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Prior works have shown that semi-supervised learning algorithms can leverageunlabeled data to improve over the labeled sample complexity of supervisedlearning (SL) algorithms. However, existing theoretical analyses focus onregimes where the unlabeled data is sufficient to learn a good decisionboundary using unsupervised learning (UL) alone. This begs the question: CanSSL algorithms simultaneously improve upon both UL and SL? To this end, wederive a tight lower bound for 2-Gaussian mixture models that explicitlydepends on the labeled and the unlabeled dataset size as well as thesignal-to-noise ratio of the mixture distribution. Surprisingly, our resultimplies that no SSL algorithm can improve upon the minimax-optimal statisticalerror rates of SL or UL algorithms for these distributions. Nevertheless, weshow empirically on real-world data that SSL algorithms can still outperform ULand SL methods. Therefore, our work suggests that, while proving performancegains for SSL algorithms is possible, it requires careful tracking ofconstants.

Can semi-supervised learning use all the data effectively? A lower bound perspective