Geometry-Aware Decentralized Sinkhorn for Wasserstein Barycenters

Abstract: Distributed systems require fusing heterogeneous local probability distributions into a global summary over sparse and unreliable communication networks. Traditional consensus algorithms, which average distributions in Euclidean space, ignore their inherent geometric structure, leading to misleading results. Wasserstein barycenters offer a geometry-aware alternative by minimizing optimal transport costs, but their entropic approximations via the Sinkhorn algorithm typically require centralized coordination. This paper proposes a fully decentralized Sinkhorn algorithm that reformulates the centralized geometric mean as an arithmetic average in the log-domain, enabling approximation through local gossip protocols. Agents exchange log-messages with neighbors, interleaving consensus phases with local updates to mimic centralized iterations without a coordinator. To optimize bandwidth, we integrate event-triggered transmissions and b-bit quantization, providing tunable trade-offs between accuracy and communication while accommodating asynchrony and packet loss. Under mild assumptions, we prove convergence to a neighborhood of the centralized entropic barycenter, with bias linearly dependent on consensus tolerance, trigger threshold, and quantization error. Complexity scales near-linearly with network size. Simulations confirm near-centralized accuracy with significantly fewer messages, across various topologies and conditions.