  Meta-learning involves training models on a variety of training tasks in a
way that enables them to generalize well on new, unseen test tasks. In this
work, we consider meta-learning within the framework of high-dimensional
multivariate random-effects linear models and study generalized
ridge-regression based predictions. The statistical intuition of using
generalized ridge regression in this setting is that the covariance structure
of the random regression coefficients could be leveraged to make better
predictions on new tasks. Accordingly, we first characterize the precise
asymptotic behavior of the predictive risk for a new test task when the data
dimension grows proportionally to the number of samples per task. We next show
that this predictive risk is optimal when the weight matrix in generalized
ridge regression is chosen to be the inverse of the covariance matrix of random
coefficients. Finally, we propose and analyze an estimator of the inverse
covariance matrix of random regression coefficients based on data from the
training tasks. As opposed to intractable MLE-type estimators, the proposed
estimators could be computed efficiently as they could be obtained by solving
(global) geodesically-convex optimization problems. Our analysis and
methodology use tools from random matrix theory and Riemannian optimization.
Simulation results demonstrate the improved generalization performance of the
proposed method on new unseen test tasks within the considered framework.
