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 at https://anonymous.4open.science/r/SpanningTreePushPull-5D3E.