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Self-training is a classical approach in semi-supervised learning which is successfully applied to a variety of machine learning problems. Self-training algorithm generates pseudo-labels for the unlabeled examples and progressively refines these pseudo-labels which hopefully coincides with the actual labels. This work provides theoretical insights into self-training algorithm with a focus on linear classifiers. We first investigate Gaussian mixture models and provide a sharp non-asymptotic finite-sample characterization of the self-training iterations. Our analysis reveals the provable benefits of rejecting samples with low confidence and demonstrates that self-training iterations gracefully improve the model accuracy even if they do get stuck in sub-optimal fixed points. We then demonstrate that regularization and class margin (i.e. separation) is provably important for the success and lack of regularization may prevent self-training from identifying the core features in the data. Finally, we discuss statistical aspects of empirical risk minimization with self-training for general distributions. We show how a purely unsupervised notion of generalization based on self-training based clustering can be formalized based on cluster margin. We then establish a connection between self-training based semi-supervision and the more general problem of learning with heterogenous data and weak supervision.