Contractive Diffusion Policies
Authors: Amin Abyaneh
Charlotte Morissette
Mohamad H. Danesh
Anas Houssaini
David Meger
Gregory Dudek
Hsiu-Chin Lin
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion Policy
Contraction Theory
Robot learning
Primary area: applications to robotics, autonomy, planning
TL;DR: We improve the performance of diffusion policies for offline learning by promoting contraction in the diffusion sampling process.

Abstract: Diffusion policies have emerged as powerful generative models for offline policy learning, whose sampling process can be rigorously characterized by a score function guiding a Stochastic Differential Equation (SDE). However, the same score-based SDE modeling that grants diffusion policies the flexibility to learn diverse behavior also incurs solver and score-matching errors, large data requirements, and inconsistencies in action generation. While less critical in image generation, these inaccuracies compound and lead to failure in continuous control settings. We introduce **C**ontractive **D**iffusion **P**olicies (CDPs) to induce contractive behavior in the diffusion sampling dynamics. Contraction pulls nearby flows closer to enhance robustness against solver and score-matching errors while reducing unwanted action variance. We develop an in-depth theoretical analysis along with a practical implementation recipe to incorporate CDPs into existing diffusion policy architectures with minimal modification and computational cost. We evaluate CDPs for offline learning by conducting extensive experiments in simulation and real world settings. Across benchmarks, CDPs often outperform baseline policies, with pronounced benefits under data scarcity. Project page: https://contractive-diffusion.github.io

Published as a conference paper at ICLR 2026
CONTRACTIVE DIFFUSION POLICIES
Amin Abyaneh!.s 1.3, Charlotte Morissette Mohamad H. Danesh Anas El Houssaini 2.3
David Meger Gregory Dudek2.3 Hsiu-Chin Lin 2 DDpaarment of Electrical and Computer Engineering, McGill University
2 DDepartment Computer Science, McGill University Al Institute
Correspondence: anin.abyanchh@miil.mme...
ABSTRACT
Diffusion policies have emerged powerful generative models for offline pol- icy learning, whose sampling process can be rigorously characterized by score function guiding stochastic differential equation (SDE). However, the same score- based SDE modeling that grants diffusion policies the flexibility to learn diverse behavior also incurs solver and score score-matchine matching errors, large data requirements, and inconsistencies in action generation. While less critical in image generation, these inaccuracies compound and lead to failure continuous control settings. We intro- duce Contractive Diffusion Policies (CDPs) to induce contractive behavior in the diffusion sampling dynamics. Contraction pulls nearby flows closer to enhance ro- bustness against solver and ccore-matching errors while reducing unwanted action variance. We develop an in-depth theoretical analysis along with practical imple- mentation recipe to incorporate CDPs into existing diffusion policy architectures with minimal modification and computational cost. We evaluate CDPs for offline learning by conducting extensive experiments in simulation and real world settings. Across benchmarks, CDPs often outperform baseline policies, with pronounced benefits under data scarcity. Project page: contractive-diffusit github.io
INTRODUCTION
Diffusion policies have substantially advanced offline policy learning, particularly in robotics and control, where they are trained with reinforcement learning (RL) or imitation learning (IL) (Chi et al., 2023; Ke et al., 2024; Huang et al., 2024; Wang al., 2024; Sridhar et al., 2024). These improvements stem from stable training and strong representational capabilities of conditional diffusion models in capturing long-horizon and multimodal behavior "Sohl-Dickstie
et al., 2015; Ho al., 2020). Essentially, diffusion models corrupt actions with noise and learn a score function that, conditioned on the state, iteratively denoises the corrupted actions to reconstruct the underlying conditional action distribution. This coruutiion-eeoising process can be formally described by stochastic differential equation (SDE), where actions are obtained through differential equation solvers (Song et al., 2021b).
Despite recent advances, diffusion policies still demand sub- stantial samples to learn accurate score functions and are vul- nerable to solver enrors. Iterative diffusion sampling causes inaccuracies in (i) score estimation, (ii) discretization, and accumulate across denoising
Limited
Standard data
Kiteleg
Transport
60
(iii) numerical integration steps (Zheng et al., 2023; Tang & Zhao, 2024). In practice, the sampling process also suffers from inconsistency in action generation given the same state (Gao et al., 2025). Unlike image generation, inaccuracies of this sort are detrimental robotics and control, even small deviations from the data distribution compound and eventually push the policy away from the dataset support (Kumar al., 2020; Fujimoto & Gu, 2021). The negative impact on performance and safety is more pronounced in physical robotic systems (Ada et al., 2024; Zheng et al., 2024; Mao et al., 2024), which necessitates curated low-level safeguards (Chi et al., 2023).
S2
Average reward
43
24
20
0
EDP
CDP-RL
CDP-IL
Figure 1: CDP's performance com- pared to diffusion policy baselines in offline RL and IL tasks.

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Published as a conference paper at ICLR 2026
Noise distribution
Reverse diffusion ODE
Action variance
Action distribution
dla Fola, ()d
44
P1
ap
Solver und score matching errons
a
Probubility
low
high
Sampling flows
Standard
Figure 2: Contraction in diffusion sampling. We observe that contraction pulls nearby diffusion flows closer. Contraction plays a critical role in diffusion sampling for offline learning: it dampens
solver and score-matching errors while reducing unwanted variance the generated actions.
While the score- boore-baeed and iterative sampling crucial for capturing diverse action distributions, introduces inaccuracies that must be addressed. Too this end, we introduce Contractive Diffusion Policies (CDPs) to learn conditional action distributions and encourage contraction in the diffusion sampling process. Contraction theory studies whether the solutions of differential equation converge over time, enabling the system to quickly forget small perturbations in its initial conditions and naturally suppress error growth (Tsukamoto et al., 2021; Dawson et al., 2023). Theoretically, contraction is proven to reduce solver and score- score-matchine matching imprecisions in diffusion modeling under step-size bounds (Tang & Zhao, 2024). As shown in Figure 2, CDPs leverage contraction tackle solver and score-matching errors in offline policy learning while reducing unwanted action variance by pulling the nearby solver flows closer to the dominant modes of the action distribution.
Built on our rigorous theoretical analysis, we derive a practical recipe by adding only one tuned hyperparameter and computationally efficient contraction loss to promote contraction in the sampling process of diffusion policies. As a result, CDPs are straightforward to implement and seamlessly integrate with existing architectures. We empirically validate CDPs through detailed experiments on D4RL (Fu et al., 2020) and Robomimic (Mandlekar et al., 2021) benchmarks, as well as on physical robotic manipulation tasks. As highlighted in Figure 1, CDPs often outperform non-contractive diffusion policies, particularly in low-data regimes. Such results are consistent with the hypothesis that contraction mitigates the impact of acore-matchine and solver errors, ultimately enhancing the performance of standard diffusion policies.
Summary of related work. Recent studies have extensively applied diffusion policies to offline learning. In offline RL, DQL (Wang et al., 2023), EDP (Kang et al., 2023), and IDQL (Hansen- Estruch et al., 2023) combine behavior regularization through diffusion loss with value maximization, while hierarchical approaches (Ma et al., 2024) and divergence minimization methods (Gao et al., 2025) have been proposed to enhance scalability and stability. In II, DP (Chi et al., 2023) and DBC (Pearce et al., 2023) model conditional action distributions, and 3D-aware variants integrate visual, language, and proprioceptive modalities (Ze et al., 2024; Ke et al., 2024). The emphasis in these works has largely been on training pipelines and multimodal integration rather than the diffusion process dynamics. Contraction-basee perspectives remain relatively underexplored, with methods like contractive autoencoders (Rifai al., 201 demonstrating reduced sensitivity to input perturbations in representation learning. More recent contractive diffusion probabilistic models (Tang & Zhao, 2024) show that enforcing contraction can mitigate score-matching and discretization errors. While insightful, the method enforces contraction globally, potentially reducing diversity and hindering efficient integration in offline learning. more in-depth literature review is presented in Appendix A.
2 BACKGROUND: OFFLINE LEARNING, DIFFUSION, AND CONTRACTION
Consider an environment modeled as a Markov Decision Process (MDP). The environment dynamics are governed by an unknown transition distribution P(s' s, a), where taking an action A C R while in state E S C Rd results in transition to the next state s' E S. The transition yields a reward signal, given by R(s, a). A policy ग S x AA + [0, 1] assigns a probability distribution over actions given state. When parameterized by 0, ae s) represents the probability of selecting action a given state s under the parameterized policy. We assume the existence of a dataset, D [(s, s', r)), collected from the interactions of behavior policy ma with observed rewards
2

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IL methods often consider the behavior policy тy expert and aim to learn ne to imitate the expert's 's behavior (Hussein et al., 2017). On the other hand, offline RL makes no such assumption, Tb may be unknown suboptimal. The goal is to learn To from the offline dataset D that maximizes the expected discounted return for the environment (Levine et al., 2020).
2.1 DIFFUSION MODELING USING DIFFERENTIAL EQUATIONS
Diffusion policies have been employed learn the true underlying policy distribution ne p(a s) from offline data D (Chi et al., 2023; Dong et al., 2025). The forward diffusion process is first designed to gradually corrupt the dataset action step & [0, 1]. The process is modeled by an SDE:
ag into noisy version as, where the diffusion
(1) where f IRd++1 + R is the drift function, g I - R is the diffusion coefficient which scales the noise, W, the increment of a Wiener process (Song al.. 2021b), and p(a) f p(a s)p(s) ds is the true action marginal distribution. The forward SDE seen in Equation has an equivalent reverse sampling process that maps the noisy input a back to the original sample ag = a, described by:
da: f(a., t) + g())d,,
p(a),
da,- = [ t) g(t)2 Pe {s] dt + g(t dw,,
(2)
where dw, is Wiener process in reverse time. Note that the score function is the guiding hand which steers the denoising flows towards high-densit regions of the true distribution based on the gradient of the log- likelhool of actions given state each step t. We focus on an equivalent ordinary differential equation (ODE) diffusion sampling for efficient sampling.
Theorem 2.1. (Song et al., 2021b) There exists an equivalent probability flow ODE with the same marginal distribution, but better sample efficiency as the SDE in Equation 2, given by:
1 f(ar. g a. ao Pr(a
da.
dt.
Note that the only unknown component here is the score function, as all other terms in the reverse process are determined by the forward diffusion parameters (Dong et al., 2024a). Equation 3 generates reverse flows a₁ ... a, policy distribution. The objective is then to learn a scaled score function, by optimizing
an = which, given accurate score, approximates the true
(4) where or is the known standard deviation of the noise at time t in the forward process. In practice, we only have samples of Pr(a; s), and expect to train a highly accurate scaled score, co s) x -o, log (a, s) to approximate the true action distribution.
La == E (., ||ea s, o Va, log Pr s)]l
Limitations of score-based learning. In practice, limitations in sample count and diversity directly impact the quality of the learned score function. Additionally, the sampling process described in Equation 3 typically relies on SDE/ODE solvers (Zheng et al., 2023) or discrete samplers (Song al., 2021a) for generating the iterative diffusion flows. These solvers are prone discretization and integration errors due to finite step sizes, with a potential accumulation of approximation errors over denoising iterations.
Offline learning with diffusion policies. Offline learning methods adopt diffusion models to learn Ty in which the actions are generated through a state-conditioned reverse diffusion process. Offline RL methods ( (Wang et al., 2023; Kang et al., 2023) enhance the score- moore-mathhing loss in Equation 4 by incorporating value maximization, encouraging actions that balance proximity ne and high expected returns. Diffusion policies (Chi et al., 2023; Ke et al., 2024), meanwhile, directly model Ta through maximum likelihood training on state-action pairs. Since both paradigms build on diffusion models, they are reliant on accurate score functions and efficient ODE solvers for best performance.
2.2 CONTRACTION THEORY IN DIFFERENTIAL EQUATIONS
Consider an ODE in continuous time da, = F(as, t)dt. Contraction theory characterizes the convergence behavior of ODE flows a as increases (Lohmiller & Slotine, 1998; Tsukamoto et al., 2021). By regulating the rate of contraction, trajectories of a contractive ODE become robust to perturbations and compounding errors. Contraction is formally defined as follows.
3

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Published as a conference paper at ICLR 2026
Epoch: 10
Epoch: 30
Epoch: 100
Epoch: 150
Epoch: 200
Standard
Solver and matching
agy
Contractive denoising
Actions closer mean
CDP
Q2
Action a
Sampled
Ground truth
Figure 3: 2D toy experiments. As we train CDP and a standard policy, both methods produce increasingly accurate actions. However, the actions generated by CDP tend concentrate near the mean of distinct action modes, and show signs of mitigating solver and score-matching drifts.
Definition 2.1. (Lohmiller & Slotine, 1998) An ODE, F(ar, t). is contracting with contraction rate 7) E Rt, for any two initial conditions, a aB R's, and some constant c E R4, the {ax too) flows generated by F for each initial condition satisfy
a at S ce " a agl
Vt O,
where ||
denotes an Ly norm.
Contraction can be analyzed by examining the Jacobian JF distance between nearby ODE flows evolves (Tsukamoto et al., 2021). As shown in Appendix D.3, sufficient condition for contraction is
8 F F(ae, (), which captures how the
Amax (Jr Jb) < 2 0.
(5)
where Amax returns the largest eigenvalue of the symmetric Jrr for all and t.
3 CONTRACTIVE DIFFUSION POLICIES
As discussed in Section 2.1, diffusion policies can generate state-conditioned actions by progressively steering noise toward actions through solving the reverse diffusion ODE. Our key insight to promote contraction while sampling from the reverse diffusion ODE to mitigate score-matching and solver errors. as highlighted by the toy example Figure We pursue this goal in two steps: (i) providing theoretical analysis of contraction in the diffusion ODE, and (ii) by introducing practical training techniques designed actively promote contraction during learning.
3.1 CONTRACTION IN REVERSE DIFFUSION ODE
According to Section 2.2, we need to compute the Jacobian of the reverse diffusion ODE with respect to actions. Following the literature (Zhang & Chen, 2022), we reformulate the reverse diffusion ODE in Equation 3 to have linear drift term f (t)a, and view the approximate reverse process
da. [f(t)a, + ((,))) s, t)] dt Fe (a, t)dt,
(6)
where h(() - (20)) accounts for both the diffusion and noise schedule terms. Note that
consider Equation with a fixed state without loss of generality, as s is only set once in the iterative reverse process to generate each action. In the step, calculate the Jacobian with respect to a, which reveals the relationship between the score Jacobian and that of the reverse diffusion process:
② Fo(ai, f()) h(t) Da.
dco St,()
Jr. da.
f(t) h(t)
4

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Published as a conference paper at ICLR 2026
Sec 2.1
Denoie:
Jacobian computation
Diffusion
Dey (a,, Si. () Ehute
Obe
Act
Predict
Update
Policy deployment
Sec.
Policy learning
Sec.
Offine dataset
Loss optimization
Contraction
Batch
Offline
Be
Reward
Diffasion
Ee ( 8, f)==
Figure 4: Methodology overview. The policy is trained on offline data to minimize contraction and diffusion losses. For each batch of data, the score Jacobian Jea is efficiently computed for all denoising steps, and is then penalized with the contraction loss. At deployment, the diffusion policy frozen, and the ODE sampling process generates the actions given observations.
The decomposition in Equation 7 is insightful. It shows that for any state, the contraction expansion behavior of the diffusion ODE is governed by [()) and ltt). These are determined by scheduling parameters of the forward process, and Jey. which modulates the reverse diffusion flows based the learned structure in the data. Notably, only Jep is trainable among these terms, signaling that proper constraints on Jep during training can alter the contractive behavior of the sampling process.
Noting the sufficient condition for contraction in Equation 5, we can examine the symmetric part (denoted by sym) of the matrix Jrp through J 1 (JF, + J.F.) f(t) h(t) The next theorem solidifies the results by highlighting the connection between the eigenvalues of Jcp to the contraction of the sampling ODE.
Theorem 3.1 (Interplay of score Jacobian and contractive sampling). Given a state S and diffusion ODE, da, Fo so, s)dI, Fo is contractive, iee, Jom < 0, iff the score Jacobian satisfies
(mmr < -(t))((t)-, Vt V [0,
where Amax denotes the largest eigenvalue, and J, h are defined by the forward process.
The proof is outlined in Appendix D.2. Theorem 3.1 shows that the drift term s(()/ is often contractive and pulls nearby flows closer together at each solver iteration. In contrast, the score term can locally exhibit either expansive or contractive behavior. Theorem 3.1 ensures that any local expansiveness of the score term is dominated by the contractive effect of the drift.
Corollary 3.1.1 (Bounded action variance under contraction). For any two flows at and at initialized at a and a the difference in actions generated by Fo. denoted by da, = a a is bounded by
|aall
< exp (F.
||ãao|].
where JET is the sampling ODE's Jacobian at solver step
Appendix D.3 contains the proof. Corollary 3.1.1 describes natural by by-product of Theorem 3.1 as an upper bound for sensitivity to initial seed. Bounded sensitivity results more robust action generation given similar states.
3.2 LEARNING RECIPE FOR CDPS
In Theorem 3.1, we have established that the contractive behavior of the reverse ODE process is linked to the largest eigenvalue of the negative score Jacobian, (Ja) Yet, direct penalization of

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Published as a conference paper at ICLR 2026
Amax is computationally expensive, and critically, excessive penalization could fuel mode collapse in sampled action. As illustrated in Figure 4, we design a loss to encourage contractive behavior in diffusion ODE while avoiding direct eigenvalue caleulation and over- over-penalization. As a result, contraction is promoted by augmenting a penalty on (Jea into the diffusion loss to strike a perfect balance between denoising accuracy and encouraging contraction.
(I) Efficient eigenvalue computation. The first step is to compute the largest eigenvalue of Jep We avoid direct and expensive computation of all eigenvalues using the following approximation.
Lemma 3.1 (Power iteration (Austin et al., 2024)). Let (Jay") Rdxd be symmetric with Xmax the largest eigenvalue. For an initial vector vo E Rdxx NO(,, T):
(J
T (kt"( ck+1, k= = -I
Uk++
[ee uk l12"
Then,for for every k 2 O, Ad Amax| - O at a linear rate.
Each catrropaggatio-frienntt iteration of Lemma 3.1 costs only a single Jacobian-vector product. In practice, Appendix C.5 establishes that K" = or 4 suffices for a stable penalty term, while significantly reducing computational and memory overheads.
(II) Contraction loss. With computational efficiency ensured, the next step is to formulate the contraction loss. Let Åmax be the inwer-iteration estimate of (J ( ), and let AJa
(J.gg) denote its value for a specific denoised action and step. Therefore, the per-sample contraction loss becomes
(0) == max(-3, "Ja + f(t)h(t)=))
where is a desired margin to enforce stricter contraction. An efficient alternative loss for penalizing the largest eigenvalue is the Frobenius norm, formulated as
(0) m1 Flob / /trace ((J2) + ( + +1))..
The link between the Frobenius norm and eigenvalues comes from the fact that the Frobenius norm is tied to matrix's eigenvalues for symmetric matrices, as further explained in Appendix D.5. Therefore, penalizing large Le pushes the Jacobian towards negative curvature, while a conservative prevents extreme levels of contraction by avoiding excessively negative eigenvalues.
(III) Training loss. We now build the training loss by adding the score-matching loss La. described in Equation 4, the contraction loss L. as
(()) E (a.s)-D E (.2 Ileo + V. pr + 7 [o0)]]
(9)
with a single positive hyperparameter to determine the contraction strength. Note that the score Jacobian needs to be calculated at each solver step during the sampling process, and for each state- action pair in a batch of data. While a Jacobian penalty on the score function alone could drive the model toward a trivially contractive field, Equation prevents by penalizing the score to be accurate, thereby striking balance between contraction and meaningful flows in the diffusion ODE. Hence, (.) is regularized to enforce local contraction, pulling nearby trajectories together to reduce variance and improve robustness, but it does not collapse distinct, well-separated modes.
(IV) Offline learning integration. Since the added contraction loss term calculated from dif- fusion ODE, the implementation is independent of the choice of policy learning method and is straightforward as long as a differentiable score function is accessible. Here, our method builds on two computationally efficient learning approaches: EDP (Kang al., 2023) for offline RL and DBC (Pearce et al., 2023) for IL. We leave the details to Appendix C.
EXPERIMENTS
Our theoretical results and prior findings in the literature (Tang & Zhao, 2024) emphasize that contractive sampling dampens solver and score-matching errors and reduces unwanted sampling variance, thereby improving the consistency action diffusion. Building on this foundation, we aim to empirically address three key questions:

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Published as a conference paper at ICLR 2026
Table Offline RL experiments with D4RL data. CDP compared to diffusion -ased offline RL methods trained on the D4RL benchmark. Average and standard deviation of normalized episode return for each environment, and overall average across all environments are reported (higher is better). The best performing results for each environment, determined by the mean, are highlighted.
Dataset
Environment
BC
IQL
DQL
EDP
IDQL
CDP
Medium Expert (ME)
HalfCheetah
49.1 2.5
84.9 - 0.3
91.2 0.2
934 0.3
8..
94.8
0.3
Hopper
45.9++3.1
101.5
106.0 8.2
110.0 +28
107.4
110.0
+1.5
Walker2D
56.6:13.55
109.44:2.0
105.0
109.8
0.4
110 1.0
109.4
0.6
HalfCheetah
44.5
46.4 5.0
49.1 2.1
46.2
48.3++2.6
46.0
Medium (M)
Hopper
43.4
02.2
7.4
61.1=34
51.2
62.8 2.6
Walker2D
6..5
76.2 2.5
78.5 18.1::..1
0.2
80.9 = 1.3
86.5 + 0.5
Medium Replay (MR)
HalfCheetah
24.6::24
0.7
47.8 0.6
43.1
4..41:.33
43.9 0.2
Hopper
18.9++-.4
48.1 + 0.3
50.4 + +-7.
55.1 +
51.5 0.1
63.5 +-4.1
Walker2D
27.8
73.8 4.9
76.8 3.5
82.0 2.6
84,6 2:244
89.7
+ 0.8
Mixed
Franka Kitchen
42.5
59.9 + 8.0
58.1 + 1.4
57.2 + 1.8
58.2 4.1
61.0
+ 1.0
Partial
Franka Kitchen
31.9
4.1
42.3 2.7
44.5 1.8
49.0 +: 2.5
48.7
1.9
Complete
Franka Kitchen
22.6 7.4
35.7
= 6.2
32.9 = 1.5
31.6 3.3
51.0 + 1.7
Play
Antmaze Medium
0.2
17.5 11.3
21.9 =: L.G
14.3 7.1
17.7 5.7
20.4 6.7
Diverse
Antmaze Medium
0.6
17.7 4: 9.9
23.5 4: 2.3
25.6 :: 10.9
19.3 ::5.0
31.8 4: 8.5
Average reward
3..1
54.9
58.8 3.6
61.2 2.6
60.3 2.33
65.7 2.2
Time (seconds/ / 100k steps)
3115 1: 12
3742 3:36
12822 1: 164
4594 +
6250 d: 35
5236 d: 71
(01) Can CDPs enhance offline learning by mitigating score-matching and solver errors? (02) Are CDPs particularly effective in low-data regimes?
(Q3) How well do CDPs transfer to and perform in the physical world?
To answer (Q1-23), we conduct offline learning experiments on standard and partial version of D4RL (Fu et al., 2020) and Robomimic (Mandlekar et al., 2021) datasets. The learned policies are evaluated in Gymnasium (Towers et al., 2023) or Robosuite (Zhu et al., 2020), depending on the task, and the results are compared against baseline methods.
Highlights of the results. We find that contractive sampling generally improves diffusion policies for offline learning. Beyond the contraction loss weight results are not highly sensitive to other hyperparameters. While contractivity does not yield universal gains, and performance is modest in some environments, we observe substantial benefits in others. Given these positive trends, together with straightforward integration and reasonable computational overhead, we view CDPs as a practical and effective choice to enhance offline learning methods.
4.1 SETUP
Datasets. We select MuJoCo (Hopper, Walker2D, and HalfCheetah), Franka Kitchen (Complete, Partial, and Mixed), and Antmaze (Medium Play and Diverse) from D4RI. datasets for offline RL. For the MuJoCo environments, we utilize three data subsets: expert, medium-expert, and medium, which represent different expertise of the behavioral policy. Additionally, we evaluate on Robomimic datasets featuring demonstrations from tabletop manipulation tasks: Lift, Can, Square, and Transport. We test both low-dimensional proprioceptive observations and high-dimensional image-based observations. Further details are provided in Appendix G.
Baselines and backbones. For offline RL, we compare CDP against DQL (Wang et al., 2023), EDP (Kang et al., 2023), and IDQL Hansen-Esttucch et al., 2023), while for IL benchmarks we consider Diffusion Policy (Chi et al., 2023) and DBC (Pearce et al., 2023) with different conditioning backbones. Across all settings, we use a diffusion SDE backbone with an ODE solver for the reverse process. To integrate state information, we adopt residual MLP (Pearce et al., 2023), which is sufficient for the low-dimensional observation spaces of D4RL. For Robomimic, we use the same MLP encoder for diw-dimensional states, while for image-based based observations we obtain an embedding vector using diffusion transformers (Dong et al., 2024b). Appendix C provides further details about baseline implementation.
7

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Table 2: IL experiments with Robomimic data. We compare CDPs against the baselines on different tasks with low-dimensional (L) and image-based (H) observation spaces. We report the average and variance of success rate. The overall average and two best performing results are highlighted.
Dataset
Task
BC-GMM
DP-DIT
DP-Unet
DBC-DiT
CDP-DiT
CDP-Unet
Lift-L
0.83 0.03
0.99 0.20
1.00
0.10
0.94=0.0
1.00 0.05
1.00 = 0.15
Robomimic (Lowdim)
Can-L
0.67 1 0.09
1.009
0.98
10133
0.82
0.97 1 0.05
1.00 0.008
Square-L
0.44 0.04
0.79
0.008
0.08
0.57
0.56 1: 0.10
0.83 0.04
Transport-L
0.21 +: 0.01
0.56
0.00
0.74
0.03
0.13 0.00
0.48 + 0.07
0.81
0.06
Lift-H
0.75 1 0.03
0.99 0.19
1.00
0.111
0.91 1: 1.188
1.00 0.044
1.00
1 L..07
Robomimic
Can-H
0.56 0.11
0.111
0.95
0.84 0.12
1.00 = 0.03
1.00 0.12
(Image)
Square-H
0.43=0000
0.46
0.88 0.16
0.62 = 0.06
0.72 + 0.13
0.82 0.09
Transport-H
0.14 0.03
0.41 0.00
0.68 + 0.03
0.29 0.01
0.52 + 0.05
0.75 0.04
Average
0.50 0.04
0.76 0.10
0.10
0.64 0.07
0.78 + 0.07
0.90 0.08
Time (seconds/ 100K steps)
2635 1: 14
7880 1: 43
8210 51
7141 1: 48
8554 1: 67
9056 53
BC
DR-CE
DI-DXT
DSCMLP
DBC-DIT
CDP
DQL
IDQI
Robomimic-
Averuge reward
2.30
200
1.90
Can-I.
Square-L.
Transpor-ll.
Lib-H
Can-H
Kinchen
Majoco-M M.
Majocc-Mll:
Majoco-MR
Antmone
Figure 5: Experiments reduced datasets. We report the average episode return across different random seeds. When training on only 10% of the original dataset size, we observe that CDP decisively outperforms the baselines. This improvement stems from the ability of contraction to dampen score- matching errors, which are amplified in low-data regimes.
Training and evaluation. Training takes 200k-500k gradient steps based on the task difficulty and observation modality. For each dataset, we train every method with 10 random seeds and report the average and standard deviation of either normalized episode returns or success rates. At train time, we monitor diffusion and contraction losses, along with policy and value loss for offline RL. We save trained policies at 20k intervals, and evaluate those policies in the same environment as the data is collected from. The hyperparameter ? is naively tuned by picking the best performing r {0.001, 0.01, .... 1003, and we directly set 7 0.1. Details are outlined in Appendix C.4.
4.2 KEY OBSERVATIONS
General performance in offline learning (Q1). We evaluate CDPs on offline RL benchmarks in Table 1. On average, CDP outperforms baselines across D4RL environments, with particularly strong gains in Kitchen and MuJoCo Medium-Replat datasets. In contrast, for tasks where all methods already achieve near-optimal rewards, CDP matches baseline performance, and in rare cases the additional robustness enforced by contractivity can slightly impede performance. In the II. setting, CDP consistently outperforms DBC, the method it builds upon, but trails DP-Unet. We attribute this gap to the architectural advantages of UNet-based models well as their ability to generate longer action sequences, which particularly beneficial in complex imitation tasks.
Performance with limited data (Q2). Reducing dataset size for each task introduces inaccuracies in score matching. Without contraction. these errors accumulate and propagate through the sampling process. By contrast, CDP maintains stability, which results in enhanced evaluation- performance relative to baselines in Figure 5. This property is particularly valuable in practice, as real world data collection for offline learning is often costly and issource-iteessve..
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Lift
Stack
Slide
Peg
100
CDP
Success rate
80
Execution time
DBC
60
40
20
Lift
Stack
Slide
Peg
Lift
Stack
Slide
Peg
Figure 6: Experiments with the physical Franka arm. We execute each policy 20 times per task,
and report the average success rate and execution time. CDP often performs better than standard DBC, particularly for the harder tasks: Slide and Peg.
Physical robot experiments (23). We evaluate CDP on four tasks: Slide, Stack, Peg. and Lift. Demonstration trajectories are collected via teleoperation, with observations consisting of agentview images and proprioceptive data, and actions represented as end-effector delta poses. As shown in Figure 6, CDP successfully solves three out of four tasks and achieves higher success rate than DBC, including on the challenging Peg task. Further details on the experimental setup and data collection are provided in Appendix B and Appendix G.3, respectively.
Supporting experiments. Appendix F presents additional studies on the impact of different con- traction loss weights, the behavior of CDP under intensified solver errors, and extended evaluation curves corresponding to the results in Table and Table 2. Together, these results provide deeper insight into how contraction improves sampling quality in offline learning.
CONCLUSION
We probed the contractive behavior of the diffusion sampling process, leading the development of CDPs, class of diffusion policies with contractive sampling. We showed that contraction can be achieved by regularizing the score Jacobian, which can be efficiently integrated into the denoising diffusion backbone of offline learning methods. Our experiments demonstrated improved policy learning in standard continuous control tasks, as well as in scenarios with limited data.
Limitations. A key limitation of this study is the sensitivity of the diffusion ODE to the choice of contraction loss coefficient. While its value can be tuned by a light hyperparameter search, poorly tuned values can adversely impact the policy's performance. Moreover, CDP is built upon EDP or DBC based on offline learning setting, and the effect of promoting contraction in other offline learning methods is not explored. Lastly, work conducts limited experiments with image-based observations in simulation, and no low-dimensional experiments real world setups.
Future work. Aside from addressing the limitations above, future studies could investigate the role of the score function's sensitivity to states given fixed actions. Additionally, we deliberately refrain from extensive hyperparameter search for contraction-related parameters, but systematic exploration could help identify theoretical ranges or bounds for these parameters, potentially linked to the structure of the data, observation and action spaces, the choice of diffusion backbone. Finally, despite a brief study in Appendix E, we leave detailed studies of contraction in alternative diffusion sampling methods to the future work.

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ACKNOWLEDGMENT
We sincerely thank Wei-Di Chang, Harley Witzler, Mahrokh Ghoddousi Boroujeni, Hanna Yurchyk, and Stanley Wu for valuable feedback and discussions. This work was supported by the FRQNT Doctoral Training Scholarships and the NSERC Discovery Grant.
REPRODUCIBILITY STATEMENT
In the spirit of full reproducibility. our codebase is attached and will be released publicly upon publication. In addition to the code, we provide the community with all configuration files and training scripts needed to regenerate every experiment, table, and figure in the paper. Each result tied to the provided configuration, and we include dependency specifications and step-by-step run instructions to enable the straightforward execution of our pipelines or adapt them to new settings. With extensive documentation and a clean and straightforward codebase, our goal to make inspect, replicate, and build
verification frictionless such that the community confidence. Lastly, in the appendix, we provide step-by-step guide to implement the method, and present exact hyperparameters utilized to reproduce the reported performance.
work with
ETHICS STATEMENT
This work investigates contractive diffusion policies for offline learning with experiments conducted in controlled lab. No human-subject data were collected beyond standard teleoperation by trained lab members; no personally identifying information
recorded released. We consider potential misuse, for instance. deploying learned policies on unsafe hardware or in uncontrolled settings, and caution that our methods must not be used without robotic experts to provide appropriate safeguarding and oversight of the deployment. Notably, we minimize environmental impact by using modest compute budgets and small hyperparameter sweeps, and we report implementation details to support reproducibility. There are no conflicts of interest or external sponsorships influencing the results of this work.
CONTRIBUTIONS OF LLMS AND GENERATIVE AI
We used LLMs primarily for proofreading and improving the clarity and flow of the manuscript. In addition, LLMs assisted in extending our core theorem to alternative diffusion solvers (Appendix E) and in verifying as well as refining the presentation of proofs (Appendix D). On the implementation side, certain utilities and generic methods were generated with LLM support, while other classes and functions were linted and partially documented by similar agents. In literature review, LLMs were used to extensively search for related work and projects. All LLM outputs cross-checked against credible references, and no key theoretical developments or experimental results were produced by LLMs.
carefully
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Contractive Diffusion Policies: Appendix
Table of Contents
A Related work
16
A.1 Action diffusion for policy learning
16
A.2 Diffusion guidance for robust and safe policies
16
A.3 Contraction theory in policy learning
17
A.4 Lipschitz regularization for generalization
17
Real-world experiments
B.1 Physical setup
B.2 Data collection and deployment
C Implementation details
20
C.I Building CDP for offline RL
20
C.2 Building CDP for IL.
21
C.3 Cantract i vevi fusaissfuiii ffusionSDD module implementation
21
C.4 Hyperparameters & Configurations
22
C,5 Power iterations vs. direct eigenvalue calculation
23
C.6 Balancing Contractivity with Multimodal Learning
24
D Mathematical proofs and derivations
25
D.1 Connecting forward diffusion SDE to scheduling functions
25
D.2 Interplay of score Jacobian and contractive sampling ODE
25
D.3 Bounded action variance under contraction
26
D.4 Power iterations full statement
27
D.5 Frobenius for penalizing the largest eigenvalue
28
E Compatibility with diffusion solvers
29
E.1 Jacobian in the DDPM Formulation
29
F.2 Jacohian in the DDIM Formulation
29
F Additional experiments
31
F.1 Offline RL experiments
31
F2 Effect of different contraction loss ................................
31
F3 CDP with fewer solver steps
31
G Datasets and benchmarks
35
G.I D4RL: Datasets for deep offline reinforcement learning
35
G.2 Robomimic: Learning from human demonstrations
36
Real world data with Franka robot
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A RELATED WORK
We present a more in-depth analysis of the related work. As our work is at the intersection of diffusion policies for offline learning, safety and robustness in action diffusion, and policy learning with dynamical systems, we divide this section accordingly.
A.1 ACTION DIFFUSION FOR POLICY LEARNING
Diffusion modeling has been extensively used in policy learning to capture complex and multimodal action distributions. In policy representation for offline RI., DQL (Wang et al., 2023) is among the first to aim for training diffusion policies with a combination of behavior cloning and value maximization losses, boosting D4RL performance but at the cost of slow iterative denoising. DQL was immediately followed by EDP (Kang et al., 2023) which approximates actions from corrupted et al., 2023) which bridges diffusion modeling and implicit Q-learning for better trade-off between behavior regularization and value maximization. Others propose hierarchical (Ma et al., 2024) and multi-agent (Zhu et al., 2024) policies, as well as divergence minimization behavior regularization (Gao et al., 2025) to further tackle challenging offline RL scenarios.
ones during training. and IDQL (Hansen-Estruts
Diffusion models have also been used for planning in offline RL. For instance, Diffuser (Janner et al., 2022) and AdaptDiffuser (Liang et al., 2023) are diffusion- trajectory planners, with the former denoising offline trajectories into goal/constraint guided plans and the latter adding an adaptive self-evolving mechanism to remain effective in changing environments. Lastly, Decision Diffuser (Ajay al., 2023) treats decision-making as conditional generative modeling by sampling trajectories conditioned on desired returns or goals.
In policy learning through imitation, the focus is on perception, long-horizon structure, and multi-task generality. Diffusion policies were initially leveraged in DP (Chi et al., 2023) and DBC (Pearce al., 2023) methods for learning conditional action distributions. DP3 (Ze et al., 2024) and 3D Diffuser-Actor (Ke et al., 2024) lift visuomotor diffusion from 2D pixels to point-clouds, improving robustness and generalization in manipulation. 3D Diffuser-Actor extends this by fusing language, proprioception, and 3D scene tokens with a denoising transformer to produce rich 6-DoF action trajectories across many tasks.
To enhance the existing diffusion policy methods, Consistency Policy (Prasad et al., 2024) leverages distillation to create a one-step policy from a pre-trained diffusion model, while OneDP (Wang et al., 2025) trains single-step policy directly via diffusion distillation without multi-step teacher. Furthermore, LDP (Xie et al., 2025) uses compact latent space representation and inverse dynamics models to build more accurate diffusion policies, while DiffAIL (Wang et al., 2024) proposes adversarial training for distributional alignment with expert demonstration. Lastly, methods such as Nomad (Sridhar et al., 2024) and Diffusion-EDD EDF (Ryu et al., 2024) explore more specific applications in robotic navigation and manipulation.
A.2 DIFFUSION GUIDANCE FOR ROBUST AND SAFE POLICIES
Safety in diffusion policies is relatively less explored. Safe Diffuser (Xiao et al., 2025) steers denoising with control-barriet guidance such that generated trajectories satisfy safety specs. Effectively, Safe Diffuser restricts the reverse diffusion process to generate samples belonging Diffusion (Mizuta & Leung, 2024) fuses control barrier and Lyapunov functions to yield collision- free, stable plans in dynamic scenes. In contrast to Safe Diffuser, CoBL-Diffusion uses reward signals corresponding to safe actions to guide the diffusion process. More recently, Constrained Diffusers (Zhang et al., 2025) formulate the constrained diffusion generating process as constrained sampling, and solve it with constrained optimization techniques. These methods make safety or constraints a property of the reverse process itself, providing safety and reliability guarantees at generation time. However, learning these constraints safety standards is still challenging problem, and enforcing them causes a considerable computational overhead in siggh-imensional spaces.
safe set. CoBL-
A different perspective is focused on feasibility and in-distribution guidance for training diffusion policies with offline RL. FISOR (Zheng et al., 2024) learns a feasibility critic to filter unsafe trajectories before execution, while searching for feasible actions with high rewards. Diffusion-
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DICE (Mao et al., 2024) builds on DICE-style methods, adding in-sample, value-aware guidance that pulls samples toward high-return, in-distribution behaviors. This is achieved using a DICE guidance term interpreted as "how much more (or less) often will the policy visit each state-action than the data did", hence, keeping the policy close to the dataset support. Trajectory-level safe diffusion (Römer al., 2024) alternates denoising with per-step constraint projections to ensure safe trajectory rollouts at test time. Lastly, SRDP (Ada et al., 2024) takes a different approach to address distribution shift by adding state representation objectives harden policies unseen states, Compared the first perspective on safe diffusion policies, this line of work offers less computational overhead, but also looser generalization guarantees.
A.3 CONTRACTION THEORY IN POLICY LEARNING
Contraction theory has been utilized learn stable dynamical systems through IL for plan- ning (Ravichandar et al., 2017; Abyaneh et al., 2025). Contraction in this scenario offers a more flexible stability criteria than using Lyapunov functions diffeomorphism (Zhang et al., 2022; Abyaneh et al., 2024). In representation learning, Contractive Autoencoders (CAEs) (Rifai et al., 2011) leverage contraction theory to learn representations that are less sensitive to input perturbations, and showcase promising results at the cost of augmenting a simple and differentiable loss term. These representations can later be utilized learn improved and more robust policies. CAEs (Rifai et al., 2011) 1) shrink the representation changes when you nudge the input. CAEs explicitly penalize the encoder Jacobian to achieve this purpose, whereas CDP penalizes the score model's local sensitivity. In both cases the goal is discouraging high local gain and brittle behavior. Notably, CAB contraction local around data, while CDPs need uniform contraction across reverse ODE trajectories.
Contractive Diffusion Probabilistic Models. Contraction theory has also been applied to improve the numerical stability of diffusion models, most notably in CDPMs (Tang Zhao, 2024), which focus extensively on diffusion SDEs rather than ODEs. The central idea is to enforce contraction in the backward sampling process, hence, provably reducing both score- matching and discretization errors. This makes CDPMs more robust inaceuracies in score estimation and time discretization. The approach can be implemented through transformations of pretrained model weights. However, contraction is imposed globally through specific design choices and assumptions on the drift and diffusion terms. Such hard constraints may completely ignore the local geometry of the score function, effectively suppressing diversity among dominant modes and limiting representational capacity. Furthermore, the formulation is not designed for conditional action diffusion or diffusion ODEs, making its integration into action diffusion for offline learning nontrivial.
A.4 LIPSCHITZ REGULARIZATION FOR GENER ALIZATION
Regularizing the Jacobian, Lipschitz regularization, has been employed outside the scope of diffusion modeling to improve generalization and performance. For instance. (Jia & Su, 2020) define information-theoretii metric based on the observed Fisher information to characterize local minima, prove generalization bound in terms of this metric, and then use it directly as a regularizer bias training toward enhanced minima. (Karakida et al., 2023) study gradient and show that a particular finite-difference scheme- implemeeipleeeentee via alternating ascent/descent steps- steps-both both reduces the computational cost of gradient regularization and strengthens a desirable implicit bias toward rich-regime solutions. Lastly, (Foret et al., 2021) introduce sharpness- aware minimization, an efficient min-me max optimizer that explicitly minimizes both loss and local sharpness by approximately optimizing the worst-case loss in neighborhood of the current parameters. Yet, these methods mostly aim to improve generalization deep neural networks and have not been extended to diffusion models.
regularization
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Policy Server
Environment
Polymetis client
Trained CDP
Actions
Aci
High-froqueccy
Gripper action
Action
Denoise
Niviay
control
End-ellector pose
a-
Torch JIT
Obs
Obe
Observations
Revene
Formard
Franka control interface
Proprioception
Agent-view image
Polymetis
Figure 7: Physical robot setup for data collection and deployment of CDP. The trained and frozen policy denoises noisy action samples into control commands. Actions are communicated to the Polymetís middleware, which provides high-frequency control through Torch JIT and the Franka control interface. The Franka control interface executes gripper and end-effector actions while returning observations, including agentview and depth images plus proprioception, which are sent back to the policy for closed-loop continuous control.
B REAL-WORLD EXPERIMENTS
B.1 PHYSICAL SETUP
The experimental setup consists of a Franka Emika Panda' robot with 7 degrees-of soaecces-offrreeeoo ff-reedom and equipped with the Franka hand: a parallel two-finger gripper. The robot is mounted onto workspace with dimensions of 1.8 m (length) x 1.22 m (width) x 0.82 m (height). As illustrated in the observation view of Figure 7, RealSense D435i camera is positioned capture the agentview perspective of the workspace. In this image, blue lines denote the camera field of view, while orange lines indicate the cropped region obtained after preprocessing, corresponding to the standard resolution of 224 224.
All models were trained on two RTX-4090s, each equipped with 24 GB of VRAM and allocated maximum of 62.5 GB of CPU RAM. Despite this, a simple GPU with GB of VRAM sufficient to run single experiment on low- www-dimesioonl observations, or to deploy an image- amage-based diffusion policy with a ResNet backbone. shown in Figure 8, the environments include Lift, Peg, Stack, and Slide, where Peg and Slide are more difficult due to precision requirements compared to the others.
Slide
Stack
Lift
Figure 8: Franka robot environments. In comparison to other tasks, Peg and Slide are more difficult and require high precision, while Lift and Stack are relatively straightforward.
hhttps://franke wetpi/aadddd/d/eeeeeeee documents
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Lin
Peg
Slide
Stack
Figure CDP deployed for of 4 real-worke tasks. The cropped footage is the real-time stream from the Intel RealSense camera, which communicates the image observations.
B.2 DATA COLLECTION AND DEPLOYMENT
To evaluate proposed method real hardware, collected teleoperated demonstrations Franka Emika Panda. CDP deployment on these tasks is shown in Figure 9, We use Polymetis 2 for high-level control and teleoperation; commands are executed the Franka via libfranks over the Franka Control Interface (FCI) at kHz, which returns current robot state to our controller. The end effector was teleoperated with 3Dconnexion SpaceMouse 3, where 6-DoF inputs were mapped to Cartesian pose commands with adjustable gains. For each trajectory, we logged time-synchronized joint states (positions, velocities), end-effector pose, gripper width and state, RGB images, and the executed actions for each trajectory at fixed 10 Hz. rate. However, we only leverage agentview image and proprioception (end effector pose and gripper state) for training and deployment.
We deploy the trained policy using two-machine client-server setup. On the robot's side, client streams observations to separate inference server that hosts the policy. The server runs inference and returns 7-dimensional action vector (delta end effector position, orientation, and gripper state), which the client executes on the Franka via Polymetis.
thotpp::/github..... com/ h htwp://rrwc//aioooeee///e///e//e y hhttps://3dconnexiinn http://ccaaaaaiooooooo//////aaaa
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Non con-contractive
Contractive
Contractive high
Random untrained
ag
Figure 10: ODE flows for toy 2D action space. The example is designed for 2D action space with fixed state, and portrays the way contraction affects the dynamics of diffusion ODE flows
C IMPLEMENTATION DETAILS
In this appendix, we present step-by-step recipe for promoting contraction in the diffusion ODE's sampling process, leveraging CDP's loss and efficient eigenvalue computation. Additional visualiza- tions of the ODE flows toy 2D action space, comparing non-contractive and contractive variants, are provided in Figure 10. While our experiments focus on building CDPs using EDP (offline RL) and DBC (IL. algorithms, our contribution targets the sampling process, and hence, is compatible with any diffusion-bused offline policy learning method. We simply pick EDP and DBC for their efficient and straightforward implementation.
Note that CDP's codebase, well all baseline implementations, are built on top of cleanDiffuser By providing a unified implementation of diffusion policy methods, the brary enables fair and unbiased comparisons, ensuring that the only differences across methods are those explicitly specified in the configuration files, or the techniques used each baseline, and not advantages such as conditioning backbones, solvers, sampling steps, and other factors related to diffu- sion modeling. To this end, we examine the CleanD fuueer library in more detail, highlighting its comprehensive support for diffusion policy architectures and state-conditioning modules, and sharing insights into the behavior of different diffusion and conditioning backbones.
C.1 BUILDING CDP FOR OFFLINE RL
Method overview. Efficient Diffusion Policy (EDP) (Kang et al., 2023) reduces the training time overhead of diffusion-based policies such as DQL (Wang et al., 2023) by (i) using high-order sampler to cut evaluation steps, and (ii) replacing multistep denoising inside the actor update with one-step action approximation.
Let ag denote the clean action and a, its noised counterpart at diffusion time under the standard forward process (equivalent to Equation 1)
ar = 0:an + oe
~~ (0,10.,
(10)
During policy optimization, EDP avoids iterative reverse denoising of a,, as explained in Equation 3, and instead approximates an in step:
ag (a S,t),
(11)
ar Te o,
(12)
at
where (11) is used with data--rediction head and (12) follows directly from (10) when using noise-prediction head (Ho et al., 2020). This replacement preserves useful gradients for the actor while eliminating the need to unroll many denoising steps inside the update. EDP also swaps the DDPM sampler for DPM-Solver (Zheng al., 2023), a high-order ODE solver, enabling accurate sampling with small number of steps which noticeably boosts the efficiency.
Building CDPs on top of EDP for offline RL
hhtpps://so thub. wwt.c//am/////ccccco f ffuserTeam Caannni ffuser
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1. Threshold. We normally assume that the contraction condition is enforced at every solver step during the denoising process. However, the parameter contr.ster allows contraction to be applied only in the later stages of denoising, leaving the high-noise iterations unconstrained. This option is not required, but can be beneficial for handling more complex action distributions.
2. Jacobian. Form the symmetric Jacobian of the diffusion actor with respect input actions
Jeg = (ag(ar, 8.
Jerm (Jes + Je.)-
Da,
This targets the part that governs the reverse ODE's contraction based on Theorem 3.1. Je, can be computed via Equation 12, given the approximated action åo
Tao
0 oa.
a. of Go co s.t)
1 (1-o(e))))
aa.
az
3. Contraction. Measure contractivity by penalizing the Frobenius norm stronger alternative by approximating the leading eigenvalue Amax( via power iteration as described in Lemma 3.1 and penalize it through Section 3.2. Note that the Jacobian needs be computed for every sampling timestep, and for all state-action pairs in the current batch.
Loss. Combine the selected terms with policy and value loss terms for offline RL the maximum likelihood loss for IL, similar to Equation 9, and tune the hyperparameter with voos-vaaiittion.
C.2 BUILDING CDP FOR IL
Method overview. Diffusion Behavioral Cloning (DBC) (Pearce et al., 2023) frames imitation as conditional action diffusion: learn a denoiser that maps (a,, to au. capturing the stochastic, multimodal nature demonstrations. With the standard forward process, the model predicts either Eg conditioned the observation s. DBC is agnostic to the perception stack: any visual encoder can feed 8 (pixels or history) into the denoiser. In this work, we plug in finetuned ResNet backbones and use diffusion transformers. Another contribution of DBC is Diffusion-X sampling which biases selection toward intra-distribution, human-like actions: it keeps denoising for extra iterations.
Recipe to build CDP top of DBC. The integration is straightforward, and generally follow the same procedure as Appendix C.I to compute the score Jacobian. Unlike EDP integration, however, approximate actions are not directly available for this purpose. Instead, we compute the Jacobian of the score function itself, which is relatively simple to implement with both MLPs and transformer-bant based architectures provided in CleanDi f fuser.
C.3 CONTRACTIVEDITE ONSDE MODULE IMPLEMENTATION
Contracti setrrattieeiiffssionn
module (implemented with the same name in codebase) is the heart CDP's implementation. It is built upon the Cont inuousDii fuusi onSDE class within CleanDif f fuser, which represents continuous-time variant of variance preserving diffusion SDE model. defines both training (adding noise to data and learning to reverse it) and inference (sampling new data points) procedures using continuous-tin solver's choice. We introduce this module to integrate contraction analysis in the diffusion SDE framework, as described in Appendix C.I and Appendix C.2. Below, we provide quick summary of major components and configurations of this implementation and how it preserves compatibility with various solvers, diffusion, and conditioning backbones.
SDEs and ODEs depending on the
Diffusion and conditioning backbones. The core components of our framework are the nn di fisiion, which serves as the primary backbone for predicting either the noise or the clean signal during the reverse process, and the nn_condite t too, which embeds the condi- tional observation inputs into a suitable representation space. Through CleanDi ffuner, the Contract vevi fus onSDE class flexibly supports range of backbone architectures. For diffusion, we primarily experiment with MLPs, which provide lightweight function approximation for dimenssiion problems, and diffusion transformers. which capture long- oong-range dependencies and complex temporal-spatio correlations critical in sequential decision-main making. For conditioning, adapt the architecture the modality and dimensionality of the observations: identity mappings
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Table 3: CDP's hyperparameters, configurations, and their importance. We list contraction- related parameters first. Among them, is the most influential, as it directly impacts evaluation performance. The parameter coss_type also plays an important role, though its effect is primarily on computational efficiency rather than performance. The parameters listed under general are shared between CDP and the baselines used in our experiments.
Parameter
Description
Importance
Range / choices
Contraction specific
Weight contraction loss
High
[10-3 100] (log-tune)
less_type # (contr_thr)
Contraction penalty type
High
Jacobian /eigian//egenn
Target cigenvalue margin
Medium
0.1
contr_stepa num
Steps with contruction penalty
Medium
or s.t.xaaamlliggsttpp
Power-iteration count
Medium
3.5
General training & inference
actor_lr, critic.lr solver
Actor/Critic learning rales
High
x 10 (tune: [3 9, 10 ode dppsollvrr+2225
Reverse ODE solver (DPM)
Medium
diffusi on.steps
Diffusion steps during training
Medium
50 (typical: [50, 100)
Sampling steps during evaluation
Medium
15 (typical: [00
ema_rate
EMA decay
Medium
0.999 (typical: [0.995, 0.9909)
an.diffusion an_condition on gamma
Diffusion backhone
Medium
DiT HAT/MIL ML.
Conditioning network
Medium
ResNet / Tdeetity
RL discount factor
Low
0.99 (lypical: [0.95, 0.995))
for low-dimensional vector states, and convolutional ResNet-based encoders when working with high-dimensional image inputs. This modular design enables us to tailor the expressivity and in- ductive biases of the backbones to the structure of the task, while maintaining consistency with our experiments.
Noise schedules. Given input sample, the implementation applies forward diffusion step using the given noise schedule. In CleanDiffuser two standard schedules are supported. The linear schedule increases the variance linearly with time, leading simple and stable forward process that has been widely adopted in early diffusion models. contrast, the cosine schedule adjusts the variance according shifted cosine function, which slows down noise growth the early steps and accelerates it later. We use the linear schedule in our experiments but our implementation of Contract www.rraatieifffff...oomm
supports both scheduling schemes.
Solvers. We employ ODE-based DPM solvers (Zheng et al., 2023), which directly integrate the reverse diffusion ODE rather than relying on stochastic updates, and are consistant with our theoretical framework. These solvers approximate the solution trajectory by applying high-order numerical integration schemes that balance accuracy with computational efficiency. In particular, we use ode_dpmso 1 +ver++_00 +_2M, a second-order multistep method that leverages two adjacent denoising steps for improved stability and precision.
C.4 HYPERPARAMETERS & CONFIGURATIONS
All critical hyperparameters and configurations of our method are summarized in Table 3. While the table lists broad set of parameters, it is important to emphasize that only two of them are critical for contractive sampling. Among these two, 7 the decisive factor that substantially influences both training stability and final evaluation performance. The second contraction-red related parameter, B, plays more supportive role, serving primarily as safeguard rather than driver of performance. To further clarify their impact, in Table 4 we report the best- cest-performin values for all parameters based on our empirical search.
CDP's hyperparameters. Our hyperparameter tuning procedure is intentionally kept simple to highlight the robustness of the approach. Specifically, for 7, we perform a search over the discrete set {0.001, 0.01, 0.1, 1.0, 10), which is sufficient to capture its effect performance across benchmarks. The value of and loss_tt loss_type is fixed to 0.1 and jacobian an, respectively, reflecting their relatively minor influence compared to the weight term. Other parameters are inherited from standard diffusion policy baselines without modification, ensuring that the only meaningful difference introduced by method stems from the contraction regularization. Therefore, ensure that the observed
inn,
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Table 4: Best performing contraction loss weights. We summarize the results of our hyperparameter tuning for the contraction loss weight here. Note that the results correspond to experiments with standard datasets for each benchmark.
Environment
Contraction loss weight (?)
Antmaze (Medium-Play Play, Medium-Diverse)
(0.001, 0.001)
Franka-Kitchen (Partial, Mixed, Complete)
(100.0, 100.0, 10.0)
Mujoco-Medium (Half- Cheetah, Hopper, Walker)
Mujoco-Mediin Mcdium-Expertt (Half-Cheetah Hopper, Walker)
(0.01, 0.001, 0.01)
Mujoco-Mediune Me--iM--------- Medium-Replay (Half-Chectuh, Hopper. Walker)
(0.01, 0.001)
Robomimic- (Lift, Can. Square. Transport)
(0.01, 0.1, 1.0. 0.1)
Rohomimic-Image (Lift, Can)
(0.01, 0.001)
Real-world tasks (Lift, Stack, Slide, Peg)
(0.001, 0.001, 0.01, 0.01)
Conputtiion time of Larget dgenvable
Estimate of largest digastaaue
30-
rich foutz foatt-tyaa
16
of power ltertions
Figure 11: Efficiency of power iterations for calculating the leading eigenvalue. (Left) Computa- tion time for n matrices compared against PyTorchh's 's built-in eigenvalue method. (Right) Number of iterations required obtain a stable estimate of the largest eigenvalue for different matrix sizes.
performance gains are not the product of extensive hyperparameter optimization, but rather direct consequence of incorporating contraction into the learning process.
Note that the configuration files in the codebase specify the correct hyperparameters for each method. The only hyperparameter that requires tuning achieve the best performance the contraction loss weight, whose values are provided in Table 4.
C.5 POWER ITERATIONS VS. DIRECT FIGENVALUE CALCULATION
The comparison between power iterations and PyToh'h's 's direct eigenvalue computation highlights clear difference in efficiency, particularly for large matrices. As shown in Figure 11, computing the largest eigenvalue via PyTorch's 's built-in routines becomes increasingly expensive the matrix size grows, since it involves solving for the full spectrum of eigenvalues. Power iterations, in contrast, specifically target the leading eigenvalue and thus achieve markedly lower runtimes, particularly for medium to large matrices. This efficiency makes them appealing in scenarios where only the dominant eigenvalue is required, as often the case in stability analysis contraction- based regularization, where repeated evaluations must be carried out during training.
Note. For each batch of data, the number of times need calculate the largest eigenvalue is
batch size x num solver steps(!),
which intensifies the differences observed in Figure 11.
We also show that only a small number of iterations are needed to reach a stable estimate of the largest eigenvalue, and the required iteration count remains modest even as matrix size increases. In practice, this means that one can obtain accurate eigenvalue estimates with just a handful of iterations, striking favorable balance between computational cost and precision.
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C.6 BALANCING CONTRACTIVITY WITH MULTIMODAL LEARNING
Our policy is trained with the standard diffusion objective in Equation 4, which anchors the score to the data distribution and thus preserves task-relevant multimodality. The contraction loss integrated in Equation 9 complements this objective by tempering the sampler's seed/solver sensitivity in locally expansive directions of the reverse dynamics, improving reliability without collapsing distinct high-density modes. The delicate balance between contraction remains as long as regu-regularization avoided. In practice, our method provides more than sufficient tools to balance robustness and diversity with the following complementary knobs.
Contraction (regularization) weight (7). As the key and only critical hyperparameter of CDP, ? controls the overall strength of contraction and is tuned coarsely with small grid (only 6 values in this work; see Section 4. 1). The experiments showcase that for Robomimic and D4RL. benchmarks, well our worldworld tasks, these values enough showcase reasonable performance gain, while all the other parameters are fixed. However, depending on the validation/test-tim performance, one could opt for more extensive search simply leave the regularization term as bounded trainable parameter to be learned with the diffusion loss.
Contraction steps Chontr-stepsee This parameter controls which denoising steps the contrac- tion regularizer is applied, allowing us to activate contraction only on a selected subset of (typically later) timesteps rather than throughout the entire sampling process. For instance, can apply contraction only in the late denoising steps, which one could leverage when handling complex, highly multimodal action distributions. This provides simple mechanism to concentrate the trajectories near high-qualit modes in the final sampling stages, while preserving expressivity and flexibility of the diffusion model at earlier steps.
Contraction threshold (3, contr.thr). Finally, we note that the parameter 3 controls the maxi- mum target contraction level by acting as an eigenvalue threshold in our Frobenius norm surrogate in Section 3.2. Concretely, we penalize violations of bound of the form (J9y)) S ||>>m/ |p S B, so that smaller values 3 enforce stronger contraction (tighter eigenvalue bounds), while larger values allow more flexible dynamics. In practice, we set { once to meaningfully reduce sensitivity without aggressively shrinking the action bandwidth or collapsing multimodality. and use only the contraction weight. Empirically, we observe that performance is reasonably stable with fixed values, which supports the robustness of our Frobenius based eigenvalue thresholding in practice.
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D MATHEMATICAL PROOFS AND DERIVATIONS
D.1 CONNECTING FORWARD DIFFUSION SDE TO SCHEDULING FUNCTIONS
The schedulers a and on originate from the non-SDE, but equivalent formulation of the forward diffusion process, where the noisy action at step is given by
a = 0g30 + JCC, C~N(0.
(13)
These scheduling parameters control how the original and noise signals evolve in the forward process, and therefore central defining the trade-off between signal preservation and stochastic corruption during training. While they are inherited directly from standard diffusion policy formulations and remain fixed across our experiments, they provide the backdrop against which contraction-related modifications are introduced.
To establish the link between and and the forward SDE functions, g and f. we start by recalling the forward diffusion SDE da. f(t) a dt + dW,
its Itō solution is
aj g(s) dW., a = exp ::==ep(pp exp(S (f. f(u) du).
a, = 0,a0
+
0 as
This implics
a ag N N(raan T)
with
ar of /
g(s)
ds.
Conversely, given a; and og 2 0, the drift and diffusion are
d
f(t)
log at
g(t)
o. 2
dt
This derivation connects the forward SDE process to scheduling functions, and enables find the
drift and diffusion functions at a given diffusion step (Song et al., 202 la; Dong et al., 2024a).
D.2 INTERPLAY OF SCORE JACOBIAN AND CONTRACTIVE SAMPLING ODE
Theorem statement. Given a state s and a diffusion ODE, da, Fo s, t)d,,, Fo contractive, i.e., Fe x 0, iff the score Jacobian satisfies
Amaxx (Joym) < <((()))(t)+,, Vt E [0,1],
where Amax denotes the largest eigenvalue, and J, are defined by the forward process.
Proof. According to Section 2.2, deterministic flow da, Fo t)dt is contractive with rate > 0 in the L2-norm, if all eigenvalues of its symmetric Jacobian are negative, i.e..
Amax (Ji. max A (J. JF.)) <0 Val, E [0, 1).
For the diffusion ODE Fg. we already know from the main text that J f(t) + h(t) J.y Specifically for any two symmetric matrices A and B, by the special case of Weyl's inequality, the following holds for their maximum eigenvalues:
Amax (A + B) S Amax (A) + (B).
Applying this to the Jacobian of the diffusion ODE, we get:
Xmnaz (Joym) Amaxx (f(1)T 1i(1) Jaym) < Amax(((1)) Amaxx (1 Joym)
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Assuming that the diffusion coefficient h(()) is positive", sufficient condition for reverse ODE contrac- tion boils down to
Xemaxe
(Jrr)
S Amax (f(t)
h(t) Amax (J O.
(14)
Drift
Diffusion
Drift. Note that since the drift term f uniformly dissipative, this is evident by
f(t) = d log az dt
Or is a dissipative schedule set in the forward process,
da
d log at <0 f()) <
< 0 7 [(1)T < 0, since log is monotonically increasing and [()) is diagonal.
dl
di
Hence, every [(I) is negative-definite, meaning that Amax( O. This is the usual situation for variance-presse preservinns (VP) schedules investigated in our work (Song et al., 2021a;b), where og a dissipative function.
Diffusion. Revisiting Equation 14, we notice that with positive h, the equation will be satisfied if
Amax )max(((())
(f(t))
-(())
Amax (Jry
(15)
h(t)
h(t)
Since /(t)/ is a diagonal matrix with f(() as every entry on the diagonal,
eigenvalues equal to f(t). In simple terms, just need an upper bound Amax (J According to
Appendix D.1, the bound in Equation 15 can be written in terms of signal and noise schedules as
- ,( d L
log ae
-5(()
Amax (Jyym
de
(16)
2 d log o7)
h(t)
17
Therefore, we can efficiently determine the bound of the forward diffusion process.
=ft) based on the known scheduling parameters 1())
Extension to variance-exploding (VE) schedules. Our proof centers VP-SDE schedules because they are the de facto choice in diffusion policies for control, they align with our experimental setups, and crucially, the probability flow ODE induced by VP has uniformly dissipative drift. This negative drift cleanly maps into a stability margin in the symmetric Jacobian, yielding transparent contraction conditions and a direct diagnostic for seed/solver sensitivity.
For VE-SDEs, the diffusion ODE lacks the dissipative drift that VP provides, so stability no longer benefits from built-in leftward spectral shift, However, it hinges entirely on curbing locally expansive directions of the learned score and the diffusion term in general. Hence, our contraction loss and the proof can still be applied: the only change required is update the contraction threshold (using in Section 3.2) to adjust for the lack of dissipative drift term. Paired with small y these controls balance robustness and multimodality under VE schedules, where the absence of dissipative drift could make the reverse flow more sensitive to score and integration errors.
D.3 BOUNDED ACTION VARIANCE UNDER CONTRACTION
Corollary statement. For any two flows a, and ai initialized at al and aij the difference in actions generated by Fe. denoted by da, = a,
ai; ai, is bounded by
| || < exp (f AJa (T))
where J JP. (T) is the sampling ODE's Jacobian solver step T.
$ "As defined Equation 6, h(() = (2o)) Hence, h(() is always positive since a represents a positive
noise variance.
For variance-exploding scheduling, the drift term loses any effect on the reverse process (Yang et al., 2023).
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Proof. We start by following the steps to outline the sufficient condition for contraction on the largest eigen value (Lohmiller & Slotine, 1998): Given two trajectories a, af the system
da. = F Fa, t),
their infinitesimal separation at step t e [0, 1]. defined by
da: a a
evolves as őa; [(a, 1) da,, where J(a.t) = (a. t). Tracking the squared distance in the Euclidean metric, da Sa, yields
d (ôa, 8ai) 20a, ba, 28a Jóa
dt
Decompose J into its symmetric part jay = (J + JT) Let Amax((r,)) 1) denote the largest eigenvalue of faym Given that for the asymmetric part of J. have oa, jasym da, 0
ba, Jóa, = da, joym S Amax (aa,.) 1) da, daz-
Hence, we obtain the differential inequality
d
di (da, da.) s 2amax(ar, (6a, dat).
and finally by applying Grönwall's inequality, this integrates to the contraction bound
||ōa,|| < | exp (% Aa)
(17)
If Amax < O, then any infinitesimal displacement da, decays exponentially to zero. Consequently, by integrating along any finite path, the total length contracts exponentially over time.
Terminal-to-initial sensitivity for reverse ODE. From the proof of contraction above, and in particular the variational bound in Equation 17, we can view the infinitesimal statement as seed- sensitivity bound between two reverse ODE solutions starting from different noise sampling, In other words, Equation 17 shows that contraction limits the action variance through:
Lower Amax + Lower ||ōau|| Less sensitive to ag sampling + Lower action variance.
Late in denoising, the signal to noise ratio gets higher, and the score field strongly points towards high-density regions which can be viewed as basins around dominant modes of conditional action distribution. Seeds started far apart often fall into the same basin for a fixed state given fully trained score function. Contraction makes this basin narrowr,, rendering action generation more consistent, yet still capable of capturing multi-modal and complex behaviors.
D.4 POWER ITERATIONS FULL STATEMENT
Let A E IRdxd be symmetric with decompoocooiooiiionn A diag(A)....,, U and orthonor- mal eigenvectors U [M,,..., ud]. Assume a simple dominant eigenvalue in magnitude:
| > 122 2 ... P
Choose any nonzero vo E Rd with 0₁ := un 40 * 0. For example, draw N Ta) and set Vo = g/ [|g112 then an # 0 with probability 1. Define the power iteration and Rayleigh quotient by
Av
uk+1
|Axxll2
d :=:=00 "I Auk.
k = 0,,,,
T =(8a² J ( 8a) 8a T da 5a (aaj)))))) = -5a" Jay 5a -- 9ER. 0.
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Let 07: L(v,, u1). Then for all l 2 0.
A2
v1
of
tan 0 v
A1
a1
and the Rayleigh-quotient error satisfies
A1x - A1 S (111++
+
A2
2k
0
A1
Consequently, since Aal/|Ar| < to tu at the same rate angle (Austin et al., 2024).
have d X1 geometric (linear) rate, and Up converges
D.5 FROBENIUS NORM FOR PENALIZING THE LARGEST EIGENVALUE
Frobenius surrogate with curvature shift. Penalizing the Frobenius norm of shifted symmetric Jacobian provides simple, efficient surrogate for directly constraining its largest eigenvalue. For any matrix J,
| (I(JJ]]= Ene,
|| max Mi.
|| S |||||
where {u:} are the singular values. When J is symmetric, 14 1& Amax (J) 11 1| Rather than shrinking J toward 0, we shrink it toward target curvature B E Rt by minimizing
J+ J
Σ ()))))
which softly drives all eigenvalues toward
- B.
Application to diffusion ODE sampling, Let J = } (Je, + )(,) denote the symmetric Jacobian of the (scaled) score field. Our shifted Frobenius penalty
(0) = | + 31
centers the regularization at -3. Indeed, for symmetric matrices,
Amaax () + | <
J + 3T
so shrinking the shifted Frobenius norm reduces an upper bound on the margin violation Amax (J B. Computationally, || || is inexpensive evaluate with automatic differentiation (sum of squares of entries), avoiding iterative eigensolvers.
only drives eigenvalues toward 0; positive Amax can persist while others move closer to 0, yielding a small Frobenius norm without satisfying contraction. In contrast, the shifted penalty ||J [T places the basin of attraction the desired curvature (c.g., negative margin), exerting greater pressure on eigenvalues that violate the target.
Remark: The unshifted penalty ||
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E COMPATIBILITY WITH DIFFUSION SOLVERS
In the main text, our focus has been on the ODE formulation of the sampling process. In this section, we illustrate with two prominent diffusion formulations, namely Denoising Diffusion Probabilistic Models (DDPM) (Ho et al., 2020) and Denoising Diffusion Implicit Models (DDIM) (Song et al., 2021a) that the Jacobian of the reverse process can be derived based on the score Jacobian and parameters of the forward process.
Kitchen-Partial
Kitchen-Mixed
Kitchen-Complete
Average reward
DDPM ODE-DPP
13
Diffisin Selver
32
Figure 12: Comparing Diffusion Solvers. We change the diffusion solver of CDP to DDPM, DDIM, and the default ODE-based solver we use throughout this work. The average and error bands are plotted here.
E.I JACOBIAN IN THE DDPM FORMULATION
DDPM overview. DDPMs are essentially predecessors to diffusion SDE and ODE formulations. Similar to diffusion SDEs, they define a forward diffusion process where clean data ag is corrupted into increasingly noisy versions a ar by adding Gaussian noise at each step, with carefully chosen variances so that ay approaches pure noise. A neural network, usually parameterized to predict either the noise given a noisy input a,, predict how to denoise it toward a;-1-
the clean sample ag, is trained to approximate the reverse transitions:
Jacobian derivation. In the discrete-tin tiecrete-timeaeeee and state-condiiioned DDPM, the reverse step is
(1 Ote). co(ar, s, t) t. {~N(0,1).
a 1111
a,
at
VI Cz.
where ā; = IT == and of 1-ne-1 (1 az). Since & is independent of and s is fixed, they do 1-61
not contribute to the Jacobian. The per-step Jacobian of the reverse mapping can therefore be derived from the score Jacobian as
da.-1
(1 an) Ōco(ar, s, t)
Jo.
Oa,
Vau
v1 Ōi.
Oai
Thus, by repeated application of the chain rule across the sequence of reverse steps, the Jacobian of the entire sampling process (conditioned on is expressed the product of the per-step Jacobians:
Jo
II Ja.
Note that even though DDPM is written discrete reverse map, it be viewed time- discretization of a continuous reverse flow ODE (Ho et al., 2020; Song et al., 2021b).
E.2 JACOBIAN IN THE DDIM FORMULATION
DDIM overview. DDIMs provide deterministic alternative to DDPM sampling by reinterpreting the reverse process non-Markovian chain. Instead of treating the reverse diffusion stochastic
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step with added Gaussian noise, DDIMs define deterministic mapping from a, to a;-1 through a reparameterization that directly leverages the network prediction of e or åy. This eliminates the sampling noise while preserving the same marginal distributions DDPMs, thereby allowing faster generation with fewer steps and enabling more controllable trajectories.
Jacobian derivation. Let dę (0, 1] with ot preserving update is
vI ât- The deterministic DDIM variance-
a vā:-1 s,, + v1 â,- c0 ac, S, t),
(18)
where the predicted clean sample ån can be also written based on the score function as
âdf ar: s,t)
vāt a. vI āe Ed(a),
(19)
After replacing Equation 19 into Equation 18, it becomes clear that we can essentially repeat the same process as DDPM and find the Jacobian of the entire process.
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F ADDITIONAL EXPERIMENTS
F.I OFFLINE RI AND EXPERIMENTS
In Figure 14, we display the evaluation graphs for CDP with two different contraction rates, compared against EDP (Kang et al., 2023), which employs diffusion sampling without any notion of contraction. Despite simplistic vector search for contraction hyperparameters (Section 4), CDP consistently improves performance across many environments, while achieving comparable or only slightly lower rewards in others. The only notable exceptions are HalfCheetah-M and Walker2D-ME, where CDP underperforms. We attribute this primarily to the limited scope of our hyperparameter search. Yet, other factors such as the choice of diffusion solver or the number of sampling steps may contribute and warrant further investigation in future work.
Figure 16 and Figure present the average evaluation reward for the Robomimic- Lowdim and Robomimic- Image environments, respectively. For Robomimic-Lowdig tasks, we report results under three settings: limited data, noisy actions, and the standard training process. Across all experimental settings, CDP demonstrates consistent improvement in average reward over the baseline, with particularly strong gains in the more challenging limited-data and noisy-action regimes. These findings mirror observations on D4RL: the benefits of contractive regularization become more pronounced in scenarios where the diffusion process is highly susceptible to score-matching inaccuracies and solver discretization errors, such when data is scarce, actions are noisy, or learning relies on high-dimensional image inputs.
F.2 EFFECT OF DIFFERENT CONTRACTION LOSS WEIGHTS
We examine how different contraction loss weights influence evaluation performance across both RL and II environments, with results summarized in Figure 15. Overall, we observe general trend in which overly strong enforcement of contraction leads drop in performance. To provide additional intuition, Figure 18 presents a toy diffusion modeling example using four log-spaced contraction weights. The resulting distributions of sampled actions across training epochs clearly illustrate how contraction shapes the diffusion sampling process. Lastly, we portray how various contraction weights change the largest eigenvalue during training to better showcase the way the contraction loss affects the model training, especially the decreasing effect on the larges eigen value.
HalfChoetah-M
Hopper-M
Walker-M
Walker&RR
Largest cigenvalue
Concratios niigtt
Figure 13: Largest eigenvalue with contraction. Higher rates of contraction naturally push the largest eigenvalues (along with the rest) to be more negative.
F.3 CDP WITH FEWER SOLVER STEPS
One of the central claims in the main text is that contraction dampens score- matchmnt and solver- related errors, including discretization errors and inaccuracies in score estimation arising in limited- data scenarios. Figure 19 illustrates this effect by comparing samples from contractive and standard diffusion processes at various training epochs under a reduced number of solver steps. Reducing the number of solver steps forces each integration step to cover a larger interval, which amplifies discretization enror and makes the reverse process less faithful to the underlying diffusion dynamics. As a result, the quality of the generated samples typically degrades when fewer steps are used. Nevertheless, the plots show that CDP remains more robust in this setting, producing samples that are consistently closer to the ground-trum -ruth actions, thereby demonstrating its ability to mitigate solver-induced inaccuracies.
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Kitchen Complete
Kitchen Portill
Kitchen Mixed
Animaze MD
Animaze MP
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MR. Mediun Reykas
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6002
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Mediem-Espprt
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Mil Malium Divere
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Figure 14: D4RL evaluation reward and contraction loss graphs. Average evaluation reward and contraction loss are reported across a range of D4RL environments and tasks. We compare CDP under different contraction loss coefficients- the baseline EDP method. To highlight the effect of enforcing contractivity, we also compute the contraction loss for both EDP and CDP rollouts, which clearly illustrates the gap between methods trained with and without contraction. Note that the contraction loss is measured for EDP but not penalized. Finally, we observe that contraction loss typically drops sharply at the beginning of training, but then rebounds and stabilizes as minimizing the diffusion loss becomes impossible with extreme levels of contraction. We plot the error bands addition average.
using the Jacobian loss formulation- aggainst
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Hopper-M
Kilchen-Partial
Antmaze-Me MP
Lin-Lowdan
1.0
Average reward
6.1
weigtts
Figure 15: Performance with changing contraction rates. We report evaluation curves and enror bands for four tasks and environments under varying contraction loss weights All other hyperparameters are held fixed across runs, and for each setting we plot the mean and variance of evaluation performance over 10 random seeds.
Square
Transport
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Normal data
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LOA
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Limited data
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Avernge roward
Noisy data
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Robomimic-Lowwdim evaluation reward graphs. We evaluate the standard Robomimic datasets, as well as two perturbed settings: a limited-data scenario containing only 10% of the original samples, and noisy dataset constructed by adding Gaussian noise with average magnitude 0.1 to the actions. While the performance gains in simpler environments with standard data modest, We plot the error bands in addition to average.
Figure 16:
challenging tasks and the limited/noisy settings highlight clearer advantages.
Lift-Image
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Life-Image
Can Lan-Image
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Figure 17: Robominic-Image evaluation reward graphs. In these environments, the observation space consists of two image modalities from the agentview and wrist cameras, augmented with low-dimensional proprioceptive data. Due computational constraints, we restrict experiments to the Lift and Can tasks. The results suggest that CDP is particularly effective when learning from visual inputs; however, we defer more thorough investigation this setting to future work.
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Epoch: 0
Epoch: 25
Epoch: 75
Epoch: 125
Epoch: 200
Contraction: 1.0
Contraction: 0.1
Contraction
Contraction: 0.001
Qg
Action a
Sampled
truth
Figure 18: CDP sampling for various contraction loss weights. The comparison illustrates the way contraction influences both the diffusion dynamics during training and the final policy performance. In this example, however, the difference between setting 7 = 0.1 and increasing it by an order of magnitude is negligible in practice. Across similar experiments, no universal rule of thumb is identified, but logarithmic hyperparameter search provides a practical and effective strategy.
Epoch:
Epoch: 25
Epoch: 75
Epoch: 125
Epoch: 200
CDP
dgy
Standard
az
2D Action
a
Sampled
Ground truth
Figure 19: Diffusion sampling with fewer steps. Reducing the number of sampling steps forces the solver to take larger updates, which decreases accuracy and amplifies discretization error. In later training epochs, observe that contraction helps dampen these solver inaccuracies and stabilizes the sampling process; however, it cannot fully climinate the errors introduced by coarse discretization.
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Published as a conference paper at ICLR 2026
Antraae
HaifCheetah
Hopper
Walker
Kitchen
Figure 20: D4RI. environments. These environments are used for offline RL. benchmarking.
G DATASETS AND BENCHMARKS
This section provides technical overview of the D4RL and Robomimic datasets used in experiments. We detail the tasks, the nature of the offline data, and the evaluation protocols for policies trained these datasets.
G.1 D4RL: DATASETS FOR DEEP OFFLINE REINFORCHMENT LEARNING
The D4RL benchmark (Fu al., 2020) is collection of standardized tasks and static datasets designed to facilitate reproducible research in offline RL and IL. The core challenge is learn effective policies from fixed batch of data without further interaction with the environment during training. Once trained, policies are evaluated in their corresponding simulated environments. An overview of the environments used in study shown in Figure 20.
Franka Kitchen. This task involves controlling a 9-DoF Franka robot arm to perform a sequence of manipulation tasks in kitchen setting. It is designed to test agent's ability learn long-horizon behaviors from complex, unstructured data. We use the following datasets:
complete: Trajectories demonstrate the successful completion all required subtasks.
partial: Trajectories include the target subtasks, but also contain unrelated actions. agent must identify and imitate the relevant segments.
mixed: No single trajectory completes the full task. The agent must learn to "stitch" together sub-trajectories from different demonstrations succeed.
Gym-MuJoCo. These classic locomotion tasks where the goal is control agents to forward as quickly as possible. We focus on the Half-Cheetah, Hopper, and Walker-2D environments. The datasets are generated from policies varying levels of proficiency:
medium um: Data collected from policy trained to moderate level of performance.
The entire replay buffer content from the training process up to the medium um policy level. This dataset larger and more diverse.
medium-expezt: A dataset composed of 50% medi um data and 50% data from near- optimal expert policy.
AntMaze. this challenging navigation task, an 8-DoF "Ant" agent must navigate a maze to target location. The reward is sparse positive reward is given only upon reaching the goal),
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Lift
Can
Transport
Figure 21: Robomimic tasks. These tasks used in our study for IL benchmarking in robosuite.
requiring the agent to Iearn from long, often unsuccessful, trajectories. The datasets we medi um-diverss and medi um-play, which feature trajectories from a moderately skilled policy attempting to reach various goals within the maze.
Evaluation Protocol. For all D4RL tasks, the trained policy is evaluated by deploying it in the corresponding MuJoCo-based simulator. Performance is measured by the average cumulative reward over several evaluation episodes. For the AntMaze tasks, this corresponds to the success rate of reaching the goal.
G.2 ROBOMIMIC: LEARNING FROM HUMAN DEMONSTRATIONS
Robomimic (Mandlekar et al., 2021) is large-scale benchmark for IL, featuring datasets collected from human teleoperation. The tasks are simulated in the robosuite framework (Zhu al., 2020). Learning from this data is particularly challenging because human demonstrations can be non-Markovian, inconsistent, and of varying quality. The tasks used in our experiments are visualized in Figure 21.
Manipulation Tasks. We evaluate our methods on four tabletop manipulation tasks:
Lift: Pick up cube from table.
Can: Pick up a can and place into a bin.
Square: Pick up a cube and place it inside designated square region.
Transport: Pick up an object from one bin and move to another.
Evaluation Protocol. Policies are trained on the provided human demonstration data and then
evaluated in the robosuit te simulator. The primary metric for all Robomimic tasks is the success rate, defined as the percentage of episodes where the agent successfully completes the task objective. The final performance is reported the mean success rate over 100 evaluation rollouts.
G.3 REAL WORLD DATA WITH FRANKA ROBOT
Information about tasks and how the demonstrations are collected is provided in Appendix B. Here, we provide a general profile the dataset sizes Table
Table 5: Characteristics of the collected real-world datasets. Observations consist of agent-view RGB images and proprioceptive states (end-effector position and orientation in xyz-rpy), while actions correspond to delta end-effector motions in the six-dimensional xy2-rpy.
Task
# Demonstrations
Trajectory Length (min-max)
Max Length
Size (MB)
Slide (cylinder)
51
148-209
209
7989.45
Stack (cubes)
41
182 882-886
286
7908.29
Lift (cube)
107-224
224
3782.33
Peg (insert in hole)
97-296
296
8976.17
36


Reviews:
[{"review_id": "Kqsc2DPo7W", "reviewer": "Reviewer_5Yr9", "summary": "This paper proposes to improve diffusion policy in the continuous control setup by inducing contractive behaviors. While the iterative sampling process of diffusion models encourages diverse outputs, it also hinders action generation in fields that require accurate signals (e.g., robotic control). The method introduces a penalty loss term on the score Jacobian to enhance robustness in the sampling process, as it effectively reduces action variance. The authors show that this offline learning method can improve the diffusion policy baseline on several robotic control problems, especially in the limited training data regime.", "questions": "Does the distillation process (e.g., consistency policy) effectively also induce a contraction? As the reduced sampling steps and the consistency distillation loss might also “pull” nearby diffusion flows closer. I suggest the authors study their proposed loss on distilled diffusion policies as well.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "+ The method is simple and intuitive. The results look promising (especially the real-world experiments).\n+ The authors provide an in-depth theoretical analysis of their proposed method for inducing the contraction of diffusion models.\n+ The work is positioned well in a detailed discussion of related work.", "weaknesses": "+ I am not fully convinced of the idea of unwanted action variance. Granted, excessive noise in the generated actions can lead to failure in continuous control. But the level of variance that is beneficial varies a lot from task to task, as pointed out in the appendix. It would be better if the authors could come up with a more principled tuning method (e.g., a self-adaptive coefficient). In particular, for tasks that require dynamic control (say, soft-body manipulation), it might be hard to justify the contraction-based regularization.\n+ I suspect that it might be harder for the contraction-regularized diffusion model to do transfer learning or further online fine-tuning. Basically, with the reduced action variance, it might be harder to adapt to various action distributions in a post-training manner. While the promise of a large diffusion policy for action generation lies in a foundation model + quick post-training fine-tuning paradigm, I wonder if the proposed approach might hinder this.\n+ The Jacobian (or in general Lipschitz) regularization approach has been popularized by a series of works in DNN generalization [1,2,3]. Please add them to the related work section.\n\nI am willing to raise my score after seeing the authors’ response.\n\n[1] Information-theoretic local minima characterization and regularization, ICML 2020\n\n[2] Sharpness-aware minimization for efficiently improving generalization, ICLR 2021\n\n[3] Understanding Gradient Regularization in Deep Learning: Efficient Finite-Difference Computation and Implicit Bias, ICML 2023", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9pRRQujMHd", "reviewer": "Reviewer_xLAA", "summary": "This paper proposes a novel diffusion policy called contractive diffusion policy. Comprehensive numerical experiments are conducted.", "questions": "The paper is clear to me. No further questions.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The approach has strong theoretical foundations. The authors clearly provide conditions for contractive diffusion policy by leveraging the results in ODE theory and diffusion models.\n2. The numerical results are very comprehensive. The paper presents results across multiple benchmarks, showing the effectiveness of the proposed approach. \n3. Writing is easy to follow.", "weaknesses": "Although I am familiar with diffusion models theory, I am not familiar with diffusion policy benchmarks. I only have one major concern:\n1. In Tang and Zhao (2024), the contractive condition excludes the application of usual diffusion model such as VP SDE. Do you face the same issue in your setup? If yes, what diffusion process are used and how to you compare with usual DDPM-based sampling process?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gE7a70n4gP", "reviewer": "Reviewer_rXr6", "summary": "The paper is improving general diffusion model policies through an additional class of losses that promote contractions in the sampling processes of the diffusion model policy. These losses are denoted Contractive Diffusion Policies (CDPs), which add a Jacobian-based loss so the reverse diffusion dynamics are contractive. The goal of this is to damp solver and score noise and reduce action variance. Experiments are done on D4RL, Robomimic, and a small real-robot setup.", "questions": "- How much action diversity is lost as the contraction weight increases? Is there a way to reconcile contraction-type losses with max-entropy RL?\n\n\n- What is the evidence that diffusion model policies suffer from the same score matching and integration errors that have been observed in e.g. image sampling?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "- Simple idea that integrates into standard diffusion-policy training without major architectural changes.\n- Well-presented theory around contraction for ODE’s and diffusion (or more generally flow-based) generative models.\n- Some positive results on D4RL benchmarks, especially where reducing variance helps.", "weaknesses": "- Contraction seems to fundamentally contradict a central reason practitioners use diffusion model polices over e.g. Gaussian policies, namely multimodality. By pulling trajectories together, CDP can collapse valid modes that diffusion policies are meant to capture. It then seems in this setting that Gaussian policies would need to be benchmarked against, or at the very least explanation provided as to why either (a) multimodality is actually not that important for RL policies, or (b) why the contraction loss does not actually eliminate multimodality when it’s efficient for optimal policies to inhibit this.\n\n\n- There is no direct evidence that the spectral condition for contraction actually holds during training and sampling. Plots of eigenvalues through various time conditioning would give future readers a better idea of the effect of the contraction loss, and what parts of the diffusion model become more strongly contractive.\n\n- As Jacobian computations are typically expensive, a computation time analysis here seems relevant.\n\n\n- While some environments show CDP giving a non-negligible boost, none of the robomimic experiments show CDP giving a clear advantage over other methods. Additionally, some appendix figures are difficult to interpret because means and confidence intervals overlap heavily (e.g. Fig. 15). This makes it hard to assess some practical differences.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WPtq9vNEaq", "reviewer": "Reviewer_T4PE", "summary": "This paper identifies a weakness in modern diffusion policies. The authors argue that the iterative nature of the diffusion sampling process introduces compounding errors from two sources 1) inaccuracy of the learned score function, especially in low-data cases, and 2) discretization errors from the ODE solver. In control and robotics, these small and accumulating errors can lead to failed actions.\n\nTo solve this, the authors introduce Contrastive Diffusion Policy (CDPs). The core idea is to leverage contraction theory to make the reverse diffusion ODE more stable. The contraction theory establishes the relation between the ODE's Jacobian and the ODE. If this condition is satisfied, it can guarantee that the ODE trajectories converge to a more concentrated target distribution. This property makes the learned ODE more concentrated and the sampling process inherently robust to small perturbations like solver or model errors. \n\nThis paper leverages the theoretical analysis of contraction theory for ODE and modifies the original diffusion or flow policy by forcing the ODE's Jacobian to satisfy the contrastive drifting condition. This is learned along with the original flow objective. For the training details, they use the power iteration method to approximate the largest eigenvalue of the score Jacobian. Then a contraction loss is applied to penalize the model to be more concentrated. The final training loss is a simple weighted sum of the original diffusion loss and the new contraction loss. \n\nExperiments on D4RL, Robomimic, and real-world Franka robot tasks demonstrate that CDPs outperform the standard diffusion policy baseline, with particularly strong performance in low-data scenarios.", "questions": "In addition to the questions in the weakness above, I have the following additional questions \n\n- Regarding Stochastic (SDE) Solvers: The paper focuses on stabilizing the deterministic ODE sampling process (and the experiments appear to use the deterministic ODE sampler). However, stochastic samplers (like DDPM) are well-known to have error-correcting properties due to their Langevin noise term, which also tends to concentrate the final distribution.\nCould the authors comment on why an ODE-based solver was chosen over an SDE-based one?\nDid the authors experiment with stochastic samplers? If so, how do they perform? Does the inherent noise of an SDE sampler already provide a similar \"contractive\" effect, making the explicit contraction loss less necessary?\n\n- Regarding Simpler Alternatives (e.g., Data Re-weighting): As an alternative to regularizing the model, could a similar \"concentrating\" effect be achieved through the training data or loss function? For example, one could identify actions that are \"central\" or \"high-quality\" (e.g., near the mean of a mode in the data) and apply a higher loss weight to these samples during training. This would intuitively \"tilt\" the learned score function to flow towards these more robust actions, without the computational cost of Jacobian-vector products. Could the authors comment on the pros and cons of their proposed method versus such a data re-weighting or distribution tilting approach?\n\n- In Table 2 (Robomimic), the proposed CDP (0.78 avg) underperforms the DP-Unet baseline (0.88 avg) significantly. Could the author provide additional results with CDP + Unet to show the improvements over different architectures?\n\n- I'm not quite sure why the method performs better for low-data regimes. if the score field ε_θ is inaccurate due to sparse data, why does forcing its Jacobian to be contractive (based on this inaccurate field) lead to a better outcome? Is there a theoretical explanation for this? \n\n*Typo: line 246 J_{\\epsilon_\\theta}^{sym} seems undefined?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper leverages a solid foundation in contraction theory and gives a formal analysis of the ODE's stability, which is insightful and provides a clear, principled target for regularization\n- The paper has a good motivation. Compounding error is a valid issue in continuous control\n- The paper includes physical robot experiments in the real world, which demonstrates its robustness in practice", "weaknesses": "- The paper describes the added cost as \"minimal\" and \"negligible.\" However, the proposed method requires computing a Jacobian-vector product (K=3 or 4 times for the power iteration) for every item in the batch at every single training step. This is definitely more computationally expensive than a standard diffusion loss. A more transparent analysis (e.g., a wall-clock time comparison) would be helpful to quantify this overhead.\n- The advantage of diffusion models is to capture complex, multi-modal distributions (like the different ways to perform a task). Forcing the system to be \"contractive\" (pulling trajectories together) seems intuitively to contradict preserving this diversity. The paper mentions that \"excessive penalization could fuel a mode collapse,\" but does not investigate this trade-off in-depth. A quantitative study on how $\\gamma$ affects the variance or modality of the final action distribution would make the paper more complete, especially on a task showing multi-modal behavior.\n- The authors acknowledge this limitation of hyperparameter sensitivity. The new contraction loss weight, $\\gamma$, is a critical hyperparameter. As shown in Table 4, the optimal value for $\\gamma$ varies dramatically across different environments (from 0.001 to 100.0). This high sensitivity could make the method difficult to apply to new tasks without a costly hyperparameter sweep", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)