convex and non-convex approaches
that address this problem. A non-convex approach based on the maximum
likelihood estimation produces an estimator with several optimal properties,
but a convex approach has an obvious advantage in optimization. We demonstrate
that in the low-dimensional setting, both estimators are consistent and
asymptotically normal, where the asymptotic variance of the non-convex
estimator is smaller than the convex counterpart. We also quantify the
efficiency gap which provides insight into when the two methods are comparable.
In the high-dimensional setting, we show that both estimation procedures
achieve $\ell_2$-consistency at the minimax optimal $\sqrt{s\log p/n}$ rates
under mild conditions. Finally, we propose an inference procedure using a
de-biasing approach. We validate our theoretical findings through simulations
and a real-data example.
