  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 $C_{\mathrm{err}}>0$
such that if a pseudolabeler $\boldsymbol{\beta}_{\mathrm{pl}}$ can achieve
classification 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 at
most $\tilde O(d/\varepsilon^2)$ unlabeled examples suffices to learn the
Bayes-optimal classifier up to $\varepsilon$ 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
$\boldsymbol{\beta}_{\mathrm{pl}}$ 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 $O(d)$ labeled
examples and $\tilde O(d/\varepsilon^2)$ unlabeled examples by way of a
semi-supervised self-training algorithm.
