  In this paper, we investigate the decentralized statistical inference
problem, where a network of agents cooperatively recover a (structured) vector
from private noisy samples without centralized coordination. Existing
optimization-based algorithms suffer from issues of model mismatch and poor
convergence speed, and thus their performance would be degraded, provided that
the number of communication rounds is limited. This motivates us to propose a
learning-based framework, which unrolls well-noted decentralized optimization
algorithms (e.g., Prox-DGD and PG-EXTRA) into graph neural networks (GNNs). By
minimizing the recovery error via end-to-end training, this learning-based
framework resolves the model mismatch issue. Our convergence analysis (with
PG-EXTRA as the base algorithm) reveals that the learned model parameters may
accelerate the convergence and reduce the recovery error to a large extent. The
simulation results demonstrate that the proposed GNN-based learning methods
prominently outperform several state-of-the-art optimization-based algorithms
in convergence speed and recovery error.
