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We provide an exact characterization of the expected generalization error(gen-error) for semi-supervised learning (SSL) with pseudo-labeling via theGibbs algorithm. The gen-error is expressed in terms of the symmetrized KLinformation between the output hypothesis, the pseudo-labeled dataset, and thelabeled dataset. Distribution-free upper and lower bounds on the gen-error canalso be obtained. Our findings offer new insights that the generalizationperformance of SSL with pseudo-labeling is affected not only by the informationbetween the output hypothesis and input training data but also by theinformation {\em shared} between the {\em labeled} and {\em pseudo-labeled}data samples. This serves as a guideline to choose an appropriatepseudo-labeling method from a given family of methods. To deepen ourunderstanding, we further explore two examples -- mean estimation and logisticregression. In particular, we analyze how the ratio of the number of unlabeledto labeled data $\lambda$ affects the gen-error under both scenarios. As$\lambda$ increases, the gen-error for mean estimation decreases and thensaturates at a value larger than when all the samples are labeled, and the gapcan be quantified {\em exactly} with our analysis, and is dependent on the\emph{cross-covariance} between the labeled and pseudo-labeled data samples.For logistic regression, the gen-error and the variance component of the excessrisk also decrease as $\lambda$ increases.

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