  Multi-level optimization has gained increasing attention in recent years, as
it provides a powerful framework for solving complex optimization problems that
arise in many fields, such as meta-learning, multi-player games, reinforcement
learning, and nested composition optimization. In this paper, we study the
problem of distributed multi-level optimization over a network, where agents
can only communicate with their immediate neighbors. This setting is motivated
by the need for distributed optimization in large-scale systems, where
centralized optimization may not be practical or feasible. To address this
problem, we propose a novel gossip-based distributed multi-level optimization
algorithm that enables networked agents to solve optimization problems at
different levels in a single timescale and share information through network
propagation. Our algorithm achieves optimal sample complexity, scaling linearly
with the network size, and demonstrates state-of-the-art performance on various
applications, including hyper-parameter tuning, decentralized reinforcement
learning, and risk-averse optimization.
