Precise High-Dimensional Asymptotics for Quantifying Heterogeneous Transfers

The problem of learning one task with samples from another task is central to transfer learning (TL). In this paper, we examine a fundamental question: When does combining the data samples from a source task and a target task perform better than single-task learning with the target task alone? This question is motivated by an intriguing phenomenon known as negative transfer often observed in the TL literature. Precise quantification of TL effects -- even within simple statistical models -- has remained elusive in the statistical learning literature. A critical challenge is that to compare TL to single-task learning, we would need to compare the risks between two different estimators in a very precise way. In particular, the comparative advantage of one estimator over another would depend on the specific distribution shifts between the two tasks. This paper applies recent developments in the random matrix theory literature to tackle this challenge in a high-dimensional linear regression setting with two tasks. We provide precise high-dimensional asymptotics for the bias and variance of hard parameter sharing (HPS) estimators in the proportional limit, when the sample sizes of both tasks increase proportionally with dimension at fixed ratios. The precise asymptotics are expressed as a function of the sample sizes of both tasks, the covariate shift between their feature population covariate matrices, and the model shift. We provide illustrative examples of our results in a random-effects model to determine positive and negative transfers. For example, we can identify a phase transition in the high-dimensional linear regression setting from positive transfer to negative transfer under a model shift between the source and target tasks. The finding regarding phase transition can be extended to a multiple-task learning setting where the feature covariates are shared across all tasks.