Guarantees for Tuning the Step Size using a Learning-to-Learn Approach

• URL: http://arxiv.org/abs/2006.16495v2
• Date: Fri, 11 Jun 2021 04:21:42 GMT
• Title: Guarantees for Tuning the Step Size using a Learning-to-Learn Approach
• Authors: Xiang Wang, Shuai Yuan, Chenwei Wu, Rong Ge
• Abstract summary: We give meta-optimization guarantees for the learning-to-learn approach on a simple problem of tuning the step size for quadratic loss. Although there is a way to design the meta-objective so that the meta-gradient remains bounded, computing the meta-gradient directly using backpropagation leads to numerical issues. We also characterize when it is necessary to compute the meta-objective on a separate validation set to ensure the performance of the learned.
• Score: 18.838453594698166
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Choosing the right parameters for optimization algorithms is often the key to their success in practice. Solving this problem using a learning-to-learn approach -- using meta-gradient descent on a meta-objective based on the trajectory that the optimizer generates -- was recently shown to be effective. However, the meta-optimization problem is difficult. In particular, the meta-gradient can often explode/vanish, and the learned optimizer may not have good generalization performance if the meta-objective is not chosen carefully. In this paper we give meta-optimization guarantees for the learning-to-learn approach on a simple problem of tuning the step size for quadratic loss. Our results show that the na\"ive objective suffers from meta-gradient explosion/vanishing problem. 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. We also characterize when it is necessary to compute the meta-objective on a separate validation set to ensure the generalization performance of the learned optimizer. Finally, we verify our results empirically and show that a similar phenomenon appears even for more complicated learned optimizers parametrized by neural networks.

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