Provably Faster Algorithms for Bilevel Optimization

Junjie Yang, Kaiyi Ji, Yingbin Liang

Keywords: Bilevel Optimization, Momentum, Recursive Gradient Estimator, Hessian Vector Computation

TL;DR: This paper proposes two bilevel optimizers that provably outperform all existing algorithms by the order of magnitude.

Abstract: Bilevel optimization has been widely applied in many important machine learning applications such as hyperparameter optimization and meta-learning. Recently, several momentum-based algorithms have been proposed to solve bilevel optimization problems faster. However, those momentum-based algorithms do not achieve provably better computational complexity than $\mathcal{\widetilde O}(\epsilon^{-2})$ of the SGD-based algorithm. In this paper, we propose two new algorithms for bilevel optimization, where the first algorithm adopts momentum-based recursive iterations, and the second algorithm adopts recursive gradient estimations in nested loops to decrease the variance. We show that both algorithms achieve the complexity of $\mathcal{\widetilde O}(\epsilon^{-1.5})$, which outperforms all existing algorithms by the order of magnitude. Our experiments validate our theoretical results and demonstrate the superior empirical performance of our algorithms in hyperparameter applications.

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Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2106.04692/code)

Code: https://github.com/JunjieYang97/MRVRBO