A framework for bilevel optimization that enables stochastic and global variance reduction algorithms
Abstract
Bilevel optimization, the problem of minimizing a value function which involves the argminimum of another function, appears in many areas of machine learning. In a large scale empirical risk minimization setting where the number of samples is huge, it is crucial to develop stochastic methods, which only use a few samples at a time to progress. However, computing the gradient of the value function involves solving a linear system, which makes it difficult to derive unbiased stochastic estimates. To overcome this problem we introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. These directions are written as a sum, making it straightforward to derive unbiased estimates. The simplicity of our approach allows us to develop global variance reduction algorithms, where the dynamics of all variables is subject to variance reduction. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O(\frac1T)$ convergence rate, and that it achieves linear convergence under PolyakLojasciewicz assumption. This is the first stochastic algorithm for bilevel optimization that verifies either of these properties. Numerical experiments validate the usefulness of our method.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 DOI:
 10.48550/arXiv.2201.13409
 arXiv:
 arXiv:2201.13409
 Bibcode:
 2022arXiv220113409D
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Machine Learning;
 Mathematics  Optimization and Control
 EPrint:
 Accepted at NeurIPS 2022