Self-training Converts Weak Learners to Strong Learners in Mixture Models

We consider a binary classification problem when the data comes from a mixture of two rotationally symmetric distributions satisfying concentration and anti-concentration properties enjoyed by log-concave distributions among others. We show that there exists a universal constant Cerr > 0 such that if a pseudolabeler βpl can achieve classification error at most Cerr, then for any ε > 0, an iterative self-training algorithm initialized at β0 := βpl using pseudolabels ŷ = sgn(〈βt,x〉) and using at most Õ(d/ε) unlabeled examples suffices to learn the Bayes-optimal classifier up to ε error, where d is the ambient dimension. That is, self-training converts weak learners to strong learners using only unlabeled examples. We additionally show that by running gradient descent on the logistic loss one can obtain a pseudolabeler βpl with classification error Cerr using only O(d) labeled examples (i.e., independent of ε). Together our results imply that mixture models can be learned to within ε of the Bayes-optimal accuracy using at most O(d) labeled examples and Õ(d/ε) unlabeled examples by way of a semi-supervised self-training algorithm.