Wasserstein Convergence of ODE-Based Samplers in Decentralized Diffusion Model via Velocity Field Decomposition

Diffusion models have achieved impressive empirical success, accompanied by substantial progress in understanding their sampling convergence. Motivated by privacy and scalability, decentralized diffusion architectures replace a single global model with local experts and a learned routing mechanism. Random expert switching introduces fluctuations beyond ordinary ODE discretization error, requiring an analysis of how local approximation errors and routing randomness jointly affect generation. We study the sampling convergence of decentralized diffusion models for finite-mixture distributions under linear flow interpolation, using a single-expert-per-step Euler sampler with normalized, state-dependent routing. Under compact support, bounded learned data predictions, and spatial and temporal regularity of the learned composite field, we establish a non-asymptotic Wasserstein-2 ($W_2$) guarantee based on time-integrated population $L^2$ errors. We measure expert error ${ϵ}_v$ under the respective component marginals and router error ${ϵ}_a$ under the mixture marginals. At a fixed cutoff $t=\delta >0$, the $N$-step sampler admits a $W_2$ upper bound of order $\mathcal{O}({ϵ}_v+{ϵ}_a+N^{-1/2})$ relative to the exact cutoff marginal. The key is to combine posterior-weighted error decomposition with cancellation of the experts' shared linear drift, allowing ordinary $L^2$ router error to enter the bound linearly without an additional spatial weight. Our analysis separates deterministic integration error from stochastic routing fluctuations and explicitly accounts for data-end truncation, connecting local population approximation accuracy to distributional guarantees for decentralized sampling.