Learning-to-Learn Stochastic Gradient Descent with Biased Regularization

The problem of learning-to-learn (LTL) [4, 30] is receiving increasing attention in recent years, due to its practical importance [11, 26] and the theoretical challenge of statistically principled and efficient solutions [1, 2, 21, 23, 9, 10, 12]. The principal aim of LTL is to design a meta-learning algorithm to select a supervised learning algorithm that is well suited to learn tasks from a prescribed family. To highlight the difference between the meta-learning algorithm and the learning algorithm, throughout the paper we will refer to the latter as the inner or within-task algorithm. The meta-algorithm is trained from a sequence of datasets, associated with different learning tasks sampled from a meta-distribution (also called environment in the literature). The performance of the selected inner algorithm is measured by the transfer risk [4, 18], that is, the average risk of the algorithm, trained on a random dataset from the same environment. A key insight is that, when the learning tasks share specific similarities, the LTL framework provides a means to leverage such similarities and select an inner algorithm of low transfer risk. In this work, we consider environments of linear regression or binary classification tasks and we assume that the associated weight vectors are all close to a common vector.