Prometheus: Endowing Low Sample and Communication Complexities to Constrained Decentralized Stochastic Bilevel Learning

In recent years, constrained decentralized stochastic bilevel optimization has become increasingly important due to its versatility in modeling a wide range of multi-agent learning problems, such as multi-agent reinforcement learning and multi-agent meta-learning with safety constraints. However, one under-explored and fundamental challenge in constrained decentralized stochastic bilevel optimization is how to achieve low sample and communication complexities, which, if not addressed appropriately, could affect the long-term prospect of many emerging multi-agent learning paradigms that use decentralized bilevel optimization as a bedrock. In this paper, we investigate a class of constrained decentralized bilevel optimization problems, where multiple agents collectively solve a nonconvex-strongly-convex bilevel problem with constraints in the upper-level variables. Such problems arise naturally in many multi-agent reinforcement learning and meta learning problems. In this paper, we propose an algorithm called Prometheus (proximal tracked stochastic recursive estimator) that achieves the first $\mathcal{O}(\epsilon^{-1})$ results in both sample and communication complexities for constrained decentralized bilevel optimization, where $\epsilon>0$ is the desired stationarity error. Collectively, the results in this work contribute to a theoretical foundation for low sample- and communication-complexity constrained decentralized bilevel learning.