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We consider a binary classification problem when the data comes from amixture of two rotationally symmetric distributions satisfying concentrationand anti-concentration properties enjoyed by log-concave distributions amongothers. We show that there exists a universal constant $C_{\mathrm{err}}>0$such that if a pseudolabeler $\boldsymbol{\beta}_{\mathrm{pl}}$ can achieveclassification error at most $C_{\mathrm{err}}$, then for any $\varepsilon>0$,an iterative self-training algorithm initialized at $\boldsymbol{\beta}_0 :=\boldsymbol{\beta}_{\mathrm{pl}}$ using pseudolabels $\hat y =\mathrm{sgn}(\langle \boldsymbol{\beta}_t, \mathbf{x}\rangle)$ and using atmost $\tilde O(d/\varepsilon^2)$ unlabeled examples suffices to learn theBayes-optimal classifier up to $\varepsilon$ error, where $d$ is the ambientdimension. That is, self-training converts weak learners to strong learnersusing only unlabeled examples. We additionally show that by running gradientdescent on the logistic loss one can obtain a pseudolabeler$\boldsymbol{\beta}_{\mathrm{pl}}$ with classification error $C_{\mathrm{err}}$using only $O(d)$ labeled examples (i.e., independent of $\varepsilon$).Together our results imply that mixture models can be learned to within$\varepsilon$ of the Bayes-optimal accuracy using at most $O(d)$ labeledexamples and $\tilde O(d/\varepsilon^2)$ unlabeled examples by way of asemi-supervised self-training algorithm.

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