A Distribution-dependent Analysis of Meta Learning

A key problem in the theory of meta-learning is to understand how the task distributions influence transfer risk, the expected error of a metalearner on a new task drawn from the unknown task distribution. In this paper, focusing on fixed design linear regression with Gaussian noise and a Gaussian task (or parameter) distribution, we give distribution-dependent lower bounds on the transfer risk of any algorithm, while we also show that a novel, weighted version of the so-called biased regularized regression method is able to match these lower bounds up to a fixed constant factor. Notably, the weighting is derived from the covariance of the Gaussian task distribution. Altogether, our results provide a precise characterization of the difficulty of meta-learning in this Gaussian setting. While this problem setting may appear simple, we show that it is rich enough to unify the "parameter sharing" and "representation learning" streams of meta-learning; in particular, representation learning is obtained as the special case when the covariance matrix of the task distribution is unknown. For this case we propose to adopt the EM method, which is shown to enjoy efficient updates in our case. The paper is completed by an empirical study of EM. In particular, our experimental results show that the EM algorithm can attain the lower bound as the number of tasks grows, while the algorithm is also successful in competing with its alternatives when used in a representation learning context.