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Training-Free Diffusion Planning with Analytical Local Scores

Michael Y. Fatemi, Jinhao Liang, Ferdinando Fioretto

arXiv:2610.01959v1cs.ROcs.LG

TL;DR

Multi-agent motion planning requires collision-free trajectories under coupled constraints, while learned diffusion planners depend on feasible trajectory demonstrations. TFDP replaces learned global scores with analytical local scores and iterative denoising; it matches or exceeds DGD’s success across configurations while running faster on average.

  • Problem

    Learned diffusion planners require feasible trajectory demonstrations and may fail to enforce safety constraints.

    Source: §1 · p. 2, §3 · p. 4

  • Method

    TFDP replaces learned global trajectory scores with decomposed analytical local scores for smoothness and feasibility, then iteratively denoises trajectories.

    Source: §1 · p. 2, §4.1 · p. 4

  • Results

    Across 12 configurations, TFDP matched or exceeded DGD’s success rates and averaged 0.59 seconds versus DGD’s 88.23 seconds.

    Source: §5.1 · p. 8

  • Takeaways & Limitations

    Local temporal neighborhoods capture most useful score information for denoising, enabling efficient refinement without demonstrations or map-specific training.

    Source: §6 · p. 9

    Learned diffusion planners may still fail to enforce safety constraints, while projection-based refinement adds computational cost.

    Source: §1 · p. 2

Abstract

from arXiv · show

Path finding and multi-robot motion planning require trajectories that are smooth, goal-directed, and collision-free in environments with complex geometric constraints. Recent diffusion-based planners have shown that trajectory generation can be cast as iterative denoising which has opened the doors to learning-based approaches that can handle multi-modal trajectory distributions and refine entire trajectories. However, a key limitation is that diffusion planners require training on large collections of feasible trajectories, rendering them map-specific, and difficult to deploy when high-quality demonstrations are unavailable. This paper introduces a training-free diffusion-based motion planner that replaces learned global trajectory scores with analytical local scores derived from obstacle, smoothness, velocity, and inter-agent feasibility terms. The proposed idea relies on a key observation: the score of a trajectory can be reconstructed by considering only local interactions between neighboring waypoints and nearby constraints. This structure exploitation yields a decomposed denoising procedure that retains the optimization structure of classical trajectory methods while inheriting the iterative refinement behavior of diffusion models. Experiments on a large collection of complex environments and large multi-agent planning tasks show that the proposed analytical score produces smooth and feasible trajectories within limited computational costs, for example in generating feasible paths for 300+ agents in environments containing 100+ obstacles in under 6 seconds on a GPU, outperforming strong learning-based and optimization baselines, while avoiding the data requirements of learned diffusion planners.

1 INTRODUCTION

Multi-agent planning is difficult because trajectories must satisfy motion and safety constraints while avoiding time-dependent collisions. The paper replaces learned global trajectory scores with analytical local scores and reports efficient, high-feasibility planning at large scales.

1 INTRODUCTION

  • Multi-agent planning is difficult because each robot must satisfy motion and safety constraints while avoiding time-dependent collisions with other agents.The passage identifies spatio-temporal coupling and nonlinear constraints as challenges that grow with the number of agents and obstacles.Source: §1 · p. 1
  • Existing diffusion planners require learned neural scores and large collections of feasible demonstrations, making them expensive to train and often map- or distribution-specific.Learned diffusion planners may also fail to enforce safety constraints; projection-based refinement adds computational cost.Source: §1 · pp. 1–2
  • TFDP achieves high success and computational efficiency across multi-agent benchmarks, scaling to scenes with 300 agents and 100 obstacles in a few seconds.The introduction reports high success on standard test configurations and few-second planning at this scale.Source: §1 · p. 2
  • TFDP replaces learned global trajectory scores with decomposed analytical local scores for smoothness, obstacle avoidance, velocity feasibility, and inter-agent collision avoidance.Updates start from coarse trajectories with fixed endpoints and use a decreasing noise schedule.Source: §1 · p. 2, Figure 1 · p. 2

2 RELATED WORK

Prior planners face trade-offs between path quality, constraint handling, computational cost, and dependence on learned trajectory scores. This paper instead uses analytical local score components to drive diffusion-style refinement.

2 RELATED WORK

  • Sampling-based planners can yield fragmented paths needing post-processing, while optimization-based planners can be initialization-sensitive and difficult to scale across coupled agents.Sampling performance may degrade in narrow passages; optimization costs grow with multi-agent space-time collision constraints.Source: §2 · pp. 2–3
  • Diffusion planners refine whole trajectories and can represent multimodal plans, but typically require learned scores from feasible demonstrations and do not guarantee constraint satisfaction.Guidance and projection approaches seek to improve feasibility while retaining learned generative models.Source: §2 · p. 3
  • The proposed method replaces learned global scores with analytical local components derived from smoothness, velocity, obstacle-avoidance, and collision constraints.This preserves iterative refinement while making denoising updates fast and tied to environment geometry.Source: §2 · p. 3

3 PRELIMINARIES

The preliminaries define multi-agent trajectories and feasibility constraints, then describe learned diffusion as iterative denoising guided by a score over noisy trajectories. That global learned score motivates the paper’s alternative.

3 PRELIMINARIES

  • The planning setup requires agents to travel from designated starts to goals while avoiding obstacles and other agents in a bounded planar workspace.Agents are modeled as disks, with obstacle avoidance, inter-agent separation, and a first-order speed bound among the stated constraints.Source: §3 · p. 3
  • Diffusion planning perturbs feasible trajectories with Gaussian noise, then iteratively denoises them using a score that points toward higher-probability trajectories.Learned planners train a neural network to approximate this score and use it to refine noisy trajectories.Source: §3 · pp. 3–4
  • A learned score network depends on the entire noisy trajectory because feasibility involves global structure, obstacle geometry, and interactions among agents across time.Learning this global score is expensive and depends on access to feasible training trajectories.Source: §3 · p. 4

4 TRAINING-FREE DIFFUSION-BASED PLANNING (TFDP)

TFDP performs diffusion-style trajectory denoising without a learned score model, using an analytical surrogate built from local planning constraints. Its local score combines smoothness, obstacle avoidance, inter-agent separation, and dynamic feasibility.

4 TRAINING-FREE DIFFUSION-BASED PLANNING (TFDP)

  • TFDP replaces the unknown global trajectory score with an explicitly constructed surrogate target and performs iterative denoising without a learned score model.Source: §4 · p. 4

4.1 A SURROGATE TARGET DISTRIBUTION OVER TRAJECTORIES

  • The surrogate is a product of four factors encoding smoothness, obstacle avoidance, inter-agent separation, and kinematic feasibility.Source: §4.1 · p. 4
  • The surrogate retains which trajectories are feasible and smooth, but discards information about which feasible trajectories are likely under demonstration data.Its factors are determined by problem geometry and the smoothness weight, so they can be evaluated without a learned component.Source: §4.1 · p. 4
  • The local score combines smoothness, obstacle avoidance, agent-agent collision avoidance, and dynamic feasibility to move each waypoint according to planning constraints.The decomposition removes long-range score dependencies while retaining terms that dominate local feasibility.Source: §4.1 · p. 5
  • At low noise, direct neighboring interactions dominate the local score approximation, while dependencies between more distant variables are higher-order effects.The cited analysis motivates short score windows at low noise and larger windows at higher noise.Source: §4.1 · p. 5

4.2 ANALYTICAL SCORE TERMS

  • Closed-form constraint scores are provided for circular and rectangular feasible sets; general non-convex sets instead use a low-noise approximation.The approximation avoids computing the convolution integral directly.Source: §4.2 · p. 6

5 EXPERIMENTS

TFDP is evaluated across multi-agent benchmarks, ablations, and a large-scale map. It shows strong success, runtime, path-length, and scalability results, while several ablations identify where locality, initialization, and noise matter.

5 EXPERIMENTS

  • The evaluation covers four map types, 6, 12, and 18 robots, and 25 instances per map type, measuring success rate, runtime, and path length against diffusion and classical baselines.A separate scalability setting uses a map with over 100 heterogeneous obstacles and scenarios of up to 300 robots.Source: §5 · p. 7

5.1 COMPARISON ACROSS METHODS

  • On Dense with 18 robots, TFDP achieves 96% success versus DGD’s 76% and MMD’s 8%; across 12 configurations, TFDP runs in 0.18–1.24 seconds.TFDP averages 0.59 seconds, compared with DGD’s 88.23 seconds.Source: §5.1 · p. 8

5.2 ABLATION STUDIES AND SCALABILITY

  • On Dense with 18 robots, TFDP achieves 96% success versus 84% for the best displayed SDF-guidance setting.The ablation replaces TFDP’s analytical scores with CHOMP-style SDF penalty guidance while retaining its denoising schedule.Source: §5.2 · p. 8, Table 2 · p. 9
  • At σ = 0.05, extending the neural estimator’s temporal window from 1 to 3 reduces mean squared error from 0.8679 to 0.3849, while a window of 7 yields 0.2988.The estimator tests locality only; TFDP itself remains training-free.Source: §5.2 · p. 9
  • On Room and Shelf, straight-line and Brownian-bridge initializations yield only 0–17% success, versus 99–100% for Voronoi and RRT.The tested coarse initializers are described as easy and cheap to compute.Source: §5.2 · p. 9
  • Removing noise matches or exceeds noisy success on standard maps, while noise improves success point estimates by 5–7 percentage points in crowded 8- and 12-agent Bottleneck and Standoff cases.Noise can also improve a weak straight-line initialization, depending on sampling parameters.Source: §5.2 · p. 9
  • On a 100-obstacle map with up to 300 robots, TFDP produces valid plans in 5.39 seconds on average; tested diffusion baselines cannot handle this scale, and MPPI fails above 50 robots.In two single-robot-width bottleneck scenarios, TFDP has much higher success than MPPI across tested robot counts.Source: §5.2 · p. 9

6 CONCLUSION

The conclusion presents TFDP as a training-free analytical diffusion planner for path finding and multi-agent planning, and identifies centralized planning and perfect plan execution as assumptions. It also outlines future extensions and includes statements on AI use, ethics, and reproducibility.

6 CONCLUSION

  • TFDP replaces learned full-trajectory scores with analytical terms for smoothness, obstacles, velocity, and inter-agent feasibility, refining trajectories without demonstrations or map-specific training.The conclusion says local temporal neighborhoods capture most useful score information for denoising.Source: §6 · p. 9
  • The method assumes a centralized planner and perfect plan executions; future work includes decentralized planning, richer kinodynamic models, harder maze-like environments, and larger multi-agent benchmarks.The authors describe higher-dimensional extensions as a natural next step.Source: §6 · p. 9

AI USE STATEMENT

  • The authors report that generative AI assisted with Figure 1, its caption, manuscript edits, proofreading, and clarity suggestions, and that they verified its contributions.The authors state that they take full responsibility for the final content.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

  • The authors frame the work as advancing multi-agent path planning, with potential applications in warehouse automation, autonomous vehicles, and robotic swarms.They state commitments to transparency, reproducibility, and responsible technology use.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

  • The paper describes its formulation, score construction, algorithm, metrics, and sampling hyperparameters; the appendix adds derivations, benchmarks, sweeps, ablations, and confidence intervals.The source code is available upon request and is planned for public release upon acceptance.Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 CONDITIONAL INFORMATION FLOW

The appendix relates conditional dependence in a Gaussian Markov random field to its precision matrix, then describes how diffusion spreads codependence across the graph.

A.1 CONDITIONAL INFORMATION FLOW

  • For a bivariate normal distribution, the conditional mutual information can be read from the precision matrix of the overall distribution.The derivation first introduces mutual information between a pair of random variables and then specializes to the bivariate normal case.Source: §A.1 · p. 13
  • In the MRF, a zero off-diagonal precision entry indicates that two variables are independent conditioned on all the others.The passage represents nonzero precision entries as adjacency between random variables.Source: §A.1 · p. 13
  • After infinitesimal diffusion time, the precision matrix changes by −Ω²; interpreting Ω as adjacency makes this a two-hop spread of codependence.The passage gives Ω_t = (Σ + tI)⁻¹ and states that the net effect is spreading codependence across the MRF.Source: §A.1 · p. 13

A.2 ANALYTICAL SCORE FUNCTION DERIVATION

This appendix derives analytical score calculations for general energy landscapes and multi-agent constraints, including circular exclusion geometry.

A.2 ANALYTICAL SCORE FUNCTION DERIVATION

  • For a general energy landscape, the score after convolution with Gaussian noise is expressed as a quotient of two integrals.The appendix then gives methods to compute these integrals for multi-agent pathfinding constraint types.Source: §A.2 · p. 13
  • For circular exclusion, the setup places the agent at the origin and models an obstacle or other agent as a circle with effective radius equal to the summed radii.The feasible region consists of points outside that effective-radius circle.Source: §A.2 · p. 13
  • The derivation bounds relevant distances by r1 and r2 and uses symmetric angular limits around the obstacle’s x-axis.It states θ_min(r) = −θ_max(r), then identifies r sin θ_max and arctan2(y, x) as quantities used in subsequent expressions.Source: §A.2 · pp. 13–14

A.3 RECTANGULAR FEASIBLE REGION SCORE FUNCTION DERIVATION

This section derives the score for a uniform distribution over a rectangular feasible region after Gaussian smoothing. It interprets the smoothed density as Gaussian overlap with the region and uses the rectangle’s structure to evaluate the score.

A.3 RECTANGULAR FEASIBLE REGION SCORE FUNCTION DERIVATION

  • A uniform density over the rectangular region is smoothed by convolution with a Gaussian at noise level σ.The resulting density is proportional to the overlap between a Gaussian centered at q and the feasible region.Source: §A.3 · p. 14
  • The score is obtained by taking the gradient of the log of the smoothed density.The passages describe this score operation but do not provide a complete displayed expression.Source: §A.3 · p. 14
  • For a rectangle, the denominator uses the Gaussian CDF, while the numerator is evaluated using an observation and the fundamental theorem of calculus.Source: §A.3 · p. 14

A.4 RECTANGULAR INFEASIBLE REGION SCORE FUNCTION DERIVATION

For a rectangular infeasible region, the derivation reuses the feasible-region score calculations by complementing the region. It negates the previous numerator and obtains the denominator by subtracting the feasible-region denominator from one.

A.4 RECTANGULAR INFEASIBLE REGION SCORE FUNCTION DERIVATION

  • The infeasible region is defined as the complement of the rectangular feasible region, and its smoothed density is treated as Gaussian overlap with that region.The passage notes that this resulting distribution is not technically normalizable, but says the equation can still give a reasonable result.Source: §A.4 · p. 15
  • The numerator is negated from the feasible-region calculation, and the denominator is obtained by subtracting that calculation from one.Source: §A.4 · p. 15

A.5 CONNECTION TO PROJECTION-BASED METHODS

Across these appendices, TFDP’s outcomes depend on step-size and initialization choices, and performance remains challenged by highly coupled traffic. The supplementary analyses also examine uncertainty, locality, scaling, and feasibility over refinement.

A.6 LARGER HYPERPARAMETER SWEEP

  • At η = 0.50, TFDP and the tested SDF guidance settings reach at least 0.95 success in every reported setting.The σ and T sweep is non-monotonic: on Dense, at σ = 0.2, increasing T from 0.0001 to 0.8 reduces success from 0.64 to 0.04; on Basic, (σ, T) = (0.05, 0.8) reaches 0.96.Source: §A.6 · pp. 17, 19, §A.11 · p. 23

A.7 SINGLE-AGENT REDUCTION (RRT-CONNECT)

  • On Dense, RRT-Connect success falls from 0.83 with 4 robots to 0.07 with 10, while mean runtime rises from 2.97 to 21.98 seconds.On the shared CPU, MPPI runtime rises from 8.18 to 32.36 seconds on Basic and from 10.84 to 40.98 seconds on Dense as robot count grows from 6 to 18; all entries exceed corresponding TFDP times.Source: §A.7 · p. 19, Table 9 · p. 19, §A.9 · p. 22

A.8 COARSE PLANNER ABLATION

  • Straight-line and Brownian-bridge initializations yield only 0–17% success on Room and Shelf for TFDP, while Voronoi and RRT restore 99–100%.MPPI shows the same simple-initialization difficulty and also reaches 98–100% success on Room and Shelf with obstacle-aware routes; initialization affects feasibility, runtime, and path quality.Source: §A.8 · pp. 20–21

A.10 EFFECT OF NOISE

  • Noise improves the observed point estimates in crowded stress tests: with 12 robots, Bottleneck rises from 0.38 to 0.45 and Standoff from 0.39 to 0.45.The reported uncertainty ranges overlap, so these trends do not establish statistical significance.Source: §A.10 · p. 22, Table 15 · p. 22

A.12 DENSITY/STRESS TESTS

  • With 8 robots, TFDP succeeds at 0.69 in Bottleneck and 0.70 in Standoff, versus MPPI’s 0.20 and 0.23, respectively.At 12 robots, success falls to 0.31 and 0.32 for TFDP, showing that narrow shared passages remain difficult, including under opposing traffic.Source: Table 18 · p. 24, §A.12 · p. 24, Table 19 · p. 24

A.13 LOCALITY OF LEARNED SCORES

  • The locality diagnostic finds that expanding the temporal window reduces score-estimation error, with the preferred context depending on noise level.At σ = 0.05, the minimum error is 0.2988 at window 7; at σ = 0.2, error reaches 0.2037 at window 9. Separately, at very low σ, a Langevin step of size σ² is precisely the projection operator.Source: §A.5 · pp. 15–16, §A.13 · p. 25, Table 20 · p. 25

A.14 CONFIDENCE BOUNDS FOR 250-STEP TFDP RESULTS

  • With 250 denoising steps, Basic and Shelf have 100% observed success at every reported robot count, while Dense with 18 robots reaches 96% with a [92.7, 97.9]% interval.The large-map experiment averages 5.39 seconds for up to 300 robots and 100 obstacles on an A100 GPU. Figure 6 shows paths across varied geometry; Figure 7 reports that TFDP finishes with the highest feasible fraction on all four maps, though SDF guidance produces feasible plans earlier.Source: §A.14 · p. 25, Table 21 · p. 25, §A.15 · p. 26
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