We consider a binary classification problem when the data comes from a mixture of two isotropic 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.
翻译:当数据来自两种异位分配的混合物时,我们考虑一个二进制分类问题。 当数据来自两种异位分配的混合物时, 我们考虑一个二进制分类问题。 当数据来自两个异位分配的混合物时, 我们发现存在一个通用的常数 $C\ mathrm{er\\ err}0美元, 这样假标签$\ boldsymbol_beta{ mathr{ pl ⁇ } 美元, 当数据来自两个异位分配的混合物时, 一个满足浓度和反浓缩特性的迭代自我培训算法, 以美元( boldsyembol_ betal_ $0:\ boldsymol_ betem} 美元。 如果假标签$\ bload\ boldsyblor_blational_lational_lational_lickrice$.