Improved Inference for Imputation-Based Semisupervised Learning Under Misspecified Setting

Semisupervised learning (SSL) has been extensively studied in related literature. Despite its success, many existing learning algorithms for semisupervised problems require specific distributional assumptions, such as ``cluster assumption'' and ``low-density assumption,'' and thus, it is often hard to verify them in practice. We are interested in quantifying the effect of SSL based on kernel methods under a misspecified setting. The misspecified setting means that the target function is not contained in a hypothesis space under which some specific learning algorithm works. Practically, this assumption is mild and standard for various kernel-based approaches. Under this misspecified setting, this article makes an attempt to provide a theoretical justification on when and how the unlabeled data can be exploited to improve inference of a learning task. Our theoretical justification is indicated from the viewpoint of the asymptotic variance of our proposed two-step estimation. It is shown that the proposed pointwise nonparametric estimator has a smaller asymptotic variance than the supervised estimator using the labeled data alone. Several simulated experiments are implemented to support our theoretical results.