Proper learning can require training error, and every properly learnable class admits a proper learner making o(m) errors on samples of size m, but every prescribed sublinear scale is necessary for some properly learnable problem

proper learning can require training error and characterize this phenomenon precisely: every properly learnable class admits a proper learner making $o(m)$ errors on samples of size $m$, but every prescribed sublinear scale $a_m=o(m)$ is necessary for some properly learnable problem

http://arxiv.org/abs/2608.26516v1