  Stochastic multi-level compositional optimization problems cover many new
machine learning paradigms, e.g., multi-step model-agnostic meta-learning,
which require efficient optimization algorithms for large-scale data. This
paper studies the decentralized stochastic multi-level optimization algorithm,
which is challenging because the multi-level structure and decentralized
communication scheme may make the number of levels significantly affect the
order of the convergence rate. To this end, we develop two novel decentralized
optimization algorithms to optimize the multi-level compositional optimization
problem. Our theoretical results show that both algorithms can achieve the
level-independent convergence rate for nonconvex problems under much milder
conditions compared with existing single-machine algorithms. To the best of our
knowledge, this is the first work that achieves the level-independent
convergence rate under the decentralized setting. Moreover, extensive
experiments confirm the efficacy of our proposed algorithms.
