DIAMOND: Taming Sample and Communication Complexities in Decentralized Bilevel Optimization

Abstract: Decentralized bilevel optimization has received increasing attention recently due to its foundational role in many emerging multi-agent learning paradigms (e.g., multi-agent meta-learning and multi-agent reinforcement learning) over peer-to-peer edge networks. However, to work with the limited computation and communication capabilities of edge networks, a major challenge in developing decentralized bilevel optimization techniques is to lower sample and communication complexities. This motivates us to develop a new decentralized bilevel optimization called DIAMOND (decentralized single-timescale stochastic approximation with momentum and gradient-tracking). The contributions of this paper are as follows: i) our DIAMOND algorithm adopts a single-loop structure rather than following the natural double-loop structure of bilevel optimization, which offers low computation and implementation complexity; ii) compared to existing approaches, the DIAMOND algorithm does not require any full gradient evaluations, which further reduces both sample and computational complexities; iii) through a careful integration of momentum information and gradient tracking techniques, we show that the DIAMOND algorithm enjoys $\mathcal{O}(\epsilon^{-3/2})$ in sample and communication complexities for achieving an $\epsilon$-stationary solution, both of which are independent of the dataset sizes and significantly outperform existing works. Extensive experiments also verify our theoretical findings.