  Traditional supervised learning aims to train a classifier in the closed-set
world, where training and test samples share the same label space. In this
paper, we target a more challenging and realistic setting: open-set learning
(OSL), where there exist test samples from the classes that are unseen during
training. Although researchers have designed many methods from the algorithmic
perspectives, there are few methods that provide generalization guarantees on
their ability to achieve consistent performance on different training samples
drawn from the same distribution. Motivated by the transfer learning and
probably approximate correct (PAC) theory, we make a bold attempt to study OSL
by proving its generalization error-given training samples with size n, the
estimation error will get close to order O_p(1/\sqrt{n}). This is the first
study to provide a generalization bound for OSL, which we do by theoretically
investigating the risk of the target classifier on unknown classes. According
to our theory, a novel algorithm, called auxiliary open-set risk (AOSR) is
proposed to address the OSL problem. Experiments verify the efficacy of AOSR.
The code is available at github.com/Anjin-Liu/Openset_Learning_AOSR.
