Semi-Supervised Triply Robust Inductive Transfer Learning

In this work, we propose a <b>S</b>emi-supervised <b>T</b>riply <b>R</b>obust <b>I</b>nductive trans<b>F</b>er <b>LE</b>arning (STRIFLE) approach, which integrates heterogeneous data from a label-rich source population and a label-scarce target population and utilizes a large amount of unlabeled data simultaneously to improve the learning accuracy in the target population. Specifically, we consider a high dimensional covariate shift setting and employ two nuisance models, a density ratio model and an imputation model, to combine transfer learning and surrogate-assisted semi-supervised learning strategies effectively and achieve triple robustness. While the STRIFLE approach assumes the target and source populations to share the same conditional distribution of outcome <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math> given both the surrogate features <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>S</mml:mtext></mml:math> and predictors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>X</mml:mtext></mml:math> , it allows the true underlying model of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Y</mml:mi> <mml:mo>∣</mml:mo> <mml:mtext>X</mml:mtext></mml:mrow> </mml:math> to differ between the two populations due to the potential covariate shift in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>S</mml:mtext></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>X</mml:mtext></mml:math> . Different from double robustness, even if both nuisance models are misspecified or the distribution of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Y</mml:mi> <mml:mo>∣</mml:mo> <mml:mtext>S</mml:mtext></mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>X</mml:mtext></mml:math> is not the same between the two populations, the triply robust STRIFLE estimator can still partially use the source population when the shifted source population and the target population share enough similarities. Moreover, it is guaranteed to be no worse than the target-only surrogate-assisted semi-supervised estimator with an additional error term from transferability detection. These desirable properties of our estimator are established theoretically and verified in finite samples via extensive simulation studies. We utilize the STRIFLE estimator to train a Type II diabetes polygenic risk prediction model for the African American target population by transferring knowledge from electronic health records linked genomic data observed in a larger European source population.