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A key problem in the theory of meta-learning is to understand how the taskdistributions influence transfer risk, the expected error of a meta-learner ona new task drawn from the unknown task distribution. In this paper, focusing onfixed design linear regression with Gaussian noise and a Gaussian task (orparameter) distribution, we give distribution-dependent lower bounds on thetransfer risk of any algorithm, while we also show that a novel, weightedversion of the so-called biased regularized regression method is able to matchthese lower bounds up to a fixed constant factor. Notably, the weighting isderived from the covariance of the Gaussian task distribution. Altogether, ourresults provide a precise characterization of the difficulty of meta-learningin this Gaussian setting. While this problem setting may appear simple, we showthat it is rich enough to unify the "parameter sharing" and "representationlearning" streams of meta-learning; in particular, representation learning isobtained as the special case when the covariance matrix of the taskdistribution is unknown. For this case we propose to adopt the EM method, whichis shown to enjoy efficient updates in our case. The paper is completed by anempirical study of EM. In particular, our experimental results show that the EMalgorithm can attain the lower bound as the number of tasks grows, while thealgorithm is also successful in competing with its alternatives when used in arepresentation learning context.

A Distribution-Dependent Analysis of Meta-Learning