How Does Pseudo-Labeling Affect the Generalization Error of the Semi-Supervised Gibbs Algorithm?

We provide an exact characterization of the expected generalization error (gen-error) for semi-supervised learning (SSL) with pseudo-labeling via the Gibbs algorithm. The gen-error is expressed in terms of the symmetrized KL information between the output hypothesis, the pseudo-labeled dataset, and the labeled dataset. Distribution-free upper and lower bounds on the gen-error can also be obtained. Our findings offer new insights that the generalization performance of SSL with pseudo-labeling is affected not only by the information between the output hypothesis and input training data but also by the information shared between the labeled and pseudo-labeled data samples. This serves as a guideline to choose an appropriate pseudo-labeling method from a given family of methods. To deepen our understanding, we further explore two examples---mean estimation and logistic regression. In particular, we analyze how the ratio of the number of unlabeled to labeled data \lambda affects the gen-error under both scenarios. As \lambda increases, the gen-error for mean estimation decreases and then saturates at a value larger than when all the samples are labeled, and the gap can be quantified exactly with our analysis, and is dependent on the cross-covariance between the labeled and pseudo-labeled data samples. For logistic regression, the gen-error and the variance component of the excess risk also decrease as \lambda increases.