Knowledge-Guided Wasserstein Distributionally Robust Optimization

Transfer learning is a popular strategy to leverage external knowledge and improve statistical efficiency, particularly with a limited target sample. We propose a novel knowledge-guided Wasserstein Distributionally Robust Optimization (KG-WDRO) framework that adaptively incorporates multiple sources of external knowledge to overcome the conservativeness of vanilla WDRO, which often results in overly pessimistic shrinkage toward zero. Our method constructs smaller Wasserstein ambiguity sets by controlling the transportation along directions informed by the source knowledge. This strategy can alleviate perturbations on the predictive projection of the covariates and protect against information loss. Theoretically, we establish the equivalence between our WDRO formulation and the knowledge-guided shrinkage estimation based on collinear similarity, ensuring tractability and geometrizing the feasible set. This also reveals a novel and general interpretation for recent shrinkage-based transfer learning approaches from the perspective of distributional robustness. In addition, our framework can adjust for scaling differences in the regression models between the source and target and accommodates general types of regularization such as lasso and ridge. Extensive simulations demonstrate the superior performance and adaptivity of KG-WDRO in enhancing small-sample transfer learning.

Efficient Estimation and Evaluation of Prediction Rules in Semi-Supervised Settings under Stratified Sampling

In many contemporary applications, large amounts of unlabeled data are readily available while labeled examples are limited. There has been substantial interest in semi-supervised learning (SSL) which aims to leverage unlabeled data to improve estimation or prediction. However, current SSL literature focuses primarily on settings where labeled data is selected randomly from the population of interest. Non-random sampling, while posing additional analytical challenges, is highly applicable to many real world problems. Moreover, no SSL methods currently exist for estimating the prediction performance of a fitted model under non-random sampling. In this paper, we propose a two-step SSL procedure for evaluating a prediction rule derived from a working binary regression model based on the Brier score and overall misclassification rate under stratified sampling. In step I, we impute the missing labels via weighted regression with nonlinear basis functions to account for nonrandom sampling and to improve efficiency. In step II, we augment the initial imputations to ensure the consistency of the resulting estimators regardless of the specification of the prediction model or the imputation model. The final estimator is then obtained with the augmented imputations. We provide asymptotic theory and numerical studies illustrating that our proposals outperform their supervised counterparts in terms of efficiency gain. Our methods are motivated by electronic health records (EHR) research and validated with a real data analysis of an EHR-based study of diabetic neuropathy.