  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.
