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Distributed Model-Based Diffusion: Finite Horizon Contraction under Bounded Delay
Seth Golembeski, Keith L. Gibson, Alexander Gross, Shreyas Kousik, Anirban Mazumdar
TL;DR
Multi-agent trajectory optimization is difficult under nonlinearity, nonconvexity, high dimensionality, and communication latency. The paper analyzes Distributed Model-Based Diffusion, proving contraction and latency robustness and evaluating it across three multi-agent tasks, where it improves performance despite added latency.
Problem
Multi-agent trajectory optimization remains difficult for nonlinear, nonconvex, high-dimensional systems, and communication latency further increases the challenge.
Method
The paper analyzes Distributed Model-Based Diffusion, a distributed sampling-based Model-Predictive Control method that uses diffusion-based convexification for multi-agent optimization.
Results
DMBD outperforms centralized MBD and baselines across three experiments, with gradual performance degradation as latency increases.
Takeaways & Limitations
DMBD provides finite-time contraction and stability under increasing latency or network degradation, with improvements in makespan, reward, and win rate across experiments.
Takeaways & Limitations
The theory is limited to smooth convex cores, shared cost functions, weak coupling, and bounded latency.
Abstract
from arXiv · showhide
Simultaneously optimizing the trajectories of multiple agents is a challenging problem plagued by nonlinearity, nonconvexity, and the curse of dimensionality. A collection of interacting aerial vehicles or self-driving cars in an intersection are examples of complex multi-agent systems that remain difficult to solve without many simplifying assumptions. The presence of communication latency between agents further increases the difficulty. In this paper, we analyze Distributed Model-Based Diffusion: a sampling-based Model-Predictive Control method suitable for highly nonlinear, nonconvex, nonsmooth, multi-agent systems. We prove contraction and robustness to latency for multi-agent, nonconvex problems, showing applicability to real-world constraints. We test the algorithm on a circleswap task, a cooperative medium-fidelity driving task, and in an aerial combat scenario. Despite the addition of latency, our algorithm improves circleswap makespan by 31% and increases aerial combat win rate by 25% compared to centralized Model-Based Diffusion.
1 Introduction
Multi-robot coordination remains difficult for trajectory optimization and MPC because existing methods often rely on restrictive structural assumptions. This work analyzes distributed Model-Based Diffusion, proving finite-horizon contraction and latency robustness while evaluating it across increasingly complex multi-agent tasks.
- Multi-robot coordination is challenging despite benefits including robustness, complementary capabilities, and greater sensor coverage.
- Existing trajectory-optimization and MPC methods often require assumptions of linearity, convexity, or differentiability.
- DMBD extends Model-Based Diffusion to distributed multi-agent systems and addresses degraded communication caused by latency.
- DMBD globally contracts to a noise floor in finite time and is analyzed for robustness under realistic network degradation.
- The evaluation covers unicycle circleswap, multi-agent car racing, and multi-agent agile aerial combat under degraded communication.
2 Background and Related Work
The paper situates DMBD among sampling-based MPC methods that use diffusion to convexify difficult objectives. Its distributed formulation separates per-agent diffusion processes, parallelizes computation, and handles delayed communication with timestamped messages.
- Multiagent optimization partitions a shared decision variable across agents that jointly minimize a shared objective containing costs and soft constraint penalties.
- Sampling-based MPC uses Gaussian smoothing to progressively convexify problems, mitigating the convexity limitations of classical trajectory optimization.
- MBD samples the objective to estimate ∇log p_0(Y), a component of a reverse SDE whose smoothed dynamics support global convergence.
- MBD iteratively estimates and updates the trajectory through diffusion proposals, exponential weighting, gradient estimation, and gradient ascent over N steps.
- DMBD assigns each agent its own diffusion process, evaluates the global cost while holding peers’ actions constant, and runs these processes in parallel.
- Timestamped messages prevent duplicate or out-of-order updates under latency, with experiments using a maximum latency of 50ms.
3 Convergence Analysis
The convergence analysis establishes DMBD’s convergence through system definitions, stepwise contraction, and error-dynamics bounds. It also develops probabilistic guarantees for remaining in the convex region and bounds RMS error.
- DMBD’s main theoretical result is convergence.
- The proof first defines the multi-agent system and induced properties before establishing the method’s contraction behavior.
- The analysis bounds error dynamics after deriving stepwise contraction, with the key result identified as Theorem 3.6.
- The convergence analysis uses Algorithm 1, which is titled Distributed MBD.
- The algorithm processes iterations from N to 1 and accepts asynchronous messages only when their timestamps exceed the last update from the sender.
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The paper establishes finite-horizon contraction and RMS error bounds for distributed Model-Based Diffusion under bounded latency, using convexification and stepwise error analysis. The theory applies under shared-cost, weak-coupling assumptions and is supported by simulations across increasingly complex multi-agent systems.
- Distributed system setup: DMBD is analyzed for a shared objective with a unique minimizer, smooth convex core, subgaussian tails, weak agent coupling, and bounded communication latency.Latency estimates may be no more than τmax steps old.
- Distributed system setup: Diffusion smooths the objective and expands a local convex region, enabling convergence analysis while agents maintain estimates of one another’s states.The local convex radius expands linearly in σ, and peer actions are held constant in the blockwise smoothed objective.
- Contraction analysis: Stepwise error contracts up to gradient-estimator error, while recursive RMS analysis quantifies how multiple error sources interact with contraction.The analysis proceeds from a smoothed best-response gap to stepwise contraction and then to RMS error dynamics.
- Contraction analysis: The diffusion process remains in the shrinking convex region under the stated initial-condition and step-size requirements, preserving the domain needed for the error analysis.Step sizes must be large enough to remain in the shrinking region, small enough for smoothness-based descent, and drawn from a non-empty range.
- Finite-horizon guarantee: Theorem 3.6 upper-bounds finite-horizon RMS error by a recursion with probability at least p_b, combining boundedness and one-step contraction.The proof uses induction for in-region iterates, a union bound across steps and agents, and iteration of the error recursion.
- Finite-horizon guarantee: The contraction conditions expose a tradeoff between the contraction coefficient and the terminal noise floor, with the resulting error bound following from the finite-horizon recursion.The corollary assumes a bounded worst-case suboptimality forcing and verifies the resulting contraction condition directly.
4 Experiment Design
The experiments evaluate DMBD across three increasingly difficult multi-agent environments under matched rollout computation and varied latency. The suite spans circleswap, multi-car racing, and simulated aerial combat.
- Experiment environments: The evaluation uses circleswap, multi-car racing, and simulated aerial combat as environments of increasing difficulty.The aerial combat setting involves up to six six-degree-of-freedom aircraft per team.
- Common design: Distributed and centralized cases use equal total rollout computation by assigning each distributed agent n_s = n_r/M samples while centralized optimization uses n_s = n_r.This controls computational fairness regardless of parallelization.
- Circleswap: Circleswap moves agents between diametrically opposite points without collision using a fixed-velocity unicycle model and receding-horizon replanning.The setup uses a 40-timestep lookahead and 100 trials per configuration.
- Multi-Car Racing: Multi-car racing maximizes the speed of medium-fidelity cars while avoiding collisions and maintaining a prescribed formation.The experiment uses a 5-second lookahead horizon and 25 rolled-out seconds.
- Simulated Aerial Combat: Aerial combat evaluates teams of F-16 aircraft that coordinate against another team while receiving rewards for damage and penalties for being damaged.Initial headings vary in 90° increments across 16 starting conditions, with 160 trials per optimizer parameter configuration.
5 Results
DMBD outperforms centralized MBD and baselines across circleswap, multi-car racing, and dogfighting, while performance degrades gradually under latency. The latency effect is smallest in weakly coupled circleswap and strongest in multi-car racing.
- Overall results: DMBD outperforms MBD and baselines in all three experiments, including under communication latency.Its performance generally remains above centralized MBD despite latency.
- Circleswap: DMBD has a significantly shallower makespan slope as the number of agents increases than centralized MBD.The overall scaling shape is approximately linear.
- Circleswap: Latency has negligible impact on circleswap makespan because of the problem’s relatively weak coupling.All latency configurations have comparable makespan.
- Multi-Car Racing: The best distributed configuration outperforms the centralized solver on multi-car racing normalized reward.Multi-car racing has the strongest coupling and therefore the strongest latency degradation among the three problems.
- Dogfighting: Approximately 25% higher win rate is achieved in the 6 v 6 dogfighting engagement, with win rate largely unaffected by latency.The dogfighting environment improves with decentralization.
6 Conclusion
The paper concludes that decentralized Model-Based Diffusion provides strong multi-agent path-planning performance and remains stable as latency or network degradation increases. Experiments show improvements in makespan, reward, and win rate, with latency degradation linked to coupling strength.
- Conclusion: DMBD provides strong performance for multi-agent path planning and outperforms baseline MBD in all evaluated cases.The conclusion presents decentralization as effective for sampling-based Model-Predictive Control.
- Conclusion: DMBD remains mathematically stable despite increasing latency or network degradation.The paper also provides probabilistic performance bounds.
- Conclusion: The experiments demonstrate improvements in makespan, reward, and win rate across diverse multi-agent tasks.Latency degradation is most strongly influenced by coupling strength, consistent with the mathematical analysis.