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

• URL: http://arxiv.org/abs/2106.13805v1
• Date: Fri, 25 Jun 2021 17:59:16 GMT
• Title: Self-training Converts Weak Learners to Strong Learners in Mixture Models
• Authors: Spencer Frei and Difan Zou and Zixiang Chen and Quanquan Gu
• Abstract summary: We show that a pseudolabeler $boldsymbolbeta_mathrmpl$ can achieve classification error at most $C_mathrmerr$. We additionally show that by running gradient descent on the logistic loss one can obtain a pseudolabeler $boldsymbolbeta_mathrmpl$ with classification error $C_mathrmerr$ using only $O(d)$ labeled examples.
• Score: 86.7137362125503
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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