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.

提出并分析了一种基于元学习的方法，其中通过计算具有随机回归系数的高维多元随机效应线性模型的预测风险的精确渐近行为，选择广义岭回归中的权重矩阵为随机系数协方差矩阵的逆作为可扩展的度量；并通过使用从训练任务中获得的数据基于 Riemannian 优化方法提出并分析了估计随机回归系数的逆协方差矩阵的估计器，最后的仿真结果表明了在考虑的框架内所提出的方法对新的未观察到的测试任务的改进泛化性能。