Formal Bayesian Transfer Learning via the Total Risk Prior

Abstract

Existing methods for transfer learning struggle to deal with situations where the source datasets are limited and not guaranteed to be well-aligned with the target dataset. A typical strategy is to use the empirical loss minimizer on the source data as a prior mean for the target parameters. Our key conceptual contribution is to use a risk minimizer conditional on source parameters instead. This allows us to construct a single joint prior distribution for all parameters from the source datasets as well as the target dataset. As a consequence, we benefit from full Bayesian uncertainty quantification and can perform model averaging via Gibbs sampling over indicator variables governing the inclusion of each source dataset. We show how a particular instantiation of our prior leads to a Bayesian Lasso in a transformed coordinate system and discuss computational techniques to scale our approach to moderately sized datasets. We discuss connections between the Maximum a Posteriori estimate associated with our approach and the recently proposed Trans-Lasso method and demonstrate that the MAP estimator MSE-dominates the Trans-Lasso in the normal means setting when there is no regularization on the source datasets. Finally, we perform numerical experiments finding that full Bayesian inference provides superior predictive performance relative to Trans-Lasso on a genetics application, especially when the source data are limited.