Self-training Converts Weak Learners to Strong Learners in Mixture Models
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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_err>0 such that if a pseudolabeler β_pl can achieve classification error at most C_err, then for any ε>0, an iterative self-training algorithm initialized at β_0 := β_pl using pseudolabels ŷ = sgn(⟨β_t, 𝐱⟩) and using at most Õ(d/ε^2) 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 β_pl with classification error C_err 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^2) unlabeled examples by way of a semi-supervised self-training algorithm.
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