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

Abstract

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 C_err>0 such that if a pseudolabeler _pl can achieve classification error at most C_err, then for any >0, an iterative self-training algorithm initialized at _0 := _pl using pseudolabels y = sgn( _t, x) and using at most O(d/^2) unlabeled examples suffices to learn the Bayes-optimal classifier up to 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 _pl with classification error C_err using only O(d) labeled examples (i.e., independent of ). Together our results imply that mixture models can be learned to within of the Bayes-optimal accuracy using at most O(d) labeled examples and O(d/^2) unlabeled examples by way of a semi-supervised self-training algorithm.

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