We introduce contextual stochastic bilevel optimization (CSBO) – a stochastic bilevel optimization framework with the lower-level problem minimizing an expectation conditioned on some contextual information and the upper-level decision variable. This framework extends classical stochastic bilevel optimization when the lower-level decision maker responds optimally not only to the decision of the upper-level decision maker but also to some side information and when there are multiple or even infinite many followers. It captures important applications such as meta-learning, personalized federated learning, end-to-end learning, and Wasserstein distributionally robust optimization with side information (WDRO-SI). Due to the presence of contextual information, existing single-loop methods for classical stochastic bilevel optimization are unable to converge. To overcome this challenge, we introduce an efficient double-loop gradient method based on the Multilevel Monte-Carlo (MLMC) technique and establish its sample and computational complexities. When specialized to stochastic nonconvex optimization, our method matches existing lower bounds. For meta-learning, the complexity of our method does not depend on the number of tasks. Numerical experiments further validate our theoretical results.

@inproceedings{hu2023contextual,
 author = {Hu, Yifan and Wang, Jie and Xie, Yao and Krause, Andreas and Kuhn, Daniel},
 booktitle = {Advances in Neural Information Processing Systems},
 pages = {78412--78434},
 publisher = {Curran Associates, Inc.},
 title = {Contextual Stochastic Bilevel Optimization},
 volume = {36},
 year = {2023}}