Paper:
Positive-unlabeled (PU) learning addresses the problem of learning a binary classifier from positive (P) and unlabeled (U) data. It is often applied to situations where negative (N) data are difficult to be fully labeled. However, collecting a non-representative N set that contains only a small portion of all possible N data can be much easier in many practical situations. This paper studies a novel classification framework which incorporates such biased N (bN) data in PU learning. The fact that the training N data are biased also makes our work very different from those of standard semi-supervised learning. We provide an empirical risk minimization-based method to address this PUbN classification problem. Our approach can be regarded as a variant of traditional example-reweighting algorithms, with the weight of each example computed through a preliminary step that draws inspiration from PU learning. We also derive an estimation error bound for the proposed method. Experimental results demonstrate the effectiveness of our algorithm in not only PUbN learning scenarios but also ordinary PU leaning scenarios on several benchmark datasets.

Review:
The authors rst present standard binary (positive negative or PN) classica-
tion, followed by positive unlabeled (PU) classication, that they motivate with
examples, such as one-class remote sensing classication. The new setting that
they introduce and study is called positive unlabeled biaised negative (PUbN
classication) and adds a biaised negative sample to PU learning. They give
motivating examples and compare this setting to the existing literature. A con-
vincing case is made regarding the dierence between the PUbN problem and
the known problems of semi-supervised learning and dataset shift.
They start by recalling the notations and nature of standard binary classi-
cation, PU classication and the nnPU (non-negative PU) strategy, as in the
previous PU learning papers. Then, they present the semi-supervised setting
under the name PNU learning, which simply studies the minimization of a con-
vex combination of the PN risk and the PU risk. As in PU learning, a correction
exists to avoid considering the estimate of the negative risk to be negative, re-
ferred to as nnPNU.
Finally, the authors introduce PUbN learning as the problem in which we
only have access to negatives that follow the law p(x\mid y = -1; s = +1), where s
is a latent variable that formalizes the bias.
As in PU learning, the authors derive an unbiased estimator of the risk that
involves only distributions for which data is available. However, they need
to reweight the P and bN distribution by the unknown posterior probability
sigma(x) = p(s = +1\mid x) of s. Considering s as the label, the problem of learning
a probabilistic classier separating the elements for which s = +1 and s = 􀀀1
can be seen as a PU learning problem, which gives an estimator ^ of sigma,
and makes the method practical.
They derive estimation error bounds, that depend on the mean squared
difference between sigma and sigma^ and a term of order n^-1/2 where n is the cardinal
of the smallest sample. They considered the function ^ as a xed function in
their bounds, which implies that the bounds are only true if some of the data is
kept for the estimation of sigma^. Finally, they present a variant of their algorithm
for PU learning, named PUbNnN where unlabeled instances are not all given
the same weight, but weighted according to sigma hat. The experiments use neural networks with stochastic optimization, on the classic datasets MNIST, CIFAR-10 and 20 Newsgroup. They report better per-
formance using their technique on all datasets. The authors documented their
experiences thoroughly in the appendix. However, I did not find information
about the nature of the estimator of the posterior probability sigma^, which is im-
portant for reproducibility. Furthermore, in appendix B, choosing sigma^ = 0 will
minimize the criterion . Finally, they proceed to justify the dominance of
the variant of their method over usual nnPU learning.