Precise High-Dimensional Asymptotics for Quantifying Heterogeneous Transfers

The problem of learning one task with samples from another task has received much interest recently. In this paper, we ask a fundamental question: when is combining data from two tasks better than learning one task alone? Intuitively, the transfer effect from one task to another task depends on dataset shifts such as sample sizes and covariance matrices. However, quantifying such a transfer effect is challenging since we need to compare the risks between joint learning and single-task learning, and the comparative advantage of one over the other depends on the exact kind of dataset shift between both tasks. This paper uses random matrix theory to tackle this challenge in a linear regression setting with two tasks. We give precise asymptotics about the excess risks of some commonly used estimators in the high-dimensional regime, when the sample sizes increase proportionally with the feature dimension at fixed ratios. The precise asymptotics is provided as a function of the sample sizes and covariate/model shifts, which can be used to study transfer effects: In a random-effects model, we give conditions to determine positive and negative transfers between learning two tasks versus single-task learning; the conditions reveal intricate relations between dataset shifts and transfer effects. Simulations justify the validity of the asymptotics in finite dimensions. Our analysis examines several functions of two different sample covariance matrices, revealing some estimates that generalize classical results in the random matrix theory literature, which may be of independent interest.