  We consider a weakly supervised learning problem called Learning from Label
Proportions (LLP), where examples are grouped into ``bags'' and only the
average label within each bag is revealed to the learner. We study various
learning rules for LLP that achieve PAC learning guarantees for classification
loss. We establish that the classical Empirical Proportional Risk Minimization
(EPRM) learning rule (Yu et al., 2014) achieves fast rates under realizability,
but EPRM and similar proportion matching learning rules can fail in the
agnostic setting. We also show that (1) a debiased proportional square loss, as
well as (2) a recently proposed EasyLLP learning rule (Busa-Fekete et al.,
2023) both achieve ``optimistic rates'' (Panchenko, 2002); in both the
realizable and agnostic settings, their sample complexity is optimal (up to log
factors) in terms of $\epsilon, \delta$, and VC dimension.
