Locality, Not Spectral Mixing, Governs Direct Propagation in Distributed Offline Dynamic Programming

Abstract

We study the communication complexity of distributed offline dynamic programming, where a fixed batch dataset is partitioned across (M) machines connected by the data-induced dependency graph. We compare two paradigms: direct boundary-value propagation, which follows Bellman dependencies, and gossip averaging, which mixes local estimates. Our results show that **locality** is the fundamental driver of round complexity. In particular, we prove that no method can achieve (\varepsilon)-accuracy in fewer than (L_\varepsilon = \left\lfloor \log(1/2\varepsilon) / \log(1/\gamma) \right\rfloor) rounds on graphs of diameter at least (L_\varepsilon), and we show that direct propagation matches this scaling up to constants, attaining error (O(\gamma^T/(1-\gamma) + \delta/(1-\gamma))) after (T) rounds. In contrast, gossip-style fitted value iteration incurs an additional (1/\mathrm{gap}(W)) dependence in both convergence rate and asymptotic error. We also prove bandwidth-sensitive lower bounds on path topologies and extend the analysis to asynchronous systems with bounded delays. Together, these results show that spectral dependence is an artifact of gossip-based algorithms, whereas locality is the intrinsic barrier in distributed offline dynamic programming.