We show every multi-group learner in the transductive setting may incur a multiplicative penalty in its error rate on some group relative to the error rate achievable in the single-group setting, and the penalty can increasing linearly with the number of groups, up to roughly the square-root of the sample size. This stands in stark contrast to optimal multi-group learners in an analogous (group-realizable) statistical setting, where the penalty is always at most logarithmic in the sample size and independent of the number of groups.

翻译：我们证明，在直推学习框架下，任何多组学习器在某些组上的错误率可能相对于单组设定下可实现错误率承受乘性惩罚，且该惩罚可随组数线性增长，直至接近样本量的平方根。这与类比（组可实现）统计设定下的最优多组学习器形成鲜明对比，后者惩罚始终最多为样本量的对数级别，且与组数无关。