  This paper discusses the problem of weakly supervised 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.
From 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. The results highlight the importance of properness and
lower-boundedness.
