Guarantees for Tuning the Step Size using a Learning-to-Learn Approach
Authors: Xiang Wang, Shuai Yuan, Chenwei Wu, Rong Ge
Keywords: meta-learning, learning-to-learn, step size tuning, optimization, generalization
Abstract: Learning-to-learn---using optimization algorithms to learn a new optimizer---has successfully trained efficient optimizers in practice. This approach relies on meta-gradient descent on a meta-objective based on the trajectory that the optimizer generates. However, there were few theoretical guarantees on how to avoid meta-gradient explosion/vanishing problem, or how to train an optimizer with good generalization performance. In this paper, we study the learning-to-learn approach on a simple problem of tuning the step size for quadratic loss. Our results show that although there is a way to design the meta-objective so that the meta-gradient remains polynomially bounded, computing the meta-gradient directly using backpropagation leads to numerical issues that look similar to gradient explosion/vanishing problems. We also characterize when it is necessary to compute the meta-objective on a separate validation set instead of the original training set. Finally, we verify our results empirically and show that a similar phenomenon appears even for more complicated learned optimizers parametrized by neural networks.