Semi-Supervised Empirical Risk Minimization: When can unlabeled data improve prediction

We present a general methodology for using unlabeled data to design semi supervised learning (SSL) variants of the Empirical Risk Minimization (ERM) learning process. Focusing on generalized linear regression, we provide a careful treatment of the effectiveness of the SSL to improve prediction performance. The key ideas are carefully considering the null model as a competitor, and utilizing the unlabeled data to determine signal-noise combinations where the SSL outperforms both the ERM learning and the null model. In the special case of linear regression with Gaussian covariates, we show that the previously suggested semi-supervised estimator is in fact not capable of improving on both the supervised estimator and the null model simultaneously. However, the new estimator presented in this work, can achieve an improvement of $O(1/n)$ term over both competitors simultaneously. On the other hand, we show that in other scenarios, such as non-Gaussian covariates, misspecified linear regression, or generalized linear regression with non-linear link functions, having unlabeled data can derive substantial improvement in prediction by applying our suggested SSL approach. Moreover, it is possible to identify the usefulness of the SSL, by using the dedicated formulas we establish throughout this work. This is shown empirically through extensive simulations.