  This paper discusses the problem of weakly supervised learning of
classification, in which instances are given weak labels that are produced by
some label-corruption process. The goal is to derive conditions under which
loss functions for weak-label learning are proper and lower-bounded -- two
essential requirements for the losses used in class-probability estimation. To
this end, we derive a representation theorem for proper losses in supervised
learning, which dualizes the Savage representation. We use this theorem to
characterize proper weak-label losses and find a condition for them to be
lower-bounded. Based on these theoretical findings, we derive a novel
regularization scheme called generalized logit squeezing, which makes any
proper weak-label loss bounded from below, without losing properness.
Furthermore, we experimentally demonstrate the effectiveness of our proposed
approach, as compared to improper or unbounded losses. Those results highlight
the importance of properness and lower-boundedness. The code is publicly
available at https://github.com/yoshum/lower-bounded-proper-losses.
