Distributed Learning over Arbitrary Topology: Linear Speed-Up with Polynomial Transient Time

We study a distributed learning problem in which $n$ agents, each with potentially heterogeneous local data, collaboratively minimize the sum of their local cost functions via peer-to-peer communication. We propose a novel algorithm, Spanning Tree Push-Pull (STPP), which employs two spanning trees extracted from a general communication graph to distribute both model parameters and stochastic gradients. Unlike prior approaches that rely heavily on spectral gap properties, STPP leverages a more flexible topological characterization, enabling robust information flow and efficient updates. Theoretically, we prove that STPP achieves linear speedup and polynomial transient iteration complexity, up to $O(n^7)$ for smooth nonconvex objectives and $\tilde{O}(n^3)$ for smooth strongly convex objectives, under arbitrary network topologies. Moreover, compared with the existing methods, STPP achieves faster convergence rates on sparse and non-regular topologies (e.g., directed ring) and reduces communication overhead on dense networks (e.g., static exponential graph). These results significantly advance the state of the art, especially when $n$ is large. Numerical experiments further demonstrate the strong performance of STPP and confirm the practical relevance of its theoretical convergence rates across various common graph architectures. Our code is available atthis https URL.

@article{you2025_2503.16123,
  title={ Distributed Learning over Arbitrary Topology: Linear Speed-Up with Polynomial Transient Time },
  author={ Runze You and Shi Pu },
  journal={arXiv preprint arXiv:2503.16123},
  year={ 2025 }
}