A Class of Geometric Structures in Transfer Learning: Minimax Bounds and Optimality


We study the problem of transfer learning, observing that previous efforts to understand its information-theoretic limits typically ignore the role of any geometric structure of the source and target domains. In contrast, our study first illustrates the benefits of incorporating such geometric structure within a linear regression model, which corresponds to the generalized eigenvalue problem formed by the Gram matrices of both domains. We establish a finite-sample minimax lower bound, propose a refined model interpolation estimator that enjoys a matching upper bound, and then extend our framework to multiple source domains and generalized linear models. Surprisingly, as long as information is available on the distance between the source and target parameters, negative-transfer does not occur. Simulation studies show that our proposed interpolation estimator outperforms state-of-the-art transfer learning methods in both moderate- and high-dimensional settings.