Joint Model-based Model-free Diffusion for Planning with Constraints

Danfei Xu; Nadun Ranawaka Arachchige; Shreyas Kousik; Utkarsh A. Mishra; Wonsuhk Jung; Yongxin Chen

REVIEW 3 major objections 6 minor 56 references

Joint Model-based Model-free Diffusion for Planning with Constraints

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper introduces JM2D, a training-free method that jointly samples a robot plan from a diffusion model and a safety correction from an optimization module, improving task success and reducing safety-filter interventions compared with…

desk verdict JM2D is a promising and well-executed method for aligning diffusion planners with safety filters, but the proof of the score estimator assumes an exact posterior that the implementation replaces with a learned denoiser, and key hyperparameters go unreported. read the letter →

arxiv 2509.08775 v2 pith:VZUJOR7E submitted 2025-09-10 cs.RO

Wonsuhk Jung , Utkarsh A. Mishra , Nadun Ranawaka Arachchige , Yongxin Chen , Danfei Xu , Shreyas Kousik This is my paper · ORCID

classification cs.RO

keywords diffusionmodelsrobotmotionplanningsafetyfiltersconstrainedgenerationimportancesamplingjointmodel-basedoptimizationofflinereinforcementlearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Model-free diffusion planners generate diverse, multimodal trajectories, but they often ignore safety constraints that are best enforced by model-based optimization modules. This paper claims that the usual approach of first sampling a plan and then running a safety filter is wasteful and can fail, because the planner never accounts for what the filter can and cannot correct. The authors propose JM2D, which samples the plan and the safety correction jointly from a distribution weighted by an interaction potential that rewards mutual compatibility, using an importance-sampling score estimate that needs only evaluations of the potential, not gradients. On offline RL and real manipulation tasks, JM2D achieves higher safe success rates and lower intervention rates than sequential safety filters and other constrained-diffusion baselines while preserving hard safety guarantees.

What carries the argument

The load-bearing object is the Monte Carlo joint score estimator of Theorem 2, which rewrites the gradient of the joint noisy log-density as a weighted average of Tweedie scores, with weights proportional to $V(\hat{x}_0,\hat{k}_0) p^k(\hat{k}_0)$. The proposal $q(y_0|y_i)$ is the key design choice: for $x$ it is the exact posterior of the pre-trained diffusion model, approximated in practice by full denoising, and for $k$ it is a Gaussian centered on the noisy value. This estimator is what makes the method gradient-free and applicable to hard, non-differentiable constraints; its accuracy, the paper's ablations show, depends critically on the quality of the clean-sample estimates.

What would settle it

On a synthetic problem where the true joint distribution $p(x,k)$ is known and its score computable, compare the JM2D importance-sampling score estimate, with full denoising, against the true score; if the estimate does not converge to the true score as the number of MC samples grows, the unbiasedness claim behind Theorem 2 is falsified. Alternatively, if a one-step denoising estimate ever matches the performance of full denoising on the D3IL-Avoiding tasks, the paper's claim that clean-sample fidelity is the decisive factor would be contradicted.

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Extended reading notes

Core claim

The central claim is that model-free and model-based modules can be aligned without retraining by sampling from a joint distribution $p(x,k) \propto p_\theta(x)p(k)V(x,k)$, where $p_\theta(x)$ is a pre-trained diffusion planner, $p(k)$ is a prior on the optimization output, and $V$ is an interaction potential encoding compatibility, for example safety and task objective. JM2D implements this as a joint diffusion process over the concatenated variable $[x,k]$, estimating the joint score by self-normalized importance sampling with a proposal that factorizes into the model-free reverse diffusion for $x$ and a Gaussian for $k$. The resulting update uses only evaluations of $V$ on clean samples, so it handles non-differentiable and non-convex objectives. The paper shows empirically that this joint sampling outperforms sequential and Gibbs safety filters and projection- or gradient-based constrained planners, and proves that conditional generation and model-based diffusion are special cases.

Load-bearing premise

The method assumes that the internal guesses it makes about fully denoised plans and corrections are accurate; the paper's own ablation shows that when these guesses are cheap and rough, performance collapses to the weakest baselines.

Editorial extensions

If this is right

JM2D increases safe success rate and reduces safety-filter interventions compared to sequential filtering (RAIL) and Gibbs sampling, with the gap growing as inference-time obstacles diverge from training data.
The method needs no gradients, so it can steer pre-trained diffusion planners by any black-box objective, including non-convex optimizers, reachability sets, or rule-based checks.
Conditional generation is a special case: with a factorized interaction potential and uniform prior, JM2D decouples into classifier-style guidance for $x$ and model-based diffusion for $k$, unifying those two prior lines of work.
Because joint samples are already aligned with the safety module, running a single post-hoc safety-filter step on infeasible samples still gives strict safety while avoiding the overrides that hurt sequential methods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The estimator effectively performs annealed importance sampling over the interaction potential, so the paper's ablation suggests that better proposals, such as few-step samplers or learned posteriors, could cut compute without losing alignment.
The method's benefit is tied to the planner's multimodality: the authors note performance fades when the learned policy is less multimodal, which implies a testable prediction that the gain over sequential filtering scales with the effective number of modes in the plan distribution.
The paper never specifies the prior $p(k)$ used for the optimization output; a natural extension is to test sensitivity to that prior, since the estimator's weights are proportional to $V p^k$.
The joint-sampling formulation extends by composition of interaction potentials to more than two modules, for example planner, tracking controller, and safety filter, which the authors name as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Jung et al. propose JM2D, a way to sample jointly from a pretrained diffusion planner and a model-based safety module by importance-weighting an interaction potential. The idea is clean: define p(x,k) ∝ p_θ(x)p(k)V(x,k), then estimate the joint score with Monte Carlo samples of clean (x,k) pairs; the method needs only evaluations of V, not gradients. Conditional generation and model-based diffusion fall out as special cases. On the empirical side, the paper shows real gains over RAIL and Gibbs on PointMaze with a reachability safety filter, and beats DPCC, SafeDiffuser, and MPD on constrained generation. The real-robot mug pick is a nice bonus. The ablation on clean-sample estimation is honest and informative: guidance collapses when you shortcut the denoising, which pins the method's success on the quality of the clean samples.

The soft spots are real but addressable. The main one is the gap between Theorem 2 and Algorithm 2. The proof uses the exact posterior p^x_{0|i}(x0|xi) as the proposal and cancels it in the importance weight. The implementation samples x0 by full reverse denoising with the learned model p_θ. For an imperfect diffusion model those are different distributions, and the unaccounted ratio p^x_{0|i}/p_θ survives in the weight; no error bound or diagnostic is given. The u=0,1 rows in Table 3 show that when the clean samples are poor, the method degrades to the worst baselines, so the mismatch is not a technicality. That said, the empirical evidence with full denoising is strong, so the fix is to restate the theorem for the actual proposal (or at least discuss the approximation and validate it, e.g., by comparing against exact posterior samples in a toy example).

Second, the paper leaves λ and p(k) unspecified in the experiments. The model-based prior appears in the weight, so it matters. Also no runtime numbers are reported, and the method is expensive (I(I+1)/2 denoising chains). The limitations section acknowledges the cost but not the missing hyperparameters.

The derivation is not circular; the special-case reductions to conditional generation and model-based diffusion are correct, and the related work is honestly positioned. I'd describe it as a well-executed method paper with a theory section that overreaches. It is not a desk-reject: the problem is important, the experimental comparison is fair, and the flaws are fixable. A serious referee should engage with it, and I'd expect a revise-and-resubmit.

Referee Report

3 major / 6 minor

Summary. The paper proposes Joint Model-based Model-free Diffusion (JM2D), a training-free framework that jointly samples a diffusion-planned trajectory x and a model-based optimization output k from a joint distribution p(x,k) proportional to p_theta(x) p(k) V(x,k), where V is an interaction potential encoding compatibility, non-differentiable constraints, and safety objectives. The method estimates the joint score via self-normalized importance sampling with a tractable proposal, so that only evaluations of V on clean samples are needed. The authors prove a joint-score identity under this proposal, show that conditional generation and model-based diffusion arise as special cases, and validate the method in a 2D PointMaze safety-filtering task, a real Franka mug-pickup task, and D3IL-Avoiding constrained-generation tasks, reporting improved safe success and lower intervention rates relative to RAIL, Gibbs sampling, SafeDiffuser, DPCC, and MPD. The paper also includes an ablation showing that the quality of clean-sample estimation is critical to the method's performance.

Significance. If the central claim holds, JM2D is a useful contribution: it provides a principled way to integrate pre-trained diffusion planners with model-based safety modules while handling non-differentiable potentials, and it avoids additional training or privileged gradient information. The paper's strengths include an explicit derivation of the joint score and importance weights, a broad set of experiments including a real-robot demonstration, and an honest ablation that reveals sensitivity to the clean-sample estimator. However, the theoretical result is proven for an exact-posterior proposal that the implemented algorithm does not use, and key experimental components such as the model-based prior and interaction temperature are left unspecified. These issues affect the validity and reproducibility of the central claims, so the paper needs substantial revision before it can be accepted.

major comments (3)

[Sec. 4.2 (Theorem 2), Appendix A.2, Algorithm 2] Theorem 2 proves the joint-score identity for the proposal q(y0|yi) = p^x_{0|i}(x0|xi) q^k(k0|ki), where p^x_{0|i} is the exact posterior of the forward noising kernel, and the proof in Appendix A.2 relies on cancellation of p^x_{0|i} in the importance-weight ratio. Algorithm 2, however, constructs x-samples by full reverse denoising with the learned model p_theta(x_{tau-1}|x_tau). For a learned diffusion model, p_theta(x0|xi) does not equal p^x_{0|i}(x0|xi), so the importance weight actually used is proportional to V(x0,k0) p^k0(k0) times p^x_{0|i}(x0|xi)/p_theta(x0|xi), and this extra ratio does not cancel in Eq. (7). The implemented estimator is therefore not the estimator proven in Theorem 2, and no error bound, bias analysis, or empirical validation of this approximation is provided. Because Table 3 shows that the method collapses when clean-sample estimates are poor (u=0,1), this gap is load-bearing for the central claim that JM2D samples from p(x,k) proportional to p_theta(x)p(k)V(x,k). I ask the authors to extend the analysis to the learned-proposal case, or to state clearly that Eq. (7) is an approximation and provide evidence that the resulting bias is acceptable in the settings considered.
[Sec. 5.1, Sec. 6.2, Algorithm 1, Table B.1] Algorithm 1 and Eq. (7) require a model-based prior p^k0(k0) and an interaction temperature lambda in V(x,k) = exp(-J(k|x)/lambda) * 1(g(k|x) <= 0), but the main experiments never state which prior over backup actions was used or which temperature values were chosen for the PointMaze, real-robot, or Avoiding experiments. Table B.1 reports only the diffusion schedules and sample counts. Without these specifications the method is not reproducible, and the reported gains cannot be separated from tuning of lambda or the prior. The paper should report these values and, ideally, a sensitivity analysis over lambda for at least one of the benchmark tasks.
[Sec. 1, Remark after Theorem 2, Sec. 7] The Introduction states that JM2D yields trajectories that 'satisfy arbitrary, possibly non-differentiable, safety constraints by construction,' but the Remark after Theorem 2 says that infeasible joint samples are postprocessed with a model-based optimization step to guarantee feasibility, and Sec. 7 states that the approach 'cannot strictly enforce constraints' and requires a separate backup module. These statements are contradictory and affect the interpretation of the safety claims in Sec. 5.1, where the phrase 'none of these methods violate safety' is true only because of the external filter. The authors should remove the 'by construction' claim or clearly attribute hard guarantees to the postprocessing module.

minor comments (6)

[Appendix A.2, A.3] The headings 'Proof of Definition 2' and 'Proof of Definition 3' refer to Theorem 2 and Corollary 3; please correct the numbering and terminology.
[Sec. 5.1, Fig. 4] The caption of Fig. 4 says '80 trials of 5 random seeds' while the setup in Sec. 5.1 states 100 simulations per seed; harmonize these numbers.
[Sec. 4.2, Eq. (6)] After Eq. (6), the text says 'q(·|y_i) denotes (5)', but the proposal is defined in Eq. (6); the cross-reference should be corrected.
[Algorithm 1, line 5] The denoising update references 'Eq. (6) in [6]' without reproducing the equation; either make the update self-contained or include the full expression in the appendix.
[Table 1] The table uses inconsistent column labels 'SHor.' and 'THor.' for the same task-horizon metric; use one label consistently.
[Sec. 5.2] The real-robot experiment is reported only qualitatively; please include numerical success/safety statistics or explicitly state that the demonstration is qualitative.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the joint-score estimator is derived from the stated target distribution, and the CG/MBD special case is an explicit corollary, not a fitted prediction.

full rationale

The paper's central object is the defined joint target p(x,k) ∝ p_theta(x) p(k) V(x,k) (Problem 1). Equation (4) and Theorem 2 (Eq. 7) are algebraic re-expressions of the score of this target under the factored proposal (6); Appendix A.2 shows the importance weight simplifies to V p^k0 by Bayes' rule. This is a derivation from the definition, not a prediction manufactured from fitted outputs. Corollary 3 explicitly states that when V(x,k)=V_x(x)V_k(k) and p(k) is uniform, the joint score separates into classifier-style guidance for x and model-based diffusion for k (Appendix A.3). This is a sanity-check reduction of the framework to known special cases, not a presentation of those known cases as new predictions. The self-citations (RAIL [10], RTD [37,10]) appear as a baseline and a safety-filter component, not as load-bearing justification of the novel derivation. The experiments are self-contained comparisons against external baselines (DPCC, SafeDiffuser, MPD, projection). Two caveats belong to correctness rather than circularity: Algorithm 2 replaces the theorem's exact-posterior proposal p^x_{0|i} with the learned reverse process p_theta without an error bound, and the model-based prior p(k) is never specified in the experimental sections. These are approximation and under-specification issues; they do not reduce a fitted parameter to a prediction or make the derivation equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical entities. It relies on an interaction potential and a joint-sampling formulation as modeling constructs, plus several free parameters (λ, N, NK, u) that are not pinned down in the paper.

free parameters (3)

interaction temperature λ = not reported
The interaction potential V(x,k)=exp(-J(k|x)/λ)·1(g≤0) has a temperature λ that controls how strongly optimality is weighted. The paper defines λ in Sec. 3 but never states its value or sensitivity in any experiment, so it may act as an unstated tuning knob.
Monte Carlo sample counts N and NK = N=128, NK=128 in real robot; varied 4..256 in maze ablation (Fig.5)
The joint score estimator averages over N denoised plans and NK sampled optimization parameters. These are computational parameters that trade cost against performance; Fig. 5 shows intervention rate drops as they grow.
denoising depth u for clean-sample estimation = varies from 0 to full denoising (I)
The ablation in Sec. 6.2 shows that full denoising (u=I) is needed for good performance; u=0 or u=1 collapses performance. This parameter is the core of the method's computational cost.

assumptions (4)

domain assumption The joint target distribution is p(x,k) ∝ p_θ(x) p(k) V(x,k)
This definition (Sec. 3, Problem 1) is the foundation of the joint score. It assumes the modules' outputs can be combined multiplicatively and that the interaction potential is a valid unnormalized density.
ad hoc to paper The model-based prior p(k) is known and evaluable at inference time
The importance weights in Theorem 2 require evaluating p(k0). The paper never specifies p(k) for the safety-filter experiments, and treats k as a random variable even though the optimization module is deterministic (Eq. 1).
standard math The learned reverse diffusion p_θ approximates the true posterior p^x_{0|i}
The proposal q uses p^x_{0|i}(x0|xi), which the algorithm implements with the trained denoiser. This is a standard diffusion assumption but is not verified in the paper.
domain assumption The interaction potential can be evaluated on clean samples at arbitrary points
Algorithm 1 and Algo 2 evaluate V(ŷ0) on fully denoised MC samples. In robotics this requires a safety-evaluation oracle (e.g., reachability computation) at each denoising step.

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Pith. "Pith review of Joint Model-based Model-free Diffusion for Planning with Constraints." pith.science (2026). https://pith.science/paper/VZUJOR7E

@misc{pith2026250908775,
  author       = {Pith},
  title        = {Pith review of: Joint Model-based Model-free Diffusion for Planning with Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VZUJOR7E}},
  note         = {Machine review of arXiv:2509.08775}
}

read the original abstract

Model-free diffusion planners have shown great promise for robot motion planning, but practical robotic systems often require combining them with model-based optimization modules to enforce constraints, such as safety. Naively integrating these modules presents compatibility challenges when diffusion's multi-modal outputs behave adversarially to optimization-based modules. To address this, we introduce Joint Model-based Model-free Diffusion (JM2D), a novel generative modeling framework. JM2D formulates module integration as a joint sampling problem to maximize compatibility via an interaction potential, without additional training. Using importance sampling, JM2D guides modules outputs based only on evaluations of the interaction potential, thus handling non-differentiable objectives commonly arising from non-convex optimization modules. We evaluate JM2D via application to aligning diffusion planners with safety modules on offline RL and robot manipulation. JM2D significantly improves task performance compared to conventional safety filters without sacrificing safety. Further, we show that conditional generation is a special case of JM2D and elucidate key design choices by comparing with SOTA gradient-based and projection-based diffusion planners. More details at: https://jm2d-corl25.github.io/.

Figures

Figures reproduced from arXiv: 2509.08775 by the authors.

**Figure 1.** Framework Overview. We consider robotic systems composed of a model-free diffusion planner and a model-based optimization module. In this example, the diffusion policy is trained on right-skewed goal-reaching behaviors, while the model-based module can only correct plans to the right. The key challenge is to jointly sample a plan and its correction such that both modules are aware of each other’s capabilities. We pr… view at source ↗

**Figure 2.** Comparison of sampling methods on a toy domain. A model-free planner pθ (x) samples start-goal pairs in a donut-shaped region (gray), and a model-based optimization module finds the longest collision-free path connecting start and goal parameterized by k while avoiding inference-time obstacles (transparent red). We visualize 16 samples per sampling method. Sequential sampling draws x without considering k, often yi… view at source ↗

**Figure 3.** Task description of D4RL-PointMaze [43]. (a) The robot (red) must reach the goal (green) without colliding with the wall. (b) At inference time, the wall boundaries are inflated by a width w to impose additional safety constraints. The case w = 0 corresponds to the original wall thickness used during training. We use a pre-trained Diffusion Policy [3] as our model-free module to generate sequences of actions x = [at… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Joint diffusion safety filtering increases performance without sacrificing safety. A mobile robot (green dot) must navigate to a target (red dot) while avoiding collisions with walls. At test time, additional obstacles are introduced by padding walls with increasing th…

**Figure 5.** Figure 5: As the number of candidate samples (N and NK) increases, intervention of JM2D samples by the external safety filter decreases, implying higher CA. Result: Ablation on Monte Carlo Sample Size. As shown in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Real robot experiment setup We rollout the pre-trained vanilla DP in the cluttered scenario. Since there are obstacles in the scenario previously unseen by the DP, it collides with them and fails. This is expected as naive DP is uninformed action sampling. We show this…

**Figure 7.** Figure 7: (Top) Na¨ıve DP fails to execute the task of mug-pickup when it faces obstacles unseen in the training data. (Bottom) JM2D-based composition of DP and reachability-based safety filter successfully completes the task without compromising safety. Additionally, investigat…

**Figure 8.** Figure 8: Visualization of planned trajectories and safety filter collision volumes; static obstacles [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: When does projection fail? For constraint-guided diffusion in the Donut task. The gray dots show the training distribution and the red region is a test-time constraint. (a) Our method generates 60/64 valid samples within distribution. (b) Projection yields valid sampl…

**Figure 10.** Figure 10: Task description of D3IL-Avoiding [45]. (a) The robot must reach the green line without colliding with red obstacles. (b) At inference time, additional blue obstacles are introduced as safety constraints. The first three scenarios (Top Left, Top Right, Both Hard) are…

**Figure 11.** Figure 11: Planning in Cluttered Environments. In Avoiding-Cluttered, SafeDiffuser fails due to local traps; DPCC struggles to find a feasible path due to conservative obstacle padding from model uncertainty. Our method generates smooth, valid plans without privileged dynamics m…

**Figure 12.** Figure 12: Impact of clean sample quality on guidance effectiveness. (a) On the donut task, increasing denoising steps improves Distribution Fidelity (DF), leading to higher OA and lower CA. (b, c) On the Avoiding task, where DF is not directly measurable, increased denoising si…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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