Much Ado About Noising: Dispelling the Myths of Generative Robotic Control
Authors: Chaoyi Pan
Giri Anantharaman
Nai-Chieh Huang
Claire Jin
Daniel Pfrommer
Chenyang Yuan
Frank Permenter
Guannan Qu
Nicholas Matthew Boffi
Guanya Shi
Max Simchowitz
Venue: ICLR 2026 Poster (2026)
Keywords: Generative model
Flow
Control
Behavior cloning
Primary area: applications to robotics, autonomy, planning
TL;DR: We identify why generative models are successful as continuous control policies, and introduce a minimal policy parametrization which replicates their success.

Abstract: Generative models, like flows and diffusions, have recently emerged as popular and efficacious policy parameterizations in robotics. There has been much speculation as to the factors underlying their successes, ranging from capturing multimodal action distributions to expressing more complex behaviors. In this work, we perform a comprehensive evaluation of popular generative control policies (GCPs) on common behavior cloning (BC) benchmarks. We find that GCPs do not owe their success to their ability to capture multimodality or to express more complex observation-to-action mappings. Instead, we find that their advantage stems from iterative computation, provided that intermediate steps are supervised during training and this supervision is paired with a suitable level of stochasticity. As a validation of our findings, we show that a minimal iterative policy (MIP), a lightweight two-step regression-based policy, essentially matches the performance of flow GCPs. Our results suggest that the distribution-fitting component of GCPs is less salient than commonly believed and point toward new design spaces focusing solely on control performance. Videos and supplementary materials are available at https://anonymous.4open.science/w/mip-anonymous/.

Published conference paper at ICLR 2026
MUCH ADO ABOUT NOISING: DISPELLING THE MYTHS OF GENERATIVE ROBOTIC CONTROL
Chaoyi Pan', Giri Anantharaman' Nai-Chieh Huang', Claire Jin', Daniel Pfrommer', Chenyang Yuan", Frank Permenter', Guannan Qu Nicholas Boffi* Guanya Shi?.:, Max Simchowitz 1,
CCrmegie Mellon University, *Massachusetts Institute of Technology, 3 Toyota Research Institute EEqua advising.
(chaoyip, p, giria, naichieh, [lairee)@andree edu igqu, nboffi, gshi, ms msimchow) (andrew cmu.edu dpfromemit. ichenyang, yuan, frank, paame.eerenter| Stri glo.all
ABSTRACT
Generative models, like flows and diffusions, have recently emerged popular and efficacious policy parameterizations robotics. There has been much spec- ulation as the factors underlying their successes, ranging from capturing multi- modal action distribution to expressing more complex behaviors. In this work, perform comprehensive evaluation of popular generative control policies (GCPs) on common behavior cloning (BC) benchmarks. We find that GCPs do their success their ability capture multi-modality complex observation-to-action mappings. Instead, we find that their advantage stems from iterative computation, long intermediate steps supervised during training and this supervision is paired with suitable level of stochasticity. As validation of our findings, show that minimal iterative policy (MIP), a lightweight two-step regression-basee policy, essentially matches the performance of flow GCPs, and often outperforms distilled shortcut models. Our results sug- gest that the distribution-fiim fittinng component of GCPs is less salient than commonly believed, and point toward new design spaces focusing solely on control perfor- mance. Code and supplementary materials are available at our website.
express
INTRODUCTION
Why GCPs are better
Taxonomy GCPs
Minimal Policy that Matches Flow
Sistributions
Hypothesis
Finding
Learning
Relation Success
1.02
pixel-based control
XXXXXX
0.74
0.74
multi-modal data
ACT
BET
expressiviity
Flom
representation learning
SF
training atability
0aB
learning
Regresstoon
Regressson
12'
AR
KLPP
F2om
C2. Stochastieity
OOO O00 O0
Injection Iterative Compute
Figure 1: Left: After careful ablation on each component over 28 common hehavior cloning henchmarks with diverse input modalities (state, pixel, point cloud and language), architectures (with both raw and pre-trained models, like my) and tasks (standard single-task benchmarks and multi-task benchmark like LIBERO), fute number popularly held misconceptions about why generative control policies (GCPs) outperform regression policies (RCP) on these tasks. Center: We identify that the most important factor contributing to GCP success is combination of stochastic injection (C2) and supervised iterative computation (C3). Sur- prisingly, distribution learning (CI) is the least important factor, due to the absence of learned multi-m multi-modality (Section 3.2). Right: The average relative success rate flow of 7 most challenging tasks. propose simple two-step minimal iterative policy (MIP) whose performance matches that of flow-based GCPs.

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Published conference paper at ICLR 2026
Long-horizon, dexterous manipulation tasks such as furniture assembly, food preparation, and man- ufacturing have been a holy grail in robotics. Recent large rohot action models (Team et al., 2025; Black et al., 2024; Kim et al., 2024) have made substantial breakthroughs towards these goals by imitating expert demonstrations of diverse qualities. We provide more comprehensive review of related work in Section A, but highlight here key trend: while supervised learning from demonstra- tion, also known behavior cloning (BC), has been applied across domains for decades (Pomerleau, 1988), its recent success in robotic manipulation has coincided with the adoption of what we term generative control policies (GCPs): robotic control policies that use generative modeling architec- tures, such as diffusion models, flow models, and autoregressive transformers, as parameterizations of the mapping from observation to action. Given the seemingly transformative nature of GCPs for robot learning, there has been much speculation about the origin of their superior performance rela- tive to policies trained with regression loss, henceforth regression control policies (RCPs). GCPs, by modeling conditional distributions over actions, are uniquely suited to the multi-task pretraining paradigm popular in today's 's large robotic models. However, a number of hypotheses regarding the superiority of GCPs pertain even in the single task setting (Chi et al., 2023; Reuss et al., 2023):
HI. Better performance on pixel- pixel-based control
H2. Capturing multi-modality
the training data
H3. Greater expressivity due to iterative computation of the ohservation-to-actin mapping
H4. Representation learning due to stochastic data augmentation
H5. Improved training stability and scalability
The gap between generative modeling and generative control. The objective for generative mod- eling in text and image domains is fundamentally different from the goal in control task. In the former, one aims to generate high-quality and diverse samples from the original data distribution. In the latter, it suffices to select any action that leads to better downstream performance. Whereas much of the generative modeling literature has focused on the distribution of the generated variable (Lee et al., 2023), we aim to understand it is necessary to reproduce the expert data distribution- for example by capturing any multi-modality-to attain strong control performance. If not, is most salient to capture about the conditioning relationship mapping o + a?
Contributions. This paper adopts careful experimental methodology to rigorously test the key de- sign components (Section 4) that contribute to the observed success of GCPs, and to account for the key mechanisms by which they contribute to improved performance in behavior cloning (Section 5). We restrict our study to flow-hased GCPs, given their popularity and adoption in industry (Black al., 2024; Physical Intelligence et al., 2025; NVIDIA al., 2025).
We begin by first identifying which factors do not contribute to the advantage of GCPs over RCPs.
Contribution 1 (Neither multi-modality nor policy expressivity account for GCPs' success, Section 3). Through careful benchmarking, we show that RCPs with appropriate architectures are highly competitive on both state- and image-based (H1) robot learning benchmarks as well as vision- language-action (VLA) model finetuning (Section 3.1). Performance gaps only arise on certain tasks requiring high precision. However, we show that neither multi-modality (H2, Section 3.2) nor the ability to express more complex functions via multiple integration steps (H3, Section 3.3) satisfac- torily accounts for this phenomenon. In fact, GCPs do not even provide greater trajectory diversity compared to RCPs (Section l).
Essential to this finding controlling for architecture: to our knowledge, we are the first work to carefully benchmark expressive architectures popularized for Diffusion (Chi et al., 2023; Dasari et al., 2024) as regression policies. To determine what contributes to GCPs performance on these high-precision tasks (beyond architectural optimization), we parse the design space of generative control policies into three components, depicted in Figure (left).
Contribution 2 (Exposing the design space of GCPs, Section 4). We introduce a novel taxonomy that parses the three essential design components of GCPs:
C1. Distributional Learning: matching a conditional distribution of actions given observations.
C2. Stochasticity Injection: injecting noise during training to improve the learning dynamics.
C3. Supervised Iterative Computation: generating output with multiple steps, each of which ceives supervision during training.
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With this taxonomy in hand, Section 4.1 introduces a family of algorithms, each of which lies along a spectrum between GCPs and RCPs by exhibiting different combinations of the above components. While we find that neither C2 nor C3 in isolation improve over regression, we find their combination yields policy whose performance is competitive with flow, leading to our next contribution.
Contribution 3 (MIP: the power of C2+C3, Sections 4.1 and 4.2). As an algorithmic ablation that only combines C2+C3, we devise a minimal iterative policy (MIP), which invokes only two itera- tions, one-step of stochasticity during training, and deterministic inference. Despite its simplicity, MIP essentially matches the performance of flow-based GCPs across state-, pixel- and 3D point- cloud-base BC tasks, exposing that the combination of C2+C3 is responsible for the observed success of GCPs. In addition,
find that MIP often outperforms shorteut/few-step policies (Section D.8). This confirms our findings that distributional learning (which few-step policies, but not MIP, achieve) not needed in robotic control.
As described in Section D.8, MIP is substantively distinct from flow-map-basee basp- models (Boffi et al., 2025a;b), including consistency models (Song et al., 2023; Kim et al., 2023) and their ex- tensions (Geng et al., 2025; Frans et al., 2024), in that the latter do satisfy CI, and require training over a continuum of noise levels.
Contribution 4 (Attributing the benefits of C2+C3, Section 5). We identify that a property we term manifold adherence captures the inductive bias of GCPs and MIP relative to RCPs, even in the absence of lower validation loss. We explain how this property is useful proxy for closed-loop performance in control tasks. Finally, we expose how C3, through iterative computation, encourages manifold adherence, but only if stochasticity during training (C2) is present to mitigate compounding errors across iteration steps (as described in Section 5.2).
Manifold adherence in Section 5.1 measures the generated action's plausibility given out of distri- bution observations, where only off-manifold component is evaluated rather than the distance the neighbors (Pari et al., 2021). Note that manifold adherence reflects favorable inductive bias during learning, rather than brute expressivity of more complex behavior (H3). Moreover, C2 provides more of supporting role to C3, rather than enhancing data-augmentation in its own right (H4). In addition, we find that C2+C3 also enhance scaling behavior (H5), likely due to better model uti- lization through decoupling across iterations. Finally, identify that the subtle interplay between architecture choice, policy parameterization and task can affect performance by an even greater magnitude than the choice of policy parametrization (Section 5.3).
Takeaway. In robotic applications, our findings suggest that the distributional formulation of GCPs sampling from distribution of actions given observations is the least important facet that contributes to their success. Rather, our work highlights that C2+C3 offer an exciting and under- explored sandbox for future algorithm design in continuous control and beyond.
2 PRELIMINARIES
We consider a continuous control setting with observations E O and actions E A where O is the observation space and A the action space. We learn a policy x : O + ((A) from observations to (distributions over) actions to maximize the probability of success J(T) we refer to as "performance." is, supervised learning from a distribution of (observation, actions pairs) drawn from training distribution Ptrain. In applications, the actions a are often a short-open loop sequence of actions, or action-chunks, which have been shown to work more effectively for complex tasks with end-effector position commands (Zhao et al., 2023b). See Section A for an unabridged related work.
given task, which We consider the performance of policies learned via BC-thar
Regression Control Policies (RCPs). A historically common policy choice for BC is regression control policies (RCPs) (Pomerleau, 1988; Bain & Sammut, 1995; Ross et al., 2011; Osa et al., 2018), 1018) given by deterministic map π O A. In applications, it is parameterized by neural network xy and trained so as to minimize the Lz-loss on training data:
my arg ming (l) all2; (o, a) Ptrain-
(2.1)
Generative Control Policies (GCPs), Generative control policies (GCPs) parameterize a distribu- tion of actions given observation o. This is often accomplished in practice by representing the
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Relative
Best
0.8
LIGERO Spacial (VLA)
LIB800 6001 (VL4)
LIBERo object (VLA)
LIBERO (VLA)
Square (19, State)
square (PN, Steto)
Push (Stato)
Lift (199, Stete)
Life (99, State)
kitchan (Stete)
Can (PN, State)
Tramport (99, Image)
can (mN. State)
Push (Image)
Laft (P9. Image)
can (PH, Imspo)
can (991, Image)
Lafs (999 Insge)
square (PH, Ieago)
Squar (m9, Image)
Transport (e9. Imagu)
Stick -Pash (30)
Adrait poor (30)
Adrost Hammer (30)
Arsumbly (30)
bisuesumt (90)
Tonl Hong (PH. State)
Adriot Pen (30)
Transport (PI, State)
Transport (10, State)
Tool Hang (PI. Image]
Figure 2: Relative performance of RCPs compared to GCPs across common benchmarks, For single-task Sudeep-DiT and Chi For each architecture, we average performance of the best training checkpoint across seeds. For multi-lask benchmarks, we use wo as base policy and finetune it on full LIBERO benchmark (130 tasks). We then report the performance of the best- performing architecture, chosen individually for both RCPs and GCPs. For Flow, we always do 9 step Euler integrations, where its performance plateaued. For readability, RCPs success rates are plotted relative to flow, with flow normalized to performance 1 per task. Tasks are grouped by observation modality, and ordered by relative RCPs performance. Red dashed line indicates threshold at which RCP attains < 95% success of GCPs. Note that RP and Flow perform comparably on most Image, 3D-based and VLA-based multi-task benchmarks.
benchmarks. implement Chi -rrn--rranooomnno
policy Ty with a generative model such as a diffusion (Chi et al., 2023), flow (Zhang et al., 2024), or tokenized autoregressive transformer (Shafiullah et al., 2022). Given their popularity, we focus on flow-based GCPs (flow-GCPs). A flow-GCP learns conditional flow field (Lipman et al., 2023; Chisari et al., 2024; Nguyen et al., 2025; Albergo & Vanden- Bijnden, 2022; Heitz et al., 2023; Liu et al., 2022) b: [o, 1] x A x O A by minimizing the objective
by R arg ming E (1 0) 1.12,, ~~ Unif([O, 1)), N(O,
(2.2)
where again (o,a) ~ Perains I to + (1 - f) is the stochastic interpolant between the training action a and noise variable z, and where 1, = a - z is the time derivative of 4.. We note that this is special case of the stochastic interpolant framework (Albergo & Vanden-Fijnden, 2022; Albergo et al., 2023; 2024), which permits a larger menu of design decisions. A flow model then predicts an action by integrating flow. In the limit of infinite discretization steps. this amounts to sampling a ~ To(- o) by sampling = ~ N(O, I), and then setting a = 01, where {ax (4+)< solves the ODE:
d de a = (a o) with initial condition
ag = 2.
(2.3)
In practical implementation, sampling is conducted via discretized Euler integration (see Sec- tion M.2 for details). This yields a policy a = Te T0(2,0) o) which is a deterministic function of the initial noise z and the observation o. All experiments, unless otherwise stated, perform 9 integration steps. We reiterate that other GCPs, e.g. based on diffusion models and autoregressive transform- ers, have been studied elsewhere. We choose to focus on flow models due to their sate-of-the-an art performance (Chi et al., 2023; Chisari et al., 2024; Zhang et al., 2024) and deployment in industry (Black et al., 2024; Physical Intelligence et al., 2025; NVIDIA et al., 2025).
Multi-Modality in Robot Learning. Past work has conjectured that for salient robotic control tasks, Perain (a o) exhibit multi-modality, i.e. the conditional distribution of given o has multiple modes (Shafiullah et al., 2022: Zhao et al., 2023b; Florence et al., 2022). This motivated the earliest use of GCPs (Chi et al., 2023) (H2). Section 3.2 calls into question the extent to which GCPs do in fact learn modti- distributions of a on popular benchmarks.
3 MULTI-MODALITY AND EXPRESSIVITY DO NOT EXPLAIN GCPs' PERFORMANCE
This section demonstrates that neither advantages on pixel-based control (HI), nor multi-modality (H2), nor improved expressivity (H3) fully account for the GCPs performance relative to RCPs. In- stead, our analysis indicates that the advantage of GCPs is largely due to architectural innovations found in GCPs spsssfiicclllll... the adoption of powerful models like Transformers and UNets, along with the use of action chunking techniques. Sections and addresses other hypotheses, such as k-nearest neighbor approximation and the behavior diversity.
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3.1 WHEN CONTROLLED FOR ARCHITECTURE, GCPS ONLY OUTPERFORM ON FEW TASKS
We first isolate the tasks in which GCPs exhibit stronger performance by comparing across 28 pop- ular BC benchmarks including multi-task benchmarks like LIBERO (detailed in Section D.1), en- compassing diverse data quality, modalities (state, point clouds, image and language), and domains (e.g., MetaWorle ld, Robomim immi Adroit, it, D4RL, Met.a World, LIBERO). Crucially, we im- plement RCPs using the exact same architectures as their corresponding flow models by simply setting the noise level and initial noise to zero: = 0, t = O, and study three widely-used architec- tures -Chi-Trans former, Seneerruudeee Sudeep-DiT, Chi-UNe To (Black et al., 2024); detailed in Section D.2). This architectural alignment enables RCPs to ben- efit from the sophisticated network designs typically reserved for GCPs, ensuring a fair comparison.
as well as pre-trained VLA models like
Under controlled comparison, we find GCPs and RCPs achieve parity across the vast majority of BC benchmarks. Performance gaps emerge only on a small subset of tasks requiring high precision (e.g. precise insertion tasks). We report best-case results in Fig. 2 and comprehensive ablations (including worst-case architectures and loss variants) in Section D.4.
Our evaluation yields three key insights: (a)Rare Benefit of GCPs: GCPs outperform RCPs by 5% on only a handful of tasks. (b)Modality Independence: Contrary to popular belief, observa- tion modality does not correlate with GCP advantage. (c)Architectural Dominance: Architecture choice dictates performance far more than the generative vs. regression distinction.
We posit that the perceived superiority of GCPs in prior work was confounded by architectural asymmetry. To our knowledge, this the first study to benchmark Sudeep-DiT, Chi-UNet, and To backbones regression policies. In Section 5.3, we demonstrate that when equipped with these modern backbones- oo even tuned MLP baselines-RCPs are highly competitive. Furthermore, we find that hyperparameters such action-chunking horizon (Zhao et al., 2023a; Chi et al., 2023; Zhang et al., 2025) exert a greater influence on success rate than the choice of objective function (Section H.1). Design decisions like architecture and action-chunking have significant and consistent impact on control performance. In contrast, the choice between GCPs and RCPs is largely negligible outside of high-precision regimes.
3.2 GCPS' PERFORMANCE DOES NOT ARISE FROM MODAIIY
Earlier literature suggested that capturing multi modii-modality, as defined in Section 2, was precisely the root of the observed performance benefits of GCPs (Chi et al., 2023; Reuss et al., 2023). However, examining Fig. 2, we see that many tasks which have been understood to be multimodal (e.g., Push- T) do not show substantial performance gaps between RCPs and GCPs. On the other hand, RCPs and GCPs differ only on tasks that demand high precision (e.g. Tool-Hang, Transport). In this section, we provide additional evidence that multimodality is not the main factor responsible for witnessed performance ad advatteges of GCPs
1.0
Task
N(O, T)
Mean
Dataset
Flow
Reg.
0.5
Push-T 0.97
0.97
0.95
Original 0.78
0.58
Kitchen
0.99
0.97
Deterministic
(a)Push (b (b)KCitchen (c)Tool Hang Q
Tool-Hlang
0.78
0.80
0.76
Table 2: C. GCPs outperforms
Visualized action distribution with Q values. Dis- tinct modes are not observed in planned actions at symmetric and ambiguous states. (Kitchen and Too L-ang, I-SNE visualiza- tion.) In Push-T, we all trajecto- ries goes to one side. For the rest, there is no clear clustering of tions or Q.
Figure 3:
Table 1: B. Performance compar- ison of different sampling strate- gics. We compare sampling = 0.
RCPs with deterministic experts. Policy average success rate over 3 architectures, 3 seeds and 3 archi- tectures given different dataset: one from original human demonstration and another collected by rolling out flow policy in deterministic mode starting from zero noise.
z ~ N(O, 1). and mean over 64 2(0) N(O, Different sam- pling strategics show minor per- formance difference, indicating ab- sence of distinct action modes.
Evidence A: GCPs exhibit unstructured action distributions. For fixed observations, we draw multiple action samples by denoising from different initial latents and visualize the resulting action set with their Q values Q(a, o). We deliberately choose symmetry-critical or high-ambiguity states to maximize potential multi-modality: (a) Push-T the symmetry axis of the T-shape, where taking
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the left or right path is equivalent, (b) Kitchen from an initial state with multiple first-subtask choices, and (c) Too T-ang at the insertion pre-contact pose where human demonstrators pause for varying durations.
(a-c) we observe single clusters rather than distinet modes (high-dimensional actions visualized with t-SNE); see Fig. 3. Moreover, adherence to action cluster means do not correlate with performance: We color-code actions by Q-value, i.c. Monte-Carlo-estmmate rewards- to-go (Section F.1). Highest returns are distributed evenly across samples.
Evidence B: Taking mean actions does not meaningfully degrade GCPs' performance. We evaluate flow policy's 's performance with three sampling strategies: zero noise a T(2 0),,, stochastic sampling a = π (2, o), 2 ~ N(0, I). and mean action a = Eg= (2, o)] (via Monte Carlo approximation). If the learned distribution were multi-modal, or if their distributions lied on a manifold whose curvature was crucial to task success, the conditional mean would collapse modes and severely degrade performance. However, Table I shows that replacing stochastic sampling with the mean action only slightly affects performance, indicating absence of distinct action modes.
Evidence C: GCPs outperform RCPs certain tasks even with deterministic experts. To fully remove any residual multi- modti-mod,lity, we recollect the dataset with trained flow policy evaluated in deterministic mode
0) detailed in Section F.2. The new dataset fully deterministic because action labels are provided by deterministic policy evaluated in deterministic environment. While the gap performance between GCPs and RCPs shrinks somewhat, we still find that GCPs still outperforms RCPs, as in Table 2, suggesting that capturing some "hidden" stochasticity or multi- modality in the data does not suffice to explain the gap between the two. Collectively, (A) indicate that the commonly cited explanation-""GCP win because demonstrations are multi- doeel" not hold for most studied behavior cloning benchmarks.
Multi-modality and data coverage. The absence of observed multimodality is likely attributable to the large observation dimension of tasks relative to total number of demonstrations. That is, rarely see two "confficting" actions for nearby observation vectors (note: to grid space of dimension d requires 2d points). Some degree of "hidden" multi-modality may still be present, as indicated by the slight narrowing of the performance gap in Table 2. Still, our central claim that multi-modality is not sufficient to explain the full difference in performance.
3.3 LIMITATIONS OF THE EXPRESSIVITY OF GCPs IN THE ABSENCE OF MULTIMODALITY
alternative to learning explicit multimodality is to represent rapid transition between actions as the observation changes. This is depicted in Figure 6, where data that appears multi-mode can be fit with a policy that has a high Lipschitz constant, i.c. in which Vo (a o) is large. This reflects broader principle in control that need only capture the mapping from observation to a single effective action, rather than reproduce the distribution over all possible actions.
One may still conjecture that GCPs more easily higher-Lipschin policies by leveraging iterative computation, as compared to RCPs. This is because deeper networks can express larger-Lipschitz functions more easily (Telgarsky, 2016), and many have equated the multi-step computation in flow- based generative models to depth (Chen et al., 2018). Step-by-step generation is known to drastically increase expressivity in other domains as well, such as autoregressive language models (Li et al., 2024),
However, flow-based generative models use their multi-step computation express complex dis- tributions over the generated variable (Ho et al., 2020; Song et al., 2021a; Zhang & Chen, 2022; Nichol & Dhariwal, 2021). It is less clear if the iteration computation assists with represent complex observation--to-action mappings. Thus, we ask: Does the iterative computation in GCPs aid in learning more complex observation-to-acte tions for fixed observation are themselves are relatively simple (i.e. unimodal)?
mappings, even if the learned action distribu-
We now provide evidence that suggests "no." We show that in the absence of multi molliimmddlltyy (as shown in Section 3.2). GCPs cannot express more complex mappings from the conditioning variable o to the generated variable a than RCPs can. We begin by considering a ground -tuut conditional flow field by (o a). Let nă o) represent the exactly integrated 6* from initial noise to generated variable a. Given the absence of modality (Section 3.2), we assume that the distribution of is s--lg-concave (Section J), satisfied by many classical unimodal distributions. We prove

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that the Lipschitz constant of Tz(2,,) o) with respect to o, a measure of the expressivity of the o + mapping, is bounded by that of b;:
Theorem (Informal). Let || denote either the matrix operator or Frobenius norm, and suppose that the distribution of o is k-log-concave. Moreover, suppose that the flow field by (a o) is L- Lipschitz: V (a o)l I L. Then, with infinite integration steps, [[ S L vi+1
See Section for formal statement and proof. A classical example of log concave distribution is o ~ N(u(o). )) as long as the variance 1/k: is bounded above (even in the limit of a Dirac), there is at most a constant- munstaattmulliplicatty factor increase in the Lipschitz constant. When training a flow, by (a o) is approximated by the neural network. Thus, in the prototypical unimodal example of re-log-concavy distributions, GCPs are not arbitrarily more expressive than RCPs. In fewer words: more integration steps (i.e. more iterative computation), even infinitely many, need not enable greater expressivity of high Lipschitz o + mappings.
To verify our theoretical prediction, we quan- tify learned policies' Lipchitz constants with a zeroth-order proxy: starting from dataset states 8, with observation Oz, we inject small Gaus- sian perturbations in the executed action to reach a feasible nearby state (())
Method
Push-7
Kitchhn
I-Ran
State
Image
State
State
Image
Regression Flow
14.07
1.65
0.45
0.20
12.43
1.41
1.37
8t41 with observa- tion 0/ then measure input-outpun sensitivity via finite differences of the policy around the perturbed states (full algorithm and per-architecture sults in Section E). This construction (i) avoids reliance on noisy higher-order gradients in complex architectures, and (ii) keeps evaluations on feasible observation to prevent conflating expressivity with model error on dynamically infeasible states. As predicted by our theory, GCPs are not strictly more expressive than RCPs shown in Table 3. On the contrary, RCPs show increased L Lipschitz constants off the manifold of training data, ruling out the assumption that GCPs win due to express- the input variable. We note that methodology, which perturbs actions rather than states, is compatible pixel observations. To summarize: In the absence of multimodality, GCPs do not enjoy an advantage over RCPs in expressing high Lipschitz behavior, such as rapid transitions between action modes.
Table 3: Policy Lipsehitz constant comparison. Lip- schitz constant is averaged over 100 states.
ing policies with greater sensitivity
4 MINIMAL ITERATIVE POLICY (MIP): ISOLATING THE SOURCE OF GCPs' SUCCESS
In this section, we introduce number of intermediates between RCPs and GCPs that isolate which design decisions contribute to the latter's superior performance. This leads to a Minimal Iterative Policy (MIP), which matches GCPs performance, thereby identifying the source of GCPs' success.
We begin with taxonomy of the three key algorithmic components (Figure 1) present in GCPs. Section 4.1 below proposes algorithmic variants which ablate these components. We find that min- imal iterative policy (MIP, Components 2 and 3) is the reduced variant which matches the perfor- mance of flow (Section 4.2), whereas other variants match or perform worse than regression.
Component 1. Distributional learning denotes training model to fit a conditional distribution a ~ (o) of actions given observations, as opposed to deterministic predictions (i.c., a = (o)))).
Component 2. Stochasticity injection denotes the injection of additional stochastic inputs into the neural network during training time (c.g., the variable z in Eq. (2.2)).
Component 3. Supervised Iterative Computation (SIC) denotes the iterative refinement of predic- tions by feeding the previous outputs into the same network again during inference, and providing supervision signals at every step of the generation procedure at training time. For example, in flow GCPs, we integrate a supervised flow field (a, o) over time to get the final action a, and that b, receives an independent supervisory signal for each at training time (Bq. (2.2)).
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4.1 MIP: A MINIMAL INTERMEDIATE BETWEEN RCPs AND GCPS
We introduce range of policies which lie along the spectrum between RCP and flow-based GCPs via varying combinations of Components and 3, culminating in the Minimal Iterative Policy (MIP). These policies do not satisfy Component 1, because Sections 3.2 and I suggests that this is not needed. In particular, we consider networks To (o, I,,1) 1) that predict actions, not velocities, and given observations o, time indices t, and interpolants I corresponding to noising actions. We state all networks below of L2 minimization, but our findings remain consistent when minimizing L error instead (Section H.2).
Two-Step Denoising. As a simplification of flow-based GCPs, we consider a two-step denoising (TSD) policy. As discussed in Section D.8, this parametrization is superficially similar to. but substantively different than, popular TSD performs two steps of denoising, one from zero, and a second from a fixed index t, .9:
models (Boffi et al., 2025b).
" = arg ming E [To Iu 2, 0) (t.)
[l( .t) - a) (4.1)
where a) ~ Perain; 2 N(O, I), and ta + (1 - 1)= the same interpolant used in flow models, and where 4 = -9 is fixed. The normalization by f. in Eq. (4.1) comes from the identity t_a = E [..]. We then sample ogso +- n 2, 0) and à TSD <- t,, + (1 - t.),,,t...
Minimal Iterative Policy. We find that π TSD performs equivalently to a minimal policy which only adds training noise in the second step and has no stochasticity at inference time, which we call the minimal iterative policy.
Minimal Iterative Policy (MIP; ours)
Minimal Iterative Policy (MIP), representing Components 2 and 3, is trained via
MIP n
R arg min [[] (T)( To = 0, 0) a)ll + [o (,,44) - a))l))
(4.2)
0
where (o, a) Perain; & N(O. 1), 4, == .9. At inference time, we compute:
MIP MIP + no (0,0, 0, 0), MIP - n (o, t.ag MIP t.).
(4.3)
Minimal iterative policy provides a minimal implementation that still exhibits competitive perfor- mance with flow. Starting, with TSD and replace (t.)- in the first term of the loss in Eq. (4.1) with its expectation a = -EE,) We set the initial noise Io = 0 to be zero, so that 2 only contributes to the second training loss. Finally, stochasticity at training time, without stochasticity at inference time (c.f. Table 1). Since we provide supervision for both first step = (o, Io 0, L = 0) and second step MIPP (o, It, L (.) with ground truth action a, MIP also exemplifies SIC in its simplest form. We compare MIP to Shorctu Models in Section D.S.
sample with = 0 to isolate the effect of adding
Additional methods. Straight Flow (SF, ours), representing only Component 2, further simplifies MIP a single stage by setting the interpolation index t* 1 and removing the second term: TsE R E [| 2, -) - 2 Finally, we study residual regression (RR), which replaces I. in Eq. (4.2) with its expectation over z: E[[,,] 44a. This preserves SIC (Component 3) yet removes stochasticity injection. Full details are provided in Section C.
To summarize, minimal iterative policy (MIP), straight-flow (SF) and residual regression (RR) rep- resent all combinations of Components and without exhibiting Component
4.2 COMPONENTS 2 AND DRIVE PERFORMANCE: MIP MATCHES FLOW
Based on the design space parsing in Section 4, we are able to systematically ablate different de- sign components' contribution to the final performance in Fig. 4 and Table 9, Our evaluation shows that either stochasticity injection (Component 2, exhibit by SF) supervised iterative computation (Component 3, exhibited by RR) in isolation do not match the success of GCPs. MIP, being the
Note that Component refers to training a model to fit a conditional distribution, not necessarily to the sampling. For example, training by via flow model but conducting deterministic inference with Φ0 0 o) is still considered distributional learning.
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Average Rel Relative Success
1.8
Best Rel Relative Success
1.0
0.8
0.0
Tool. Hang (PH. 5.)
Transport (PH, S.
Transport (MH,
Push T (5.)
Tool Hang (PH,
Ki tcthen (5.)
Transport (PH, I.)
Toet Hang (PH. 5.)
Transport (PH.
Transport (MH, 5.1,
Push
Tool Hang (PH. I.)
Ki Kitchen (S (S.)
Transport (PH, 1.)
Regression
Straight Flow
Residual Regression
MIP
Flau
Figure 4: Performance comparison between MIP and its variants on single-task benchmarks. Average relative success rate worst architecture and the best relative success rate on optimal architecture reported. "S": state: "I" image.
only method which combines supervised iterative computation and stochasticity injection, achieves success on par with flow. Thus we conclude: the performance of GCPs comes from combining stochastic injection and iterative computation. Distributional training appears to be the least impor- tant factor. To further rule out the effect of distributional training and demonstrate the computation efficiency of MIP, we compare it with consistency models in Section D.8, where we find that MIP matches the performance of common consistency models with half of the training time.
Remark 4.1. Section C.3 exhibits two further variants which preserve Components 2 and 3: one that does not supervise intermediate steps, and second which does not condition a time step t.. The latter does not enable network to learn separate functions across time steps. Both perform even worse than regression, confirming the importance of supervision of intermediate steps and decoupling network behavior across time steps.
5 INDUCTIVE BIAS, NOT EXPRESSIVITY, EXPLAINS MIP'S PERFORMANCE
5.1 MANIFOLD ADHERENCE, NOT RECONSTRUCTION, DRIVES PERFORMANCE
MIP, and the absence of multimodality, sug- gest better ability approximate the expert more accurately on training data. We test this by evaluating the Ly-error, i.c., reconstruction error, on validation set. Surprisingly, we find that MIP, Flow, and RCP exhibit the same val- idation loss; hence validation loss does predict their relative performance. Section G.I reveals
Metric
Regression
SP
RR
MIP
Flow
Off-manifold La
0.067
0.063
0.062
0.054
0.042
Validation Le
0,290
0.234
0.224 0.195
0.217
Table 4: Comparison of different methods on mani- fold adherence and reconstruction error. Results are averaged across 3 different architectures and 32 states state-based Fooi-Hang with deterministic dataset. that validation loss doesn't It correlate with performance across other axes of variation. Indeed, policy performance requires taking good actions on o.o.d. states under compounding error at deployment time (Simchowitz et al., 2025).
Thus, we study a proxy which reflects performance in o.o.d. situations. We perturb expert trajecto- ries in dataset described in Section 1, and evalute novel metric that call the off-manifold norm. Informally, this measures the projection error of a predicted action a onto the space spanned by expert actions neighboring states; see Section G.2 for formal definition. Our metric assesses the quality of actions under simulated compounding error. Table 4 reports both Le validation loss and off-manifold L2 norm for different methods: while all methods achieve low validation loss, only MIP and Flow are able to achieve low off manifold L2 norm, indicating their better manifold adherence. As SF does not exhibit the same benefit, we conclude that supervised iterative com- putation facilitates projection onto the manifold of expert actions by refining the prediction across sequential steps. Section I. provides additional confirmation of this hypothesis on comprehensive toy experiments: GCPs are better than RCP fitting high frequency functions, but exhibit lower on-manifold error, suitably defined.
Why manifold adherence matters for control. We conjecture that, for high-precision tasks, the sensitivity to errors is not homogeneous across eтror directions in action space. Our findings present preliminary evidence that some form "on-manifold inductive bias" directly aligns with minimiz- ing error along relevant directions, yet is permissive to error in directions of lesser consequence. We think that rigorously establishing this hypothesis is an exicting direction for future work.

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No known mechanism accounts for greater manifold adherence in GCPs vs. RCPs. There is a growing body of literature that shows that, if training data are supported on a given low dimensional manifold M, then generative models learn to project onto M (Boffi et al., 2024; Permenter & Yuan, 2024). However, to knowledge. there is no work that explains why this inductive bias would be stronger than what would be achieved with a well-trained regression model. Specifically, if lies in some (local) manifold, regression too should learn to project onto it.
One might conjecture that the iterative computation provides many changes to predict an action that "stick" to the action manifold. However, such mechanism would require that once an on-manifold action is predicted, subsequent predictions do not nudge the prediction off- manifold. In Section K, show that simple arguments based implicit regularization in linear models do not suffice to explain this hypothesis, at least for MIP. Much like the usefulness of manifold adherence for control described above, the mechanism behind manifold adherence remains mystery for future study.
5.2 STOCHASTICITY STABILIZES ITERATIVE COMPUTATION
We recall from Figure 4 that SF matches regression, whilst RR under under-perform performs regression. This suggests that sequential action generation highly brittle in the absence of stochasticity (Permenter & Yuan, 2024). Our findings support the hypothesis that stochasticity injection serves to provide "coverage" of the generative process. Note that this is different from task MDP-level augmentation like image augmentation exploratory data collection since the augmentation happens in iterative generative process. Specifically, we can think of learning to perform two-stage action generation as an "internal" behavior cloning problem (Ren et al., 2024) under the dynamics induced by the generative process. Injecting stochasticity amounts to enhancing coverage of the action ây in the first step of MIP, thus enable iterative improvement with more NFEs (Section D.7). Its benefits analogous to trajectory noising effective in other behavior cloning applications (I (Laskey et al., 2017; Block et al., 2023; 2024: Simchowitz al., 2025; Zhang al., 2025). Similar benefits are found in the improved sensitivity analysis of diffusion relative to flows (Albergo et al., 2024).
5.3
ARCHITECTURE REMAINS ESSENTIAL FOR SCALING
Transport (PH, State)
Tool Hang (PH, State)
Transport (PH, State)
Tool (PH, State)
Average Success
0.75
0.75
0.6
0.6
8.58
0.50
8.4
8.25
0.25
0.2
8.2
8.08
0.80
Sudeop- DDT Chi-1 Thi-TPH Chi UlNet
MLP
RNN
Sudeep- D1T
Chi-1 TPM Chi UWet
MLP
RAN
0.3 1.2 5
20 80
0.3 1.2
20 80
Model Size (M)
Model Si.ze (M)
Architecture
Archirecture
Regression
MIP
Flow
Figure 5: Architecture and model size ablation. Success rate are averaged across 3 seeds and 5 checkpoints on Tool-Hang and Transport tasks. Left 2 plots: architecture ablation. Right 2 plots: Model size ablation.
Regression, enjoys stronger relative performance at the smallest model sizes but scales more poorly than flow and MIP with increased model capacity (Fig. 5). We conjecture that supervised iterative computation can better utilize larger models, both by introducing more supervision steps train- ing, and by providing more parameters to represent different computations at successive generation steps. Nevertheless, architecture design plays an incredibly significant role. To showcase its im- portance, we ablate the performance of different method's 's average performance across both the 3 architectures above, and the more traditional MLP and RNN architectures, implemented with modern best practices including FiLM conditioning (Perez et al., 2018), and skip-connections (He et al., 2016/LayerNorm (Ba et al., 2016) where appropriate (details in Section D.2). As demonstrated in Fig. 5, the combination of training method and architecture design has strong yet somewhat erratic effect on both GCPs and RCP performance. In Too Tool-Hang, RCP achieves the best performance with an MLP architecture. In Transport, MLP with flow even outperform more expressive architectures like Chi Chi-Trande Transfrrrnrreeee The coupling between training and architecture choice highlights the importance of controlling architecture design when comparing across methods.
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ACKNOWLEDGEMENTS
Max Simchowitz and Giri Anantharaman acknowledge TRI University 2.0 Fellow and Google Robotics Research Award. Max Simchowitz and Chaoyi Pan thank Nur Muhummad (Mahi) Shuf- fiulah for his insightful feedback, and Max Simchowitz also thanks Aviral Kumar, Sarvesh Patil, and Andrej Risteski for their thoughtful suggestions. Guannan Qu acknowledges support from NSF CAREER Award 2339112, 12, NSF Award 2512805, and the Pennsylvania Infrastructure Technology Alliance. Guanya Shi holds concurrent appointments as an Assistant Professor at Carnegie Mellon University and as an Amazon Scholar. This paper describes work performed at Carnegie Mellon University and is not associated with Amazon.
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Brianna Zitkovich, Tianhe Yu, Sichun Xu, Peng Xu, Ted Xiao, Fei Xia, Jialin Wu, Paul Wohlhart, Stefan Welker, Ayzaan Wahid, et al. Rt-2: Vision-language-aetie models transfer web knowl- edge to robotic control. Conference on Robot Learning. PMIR, 2023.
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A RELATED WORK
Robotic Behavior Cloning. Behavior cloning (BC), also known as learning from demonstrations (LfD), has become a popular paradigm to enable robots to conduct complex, diverse and long- horizon manipulation tasks by learning from expert demonstrations (Argall al., 2009; Zhu & Hu, 2018: Zhao et al., 2023b; Chi et al., 2024; Lin et al., 2024). In parallel, "robot foundation mod- els" scale BC with internet-pretrained vision-language transformer-base backbones (Brohan al., 2022: Zitkovich et al., 2023; O'Neill et al., 2024) and large-scale teleoperation datasets (Kim et al., 2024; Team et al., 2024). More recently, to better model continuous actions, generative models like diffusion and flow have been adopted to replace the tokenization method in transformers to achieve more expressive policies (NVIDIA et al., 2025; Black et al., 2024; Physical Intelligence et al., 2025; Liu et al., 2024). This work focuses on the generative modeling part of the behavior cloning pipeline, ablating the key design choices that lead to the success of generative control policies.
Generative Modeling. The recent success of behavior cloning policies is built upon rapid evo- lution of generative modeling techniques, starting from tokenization methods (Brown et al., 2020; Chen al., 2021; Pertsch et al., 2025) and adversarial methods (Brock al., 2019; Goodfellow et al., 2020; Ho & Ermon, 2016). Later, probabilistic generative models with iterative computation like diffusion models (Ho al., 2020; Song et al., 2021b; Lu et al., 2025; Song et al., 2022; Nichol & Dhariwal, 2021; Karras et al., 2022) became a popular choice for generative modeling thanks to their better training stability and sampling quality. Flow models (Lipman et al., 2023; Albergo & Lunden-Eijnden 2022; Liu et al., 2022) and consistency/shorteur models (Song et al., 2023; Song & Dhariwal, 2023; Meng et al., 2023; Boffi et al., 2025a; Geng et al., 2025) were later developed to achieve faster sampling while maintaining the expressivity of diffusion models. Though there have been extensive studies on probabilistic generative modeling's effectiveness in image and text generation (Lee et al., 2023; Chen al., 2023), its mechanism in control, especially the key design choices, are still opaque in decision making.
Generative Control Policies. To model diverse and complex behaviors, GCPs parameterize the relationship between observations and actions as a distribution rather than a deterministic function. Early works use transformers with tokenizers (Chen et al., 2021; Shafiullah et al., 2022), energy functions (Florence et al., 2022; Dasari et al., 2024) and VAFS (Zhao et al., 2023b) to parameterize the distribution. Diffusion models (Reuss et al., 2023; Chi et al., 2023; Ke al., 2024; Dong al., 2024; Janner et al., 2022; Yang et al., 2024) were introduced for their better expressivity of complex and multi-modal behaviors, followed by flow-based (Zhang al., 2024; Black al., 2024; Phys- ical Intelligence et al., 2025) and methods (Hu et al., 2024; Prasad al., 2024; Sheng al., 2025).
miaiiistencyyoooo///oottttt based acceleration
Theoretical Literature on GCPs. Block et al. (2024) established that GCPs can imitate arbitrary expert distributions. Given our findings on the absence of multi-modality, a more closely related theoretical findings is that of Simchowitz et al. (2025), which elucidates how GCPs can circumvent certain worst-case compounding error phenomena in continuous-contre imitation learning. Though the proposed mechanism is different, that finding is conceptually similar our own: GCPs benefits arise from their favorable out-of-distribution properties, rather than raw expressivity of fitting in- distribution expert behavior.
A. PREVIOUS WORKS' CONNECTION WITH GCP's TAXONOMY.
We classify GCPs into three components: distributional learning, stochasticity injection, and su- pervised iterative computation. Starting from regression, it has none of the three components. To model a more complex distribution. Gaussian Mixture Model (GMM) (Zhu & Hu, 2018) was used to parameterize the distribution, trained with cross entropy loss. To make the network be able to represent more complex distirbutions, prior to diffusion, non-parametric method like VAEs (Zhao et al., 2023b) was used to parameterize the distribution, trained with reconstruction loss. During the training, a latent variables is predicted to predict the style the motion by mapping it from a noise z. Another line of work try to improve the policy expressivity by introducing iterative compute, like implicit behavior cloning (Florence et al., 2022; Dasari et al., 2024) and behavior transformer (Shafi- ullah et al., 2022). In IBC, the idea is to allow the network predict the energy function of the action rather the action itself. Compared to diffusion, the major difference is that they do not explicitly injecting noise during training and no intermediate supervision is provided for the intermediate
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sults. Similarly, behavior transformer, two step policy is introduced first predict the policy class and then refine with another network to achieve higher precision control. Lastly, flow-based GCPs (Zhang al., 2024; Black et al., 2024; Physical Intelligence al., 2025), which holds all the three components and demonstrate state-of-the-art performance popular benchmarks. In this paper, we look into a new combination that haven't been explored before, which the combination of stochasticity injection and supervised iterative computation.
B LEARNING DYNAMICS OF GCPS AND RCPS GIVEN DIFFERENT TYPES OF DATA
(a) training dense/multi-modal data
(b) training w/ sparse/uni data
T
(a.1) dense denonstrations
(a.2) policy learns
(b.1) sparse
(b.2) policy learns
(b.3) policy learns high-Lipschitz
multi modal behavior
demonstrations
multi- madiity
Demonstrations
GCPs Prediction
RCPs Prediction
Figure 6: GCP behavior given different types of data. (a) Given true multi-modal data (a.1), GCPs leam both modes while RCPs collapse (a.2). (b) Given sparse data (b.1) in high-dimensional space, both policies learn high-Lipschitz policies quickly switch between modes (h.3). We find both GCPs and RCPs learn (b.3) high-dimensional tasks (Section 3.2). Theorem suggests that GCPs have limited advantage over RCPs from pure expressivity perspective.
C ADDITIONAL POLICY PARAMETRIZATIONS
This section further claborates the design space of MIP in stochasticity injection, iterative computa- tion and intermediate supervision.
C.1 FULL ABALATION OF VARIANTS
This section formally describes the training process of all MIP with different stochasticity injection and supervised iterative computation design.
Residual Regression (RR) removes stochasticity in training and the training objective is:
30 ss arg ming Eo.
(C.I)
((( to = 0, = 0) t. +
= 0, 0)), - -))
(C.2)
Two-Step Denoising (TSD)
The training objective is:
π ~ arg ming Eio.a))ppiminn
(I)(To(o, Io. = 0) t.aa)l² + [|(Ta (g, 0)) + +(1 (1-t.)2,t,) a)l
where Compared to MIP, TSD adds stochasticity to both first step training.
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MIP with Data Augmentation (MIP-Daggez) To understand the importance of decoupling for enabling iterative computation, we propose an additional variant of MIP that lies between MIP and RR, where the two steps are partially coupled. Since the training method of second iteration similar to data augmentation, call this variant MIP-Dagger:
"
MIP-Dagger arg min (0.0) P (Tl (o, To 0, f,))l (( ( 0, + (4),,4) -)]l
E
where the major difference compared to MIP is the second step takes in the interpolant between first step output and noise rather than the action and noise.
MIP without intermediate supervision (MIP soip-Nosupervvsson)) intermediate supervision on iterative computation, we propose one variant of MIP that removes the supervision of intermediate computation steps while preserving stochasticity injection at training time, named MrP-NoSupervision:
understand the effect of
"
MIP- MIp-MoSupervision
arg min
( P
a)ll
where the first step's output is unsupervised.
MIP without conditioning By removing f conditioning in MIP, degenerates to SF. Here we present the multi-step integration process for straight flow when action distribution is Dirac delta. The integrator from to t is:
t -- To(o, a.) 1 -
0g 1
0g
C.2 ADDITIONAL NOISE INJECTION METHODS
While MIP only injects noise to action, we also explore the possibility of injecting noise obser- vation. We propose two variants of MIP: MIP-Obs and MIP-Daager--Os. In MIP-Obs, we perturb the first step's observation with noise z, while the second step's training the same as the original MIP with decoupled training. In MIP-Dagger-Obs we perturb the first step's observa- tion with noise z, and the second step's training conditioned on the first step's output, making similar to Dagger. Major differnce compared to the original MIP: perturb the first step's observa- tion. both variants, we fixed i,, 0.9 and all observation perturbation happens at observation embedding space with normalized features.
T
MIP MIP-OBS 2s arg argmin min
( (1-,,,,,,1 To = 0, = 0) ttaa 12
20.1
1)
T
MIP-DAGGR DAGGER OBS
arg min
0) t.a)]l²
(o.ii (0.)
+ (ttp::/ssooogrddoo(00111 1ell-t)2,,10 0, = 0)) + 1 - tt) -
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We find that perturbing observations introduces data conflicts and degrades performance (Table 5).
In a two-step model, selecting noise levels that prevent observation overlap becomes challenging and brittle, leading to training instability across architectures.
Architecture
Method (L2)
Transport (ph)
Tool-Hang (ph)
18l-Tre Tra-Transfrrer
Regression MIP
0.50/0.45
0.50/0.37
Thi-Tran SillTransform:
0.79/0.69
0.92/0.85
Chi Thi-Transforme
MIP-Dagger-Obs
0.00/0.00
0.00/0.00
Tri-Transforme
MIP-Obs
0.61/0.46
0.13/0.08
Chi-Trans
Flow
0.81/0.71
0.89/0.75
Sudeep-DiT
Regression
0.65/0.54
0.31/0.25
Sudeeb-DiT
MIP
0.80/0.69
0.80/0.72
Sudeep-DiT
MIP-Dagger-Obs
0.00/0.00
0.00/0.00
Sudeep-DiT
MIP-Obs
0.00/0.00
0.00/0.00
Sudeep-DiT
Flow
0.79/0.65
0.73/0.61
Chi-UNet
Regression
0.66/0.59
0.73/0.59
Chi Chit
MIP
0.81/0.72
0.82/0.71
Chi Chi-
MIP-Dagger-Obs
0.00/0.00
0.00/0.00
Ch -UNe
MIP-Obs
0.00/0.00
0.00/0.00
Ch -UNet
Flow
0.83/0.75
0.87/0.73
state-based Table 5: Performance comparison of different methods with observation perturbation on state- based tasks, For each methods and architecture, we report the average and best performance across 5 checkpoints with 3 random seeds.
C.3 EXPERIMENT RESULTS
We benchmark all methods on the Too Tool-Hang task, given is the one with the largest gap between RCP and GCPs From Table 6, between two iterations, and intermediate supervision also important to realize the potential of iterative computation.
that the important part is to add stochasticity injection
Method
NFEs
Success Rate
TSD
2
0.80
2
0.80
MoP--oSuuervission
2
0).42
MIP-Dagger
2
0.64
RR
2
0.54
0.54
SF
3
0.55
SF
9
0.52
Table 6: Success rates across different MIP variants and !ool-Hang task 5 checkpoints architectures.
D CONTROL EXPERIMENTS
D.1 TASK SETTINGS
This section introduces all the tasks presented in the main paper. To reach sound conclusion, common benchmarks appears in previous works:
Robomimic Robomimic (Mandlekar et al., 2021) large-scale robotic manipulation bench- mark designed to study imitation learning and offline reinforcement learning. It contains five ma- nipulation tasks (Lift, Can, Square, Transport, Tool-Hang) with proficient human (PH) teleoperated demonstrations, and for four of them, additional mixed profccien/non--rofcienn human
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(MH) demonstration datasets are provided (9 variants in total). We report results on both state- based and image-based observations, since these two modalities pose distinct challenges. Among the tasks, Tool-Ha requires extremely precise end-effector positioning and fine-graines con- tact control, while Transport demands high-dimensional control and coordination over extended horizons.
Push-T Push-T (Florence et al., 2022) is adapted from the Implicit Behavior Cloning (IBC). The task involves pushing a T-shaped block to fixed target location using a circular end-effector. Randomized initializations of both the block and the end-effector introduce significant variability. The task is contact-rich and requires modeling complex object dynamics for precise block place- ment. Two observation variants considered: (i) raw RGB image observations and (it) state-based observations containing object pose and end-effector position.
Kitchen The Franka Kitchen environment is designed to test the ability of IL and offline RL methods to perform long-horizon, multi-task manipulation. It includes 7 interactive objects, with human demonstration data consisting of 566 sequences, each completing 4 sub-tasks in arbitrary order (e.g., opening cabinet, turning a knob). Success is measured by completing as many of the demonstrated sub-tasks possible, regardless of order. This setup explicitly introduces both short-horizon and long-horizon multimodality, requiring policies to generalize across compositional tasks.
MetaWorld MetaWorld is a large-scale suite of diverse manipulation tasks built in MuJoCo, where agents must perform challenging object interactions using robotic gripper. We adopt the 3D observation setting using point cloud representations, ported from the DP3 framework (Ze et al., 2024), to better evaluate geometric reasoning and spatial generalization. Tasks MetaWorld are categorized into different difficulty levels, with benchmarks testing few-shot adaptation and multi- task transfer learning.
Adroit Adroit is a suite of dexterous manipulation tasks featuring a 24-DoF anthropomor- phic robotic hand. Tasks include pen rotation, door opening, and object relocation, all of which demand precise, coordinated multi-finger control. Following DP3 (Ze et al., 2024), use point cloud observations to capture fine-grained 3D object-hand interactions. Policies are trained using VRL3, highlighting the challenges of nigh-dimensional control and sim-to-real transfer in dexterous manipulation.
LIBERO LIBERO is common multi-task benchmark to evaluate VLA's generalization ability. It composed of 130 tasks and can be categorized into multiple categories, including object, goal, spatial, and 10-task. The 10-task is long horizon and considered the most challenging solve.
D.2 ARCHITECTURE DESIGN
We study four policy backbones-Chi Schi-Trrasffomees Sudeep-DiT, Chi-UNet, RNN, and MLP undde common training recipe and data interface. Unless otherwise specified, all models are mapucity-matched to ~20 parameters to enable fair comparison.
Chi-UNet is adopted from Diffusion Policy (Chi et al., 2023) which built on top of 1D temporal U-Net (Janner et al., 2022) with FiLM conditioning (Perez et al., 2018) on observation and flow time t. Chi-UNe: has a strong inductive bias for the temporal structure of the action and tends to smooth out the action.
Chi-Transformer follows the time-series diffusion transformer from Diffusion Policy (Chi et al., 2023), where the noisy action tokens form the input sequence and a positional embed- ding of the flow time t is prepended as the first token; observations o are mapped by a shared MLP into observation-embedddng sequence that conditions the decoder stack. Compared to Chi-UNet, Chi Pei-Transff rmer uses token-wise self-attention over the whole action sequence, thus can model less-smooth and more complex actions.
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Sudeep-DiT is a DiT-style (Diffusion Transformer) conditional noise network specialized for policies adopted from DiT-Policy (Dasari et al., 2024): observation are first encoded into observa- tion vectors; the flow time / embedded via positional embedding; an encoder- decoder transformer then fuses these with initial noise to predict next action. The key ingredient of Sudeep- -ii--ii DiTTh is placing standard cross- cross-attention with adaLN-Zero blocks-adaptive LayerNorm modulation using the mean encoder embedding and the time embedding, with zero-initialized output-scale projec- tions staabiizin diffusion training at scale. Compared to Chi-Trans fo ooree, Sudeep-Di has adaLN- based conditioning (instead of vanilla tross-attention) and an explicit encoder-decoder split, yielding better training stability.
The RNN backbone processes sequences with a stacked LSTM/GRU. For each action time step in the chunk, the imput vector concatenates: the current noised action a,, time embed- ding for I, and a observation embedding for o. The RNN outputs are fed to a MLP head with LayerNorm+ U linear and recurrent weights use orthogonal initialization (biases zero), and RNN layer dropout is applied when 1epth>>1.
RNN
blocks before output the action with final linear head. All
MLP
The MLP backbone flattens the action and observation, appending the time embedding. Each mlp block has LayerNorm, ApproXGELU and Dropout blocks with residual connection and orthog- onal weight initialization throughout. Each block output is then modulated with FiLM conditioning.
DP3 built on top of Chi-UNes with extra 3d perception encoder. We use the exact same archi- tecture as 3D diffusion policy (Ze et al., 2024).
Model hy byperparameters In the main experiments, we align the model capacity to 20M parame- ters for default if not specified, with detailed hyperparameters report in Table 7.
Backbone
Heads
Layers
Embedding dim
Dropout
Sudeep-DiT
8
8
256
0.1
256
Chi-Transformer 4
8
0.1
RNN
8
512
0.1
MLP
8
512
Table 7: Model hyperparameters.
D.3 FINETUNING To ON LIBERC
For mo finetuning experiments, on LIBERO. Our flow-based finetuning experiments match their reported results. To finetune To to regression policy, we use the same architecture but the initial noise always zero and let the model directly predict the action. To finetune to MIP, we use the same practice where we modify the time step be uniformly sample from {0, t uniformly and the initial noise zero. We train all policies until convergence with 50k gradient steps on 1 node with 8 H100 GPUs.
lerobot framework (Cadene al., 2024) finetune mo
D.4 FULL RESULTS FOR FLOW AND REGRESSION COMPARISON
In the paper, we only present the aggregated results across 3 architectures. Figure 7 present the full results across all architectures with different training methods.
To further rule out the effect of training method, we also compare different methods' performance with é,, which observed to be superior for regression policy (Kim et al., 2024). We also benchmark the performance of flow model and MIP with l loss to understand the effect of loss function on the performance of GCPs. Table 8 shows the performance comparison of different methods with & and le loss, where we find that l loss generally outperforms l loss, especially for regression policy. We attribute the superior performance & loss to the fact that can capture the expert behavior better by learning the medium instead of the mean of the action. However, even with & loss, we still
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Average
Relative
Success
0.5
Transport (MI. Isage)
Disassentle (30)
Stick- Stick-Push Push (30)
Adrolt Hammar (30)
Lift (MH, State)
Push (Inage)
Assembly by (30)
Square (MH, Image)
Lift (PH, State)
Can (PH, Image)
Lift (PH. Image)
Can (PH. State)
Lift (MH, Image)
Adriot Pen (30)
can (MH, State)
Square (PH, State)
Adroit Door (30)
Square (PH. Image)
Transport (PH. Image)
Kitchen (State)
Push (State)
can (181, Image)
Square (191. State)
Transport (PH, State)
Toot hang (PH.. State)
Transport (M9, State)
Tool Hang (PH. Image)
Figure Relative performance of RCP compared to GCP across common benchmarks (worst-case ar- chitecture). For each lask, we implement Chi-Trans former, Sudeep-Di7 and Chi-UNe: For each architecture, average performance of the last 5 training checkpoints across three seeds. We then report the performance of the worst- perforeeffoming architecture, chosen individually for both RCP and GCP, demonstrate method robustness. For Flow, we always do 9 step Euler integrations, where performance plateaued. readability, RCP success rates are plotted relative to flow, with flow normalized to performance of 1 per task. Tasks are grouped by ohservation modality, and ordered by relative RCP performance. Red dashed line indi- cates threshold which RCP attains < 95% success of GCP.
observe that Regression < MIP & Flow, highlighting the importance of the stochasticity injection and iterative computation is independent of the loss function.
Architecture
Method
Transport (ph)
Tool Hang (ph)
Sudeep-DiT
Regression & Regression &
0.72/0.64
0.76/0.65
Sudeep-DiT
0.50/0.45
0.50/0.37
Sudeep-Dij
MIP {
0.81/0.73
0.91/0.84
Sudeep-DiT
MIP &
0.79/0.69
0.92/0.85
Sudeep-DiT
Flow &
0.83/0.70
0.93/0.84
Sudeep-DiT
Flow &
0.81/0.71
0.89/0.75
Chi-Transforme Regression &
0.67/0.57
0.44/0.33
Chi-Transformer
Regression &
0.65/0.54
0.31/0.25
Chi-Transformer
MIP f
0.80/0.68
0.85/0.77
Chi-Trans former
MIP &
0.80/0.69
0.80/0.72
- formerr
Flow &
0.77/0.69
0.81/0.71
Tbi-Transformes
Flow &
0.79/0.65
0.73/0.61
Chi-UNe:
Regression &
0.84/0.71
0.71/0.55
Chi-UNe:
Regression &
0.66/0.59
0.73/0.59
Chi-UNet
MIP l
0.85/0.68
0.76/0.67
Chi-UNet
MIP &
0.81/0.7
0.82/0.71
Chi-UNet
Flow &
0.85/0.69
0.87/0.71
Chi-UNet
Flow &
0.83/0.75
0.87/0.73
Table 8: Comparison vs f2 norm across different methods and architectures. Report average/hest perfor- mance across checkpoints with 3 random seeds.
Method
LIBERO Object
LIBERO Goal
LIBERO Spatial
LIBERO 10
RCP (E, loss)
92.6
94.6
97.2
78.0
RCP (E, loss)
95.2
88.0
95.8
62.4
Flow
97.4
95.0
95.8
81.6
MIP
95.8
95.2
97.6
82.2
Table Performance comparison on multi-task LIBERO benchmark. report the success rate checkpoint trained with 50k gradient steps of finetuning To on the full 1.1 LIBERO datasct. We implement MIP with 8" = 0.9 and integrate flow with 10 steps. For regression, train with both e and E loss as suggested (Kim al., 2024).
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D.5 DATASET QUALITY ABLATION
GCPs are believed to handle data with diverse quality better. To test that assumption, we manually corrupt the expert dataset and inject stochactity and multi-modality in the dataset. In Table 10, compare 4 different datasets (3 of them collected by ourselves). In the collected dataset, we manually inject noise to the policy and add delay the policy from time to time to introduce multi-modality that is common in the real world.
Architecture
Method
NFEs
Delayed & Noisy Policy (Worst Quality)
Delayed Pollcy (Mixed Quality)
Zero-Flow (Better Quality)
Proficieut Human (Good Quality)
Chi UNet
Regression SF MIP Flow
0,20/0.65
0.80/0.72
0.7649.65 65
0.76//062 6.2
Chi-UNet
0,70/0.62
0.82/0.76
0.84/9.77
0.62/0.38
-hNet Chi UNet.
0,80/0.22
0.82/0.61
0.74/9.65 64
0.80V0.68
0.76/0.68
0.74/0.50
0.76/0.54
0.84/0.70
Chi-Pransformes
Regression SF MIP
0.38/0.22
0.4080.31
0.42/0.26
0,50/U24
Chi-Transfomee
0.46/0.35
0.68/0.50
0.56/0.41
0.62/0048
-hi-Trane former
0.56/0.49
0.70/0.54 54
0.64/0.56 56
0.72/0.66 68
-hi-Transfoce Formeeerme
Flow
0.56/0.34
0,54/0.48
0.62/0.49
0.68/0.54
Budeep-Dit
Regression SP
0.42/0 29
0.36/0.25 28
0.42/0.32 32
0. 0.30/0.19
Sudeep-Dit
0.66/0.41
0.6010.54
0.72/2.57
0.68/0.50
Sudcep-DiT
MIP
0.66/0.56
0.74/0.58 58
0.7049.61
0. 0.78/0.68
Sudeep-DiT
Flow
0.56/0.45
0.66/0.58 58
0.72/0.65
Table 10: Performance comparison across different methods and data quality levels. We evaluate on the task Tool with state observations using 10M parameter networks. Success rates are reported as averages over 5 checkpoints across 3 seeds.
D.6 FULL RESULTS FOR MIP AND ITS VARIANTS
Archieeture
Methed
CLU
Squaze
Seansport
Tool-Hane Hang
Push-I
Kitchia
mth
p4
p2.
2b
Flose
1.000109
1.00F.00
0.58/0.75
1.000.94
0.8000.70
0.860.75
0.98196
Suseep- D:7
Regression
1.000.90
1.96-1.6
0.93/0.00 00
1.005.58
0.72X0.53
0.9400.56
0.133
0.500.45 44
0.5220.3 30
0.92/085
0.95/002
BuSeep-D17
Strightt Flow
1.803195
1
(19691140 961
1.080099
01728966
0.963193
0.200
11504
0.700 59
05000.0
Bufeeg-D17
Chi
Flow
1.00/1.00
1,06-108
1.992.03
1.083588
0.7800.74
0.960.80
0 34
0 64
0.680 54
0.910.89
1988196
Chi Trana cormaz
Regres di1n
1.000000
1.081199
0.03/0.0
1.000.56
0.7400.1
0.920-85
0.25-0.30
06600.51
04000 36
0.930.88
0.5m..
Ch4 TTaaa Formaz
Straishi Flow
1,0811000
1,000.09
1.000T
11
0.60073
0.960.89
0.430
030000000
0.760609
03466.92
0160196
Chi-Ukee
Flow
1.0077 m10
1,98-1119
1 66
1,997 1.00
0.900778
0598119
0.520
03309.75
0.560370 76
8.860.06
1.881197
chi UKi-UKee
Hegroscian
1.00/7.00
1,08.1.10
1
1
6.949.32
1,00091
0:230
016 55
0A801
09900 89
39
Chi Ubhes
MIP
1.00/.000
1.8/1008
1.880.98
1.005.09
0.9210.11
1.000.54
0.620146
0.900.69 69
0.5000.64
0.9748.55
1088196
Table 11: Performance comparison of Flow and Regression methods across different state-based robotic ma- nipulation tasks. For each task, we report the best checkpoint performance averraged performance over last checkpoints. Each experiment is run with 3 seeds and we report the average performance across all seeds.
Auchitictturc
Method
Lint
Can
Square
Transport
Hang
mh
pb:
Sudeeg-ni1
Flow
1,00/1.00
0.0
0:96-0..00
1.08/1009
76
0.5020
0.780055
Regression
1.00/0:59
1.00/1.00
0.52/0..1
1.00/1.00
0.7480.67
0.9440.84
0.140008
0.7400.56
0.25/0.15
0.330037
Sudlee m
Straigle Flow
1.0011159
1.80/1150 550
0:98-1.95
1.0897.95
11.336.72
1.091.933
0.2633.19
0.861085
0146-1140
018501.9
Sudac3-017
MIP
1.000.599
1,80:1.00
1.0001.96
1.09/3.50
11 0..9000.3
1.0912922
0.5000.31
0.9010.84
0.76/1.66
0.91/1.37
(hh Teanna coca
Flow
10001150
1 (00
0138-1992
1.00.0196 96
0.7000.65
0.98.8.92
0.2400.22
0.800077
(154M1.0
0.300155
Chi-Teane moee
Regression
..000.96
1,80/0.2 98
1,0000.94
1.0877.0 95
0.7640.00 70
0.9540.00
0.4000.2 27
0.9400.7
0.44m 36
0.85mL
Chi-- Yeann Corawe
Sta Straight Flow
1,001.00
1,8001.00
1.00913555
10.9000.75
0.5200.25
03860070
0.3601:25
cht-Traneto
1:56/0..
0.9000.00
0.87/155
Chi-UNet.
Hlw
1.00-0.97
1.0900.58
0.50000.79
0.9840.90
0.2400.16
0.750061
0.48/1.37
0.921137
(th) -17ee
Regression
1.0010.06
1.80/1 59
1.5.40175 70
0 109808.65
0.7440.65 66
0. 0.9448.85
0.1839 10
0.661066
0.3600.2
0.90/135
Chi-UXett
Straight Flow
1.9001544
1.80911.59
019881133
1.0913955
0.7240.65
0.9210.62
0.0000.0
0.5010.22
0.06/1.02
0.87/1.22
Chi 1'
MIP
1.0011.00
1 .00
1.00-1.00 95
1.00.00 96
0.92/1.84
0.9640.97
0.5220.37
0.9600.11
0.56.11 50
0.83/175
Table 12: Performance comparison of Flow and Regression methods across different image-based robotic manipulation tasks. For each task, we report the best checkpoint performance averaged performance over last 5 checkpoints. Each experiment is run with 3 seeds and we report the average performance across all seeds.
For Kitchen, the task has multiple stages. In the main results, we only report the performance of the last stage since it is the most challenging one. Table 14 shows the performance comparison across different design choices K tchen task.
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Architecture
Method
Aazoit
MotaWorid
Hammer
Door
Pen
Sock-Push
Assembly
Disassemble
0.96
0.02
0.60
0.11
0.92
0.08
0.72
0.14
Regression
0.97
0.04
0.52
0.16
0.47
0.08
0.95
0.06
0.98
0.03
0.78
0.08
Table 13: Performance comparison Flow and Regression methods using DP3 architecture across different point-cloud-base robotic manipulation tasks. For each task, we report the best checkpoint performance / averaged performance over last checkpoints. Each experiment is run with 3 seeds and we report the average performance across all seeds.
Architecture
Method
P1
P2
P3
P4
Flow
1.0
1.0
1.0
0.98
MIP
1.0
1.0
1.0
0.94
ONi-UNet:
Regression
0.98
0.94
0.94
0.86
Flow
1.0
1.0
1.0
1.0
MIP
1.00
0.98
0.98
0.96
Chi-Transforme
Regression
1.0
1.0
0.98
0.94
Flow
1.0
1.0
1.0
0.98
1.00
1.00
1.00
0.98
Sudeep-DiT
Regression
1.0
0.98
0.96
0.88
Table Performance comparison across different design choices on kitchen task. Kitchen task multiple stages and we report the success rate of finishing n tasks in the table. For the performance reported in the main paper and previous tables, we report the success rate of finishing 4 tasks.
D.7 DIFFERENT METHOD'S PERFORMANCE WITH DIFFERENT NUMBER OF FUNCTION EVALUATIONS
We also provide detailed evaluation on different method's s scaling behavior given different amount of online computation budgets. Table 15 highlights that only MIP and Flow benefit from iterative computate.
Method
Reg.
SF
RR
MIP
Flow
NFEs
3
9
2
2
3
S.R.
0.46
0.54
0.55
0.52
0.31
0.33
0.50
0.74
0.32
0.55
0.66
Table 15: Comparison of methods and their corresponding number function evaluations (NFEs), Eval- uated on state-based -Hain task over Chi -UNt.. Average success rate is reported across seeds and checkpoints.
D.8 COMPARING MIP WITH CONSISTENCY MODELS
Given MIP takes less integration steps compared to flow model, we compare it with consistency models which accelerate the sampling process of flow by distilling the learned flow into shortcut model. The major difference between MIP and consistency models is that the latter do satisfy C1, and require training over a continuum of noise levels. On the other hand, MIP is trained to predict the conditional mean of the interpolant, and thus, doesn't need extra distillation stage. As shown Table 16, We benchmarks MIP to common consistency model training methods including con- sistency trajectory model (CTM) (Kim et al., 2023) and Lagrangian map distillation (LMD) (Boffi et al., 2025a), where LMD only works for Chi-UNet due to its dependency on jacobian matrix computation. The benchmarking results indicates that, given best architecture, MIP outperforms consistency models. In terms of training time, MIP only takes half of the time compared to CTM, where LMD training takes even longer due jacobian matrix computation.
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Architocture
Method
Transport
Tool Hang
mh
Sudaep-DiT
0.40/0.27
0.86/0.75
Sudeep-D11
MIP
0.44/0.35
0.7640.60
0.92/0.88
Chi-Transforma
CTM
0.57/0.32
0.90/0.58
0.56/0.22
Corase
Flow
0.44/0.34
0.88/0.60 64
0.68/0.54
Chi Fransforee
MIP
0,42/0.37
0.8010.68
0.76/0.69
Chi-UNer
CTM
0.40/0.35
0.72A0.6.
0.46/0.37
Chi-UNet
Flow
0.52/0.40
0,80/0.3
0.84/0,00
Chi UNet
LMD
0,44/0,32
0.7640.68
0.74/0.52
Chi-UNet
MIP
0.62/0.46
0.8081.69
0.8000.44
Table 16: Benchmark results across different architectures and methods on state-based tasks on consistency models and MIP. Report average/best performance across 5 checkpoints with 3 random seeds. Both LMD and CTM integrate 2 steps, which the same MIP.
LIPSCHITZ CONSTANT STUDY DETAILS
E.I LIPSCHITZ EVLUATION METHOD
We note that not all inputs o are dynamically feasible, and our dataset lies only on narrow manifold of the observation space. Therefore, must carefully evaluate the Lipschitz constant the feasible observation space to avoid conflating model expressivity with errors arising from infeasible states. To ensure feasibility, instead of directly perturbing the state, we perturb the action and then roll out in the environment. This guarantees that both the perturbed state and the resulting observation remain feasible.
In practice, we identify states that exhibit the highest ambiguity of actions in the dataset, referred to as critical states. For each critical state, we inject Gaussian noise 7) (0, e²f) into the normalized action, unnormalize it, and then roll it out. We select 100 critical states from the dataset. For each state, we perturb the corresponding expert action a with 64 independent Gaussian samples.
Let denote the next nominal observation after applying the nominal action After rolling out the perturbed actions, we obtain perturbed observations o(I)..... (NNmm)) The policy then predicts the perturbed actions (()) = (o) To ensure comparability across different states and tasks, we evaluate the Lipschitz constant with respect to normalized observations ō = a-Mo and normalized actions aーHe Finally, the Lipschitz constant is estimated using zeroth-order approximation:
de
(
x||2
L Rs max
(E.I)
|v/l
Full version of above process is stated in Algorithm
1.
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Algorithm Lipschitz Constant Estimation via Action Perturbation
Require: Dataset D, policy noise scale e, number of critical states N, 100, number of pertur-
bations N,=64
Ensure: Estimated Lipschitz constant L
1: S f- identify Ng critical states from
>> Select states with highest action ambiguity
2: for all critical state S do
3: (a. t- expert action and nominal next observation for
b Get ground truth
action-observation pair
4: (a, 6) + normalize (a, o) using dataset statistics >> Ensure comparability across states/tasks
5: for 1 N, do
6:
~ (0,
Sample Gaussian perturbation b. Create perturbed action in original scale
7:
Opert unnormalize(a
8:
00) rp))) in environment
> Execute perturbed action to get feasible state Normalize perturbed observation Get policy prediction on perturbed state Normalize predicted action
9:
((0) normalize
10:
a(0) T(o(*))
11:
((() normalize (+)
f || |all
12:
P. Compute finite difference approximation
13:
La f- max; ri
Local Lipschitz constant for state Average across all critical states
L
N, NN La
return
E.2 FULL LIPSCHITZ EVALUATION RESULTS
In the main text, we only report the average Lipschitz constant on critical states across 3 archi- tectures. Here, report the full Lipschitz constant evaluation reuslt in Table 17 with different architectures and tasks.
Task
Architecture
Method
Lipschitz Constant (Policy)
Regression
0.85 + 0.58
Chi-UNet
Flow
0.31 + 0.01
Regression
0.52 0.11
Push-T (State)
Sudeep-DiT
Flow
0.22 + 0.02
Regression
1.33 1.33.1.11 1.14
Si--Transformes
Flow
0.26
Regression
13.47
Chi-UNet
Flow
13.31
Regression
15.37+
Kitchen (State)
Sudeep-DiT
Flow
12.51
Regression
13.37
Chi-Transfozmet
Flow
11.44
Regression
1.63 + 0.79
Tool-Hang (PH, State)
Chi-UNet
Flow
1.53 + 1.01
Regression
1.86 - 0.81
Sudeep-DiT
Flow
1.34 0.97
Regression
1.76 - 1.02
Chi-Trans former
Flow
1.40 + 0.99
Table 17: Detailed: Per-architecture policy Lipschitz.
F MULTI-MODALITY STUDY DETAILS
F. Q FUNCTION ESTIMATION
To rule out the possibility of hidden multi-modality, we also plot Q functions for each action to see if there is any clear clustering pattern of w.r.t. different actions in t-SNE visualization. Since
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only have access to expert actions rather than their policy, we estimate the function by Monte Carlo sampling with the learned flow policy. The detailed procedure is as follows:
Starting from one "critical state", we first sample N actions
= o =0 - 0, 1), i= = N, 20) ~ I.O.
For each sampled action ( we execute one environment step to obtain the next observation o((i) and immediate reward r(o, ) Then, starting from o(()), we rollout the learned policy for NMc
episodes until termination (horizon H), and average the cumulative returns to obtain an estimate of the continuation value. Thus, the Q-value for action (*) is approximated as:
Q4(a r((,
1 MMC
Σ
r(o (
(F.1)
We set the discount factor y 1.0 since rewards are sparse and triggered only at task completion. The reward for Tool-Hang and Kitchen defined by the final success signal (with Kitchen is success requiring all 4 subtasks to be completed). The reward for Push-T is defined by final coverage.
Algorithm 2 Q Function Estimation via Monte Carlo Sampling
Require: Dataset D, flow policy 4, reward function r, number of critical states N,=100, number of action samples N, Monte Carlo samples NMC
Ensure: For each state o, pairs {()(9) ,))))
1: S + identify N₂ critical states from
D Select states with highest action ambiguity
2: for all critical state S do
3: f observation for state
4: for i 1 N do
Sample actions and compute estimates
5:
( N(O,I)
6:
( T 200) s=0,t=1
7:
Execute (o, a (9) in env - obtain 000) ,(0) r(),a())
8: for j 1 to NMc do
Monte Carlo rollouts from o((1)
Rollout + from (1 until horizon H get cumulative return R())
9:
Qu(a(), + r() + NAc
Σ 1== R
Noc
Store ((a(0) () ( for state
The procedure above explicitly computes Q-values by rolling out trajectories separately for each sampled action.
F.2 DETERMINISTIC DATASET GENERATION
To generate a deterministic dataset that completely eliminates any potential multi-modality, we fol- low a systematic process:
First, train flow expert policy & the original dataset. Then, collect dataset by rolling out this expert policy from different initial states (using different random seeds than those used during testing). Crucially, during rollout, by setting the initial noise to zero: 2 = O. This ensures that the policy produces deterministic actions given any observation, completely removing any stochasticity from the action generation process.
always evaluate the flow policy deterministically
During data collection, we discard all failed trajectories to maintain the same success rate as the original dataset. We continue collecting until we reach the target number of trajectories Nmj-
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Algorithm 3 Deterministic Dataset Generation
Require: Trained flow policy 4, target number of trajectories Nunj, maximum episode steps Tmax Ensure: Deterministic dataset Det
1: Daet 4-0 0
2: Picollected +-
3: while Recolected Nujj do
4: Reset environment with new random seed
5: Op initial observation
6:
7 < [0o.
b Initialize trajectory
7:
for 0 max do
8:
0r - (2 = 0, or. 8 - 0,6=11
Deterministic action
9:
0c+1. rr, done - env.step(a.)
10:
+ 7 [(or,
if done then
12:
break
13:
if trajectory 7 is successful then
14:
Deet f- Datt U (T}
15:
Peollected - Recollected +
16:
return Dect
G MANIFOLD ADHERENCE STUDY DETAILS
G.1 VALIDATION LOSS NOT GOOD PROXY FOR POLICY PERFORMANCE
To investigate whether validation loss serves reliable proxy for policy performance, ine its relationship with success rates on Tool-Hang across different architectures given different training methods. Evidence that validation loss poorly correlated with success can be seen by comparing flow policies with varying numbers of function evaluations (NFEs) and their correspond- ing validation losses. Table 18 demonstrates that increasing NFEs does not reduce validation loss, yet policy performance consistently improves. We hypothesize that higher NFEs introduce stronger inductive bias and regularization, which projects actions back onto the data manifold, thereby en- hancing generalization.
Architecture
Method
NFEs
Average Success Rate
L2 Validation Loss
Regression
0.54
0.063
Flow
0.36
0.053
Chi-UNet
Flow
3
0.44
0.052
Flow
9
0.58
0.053
Regression
1
0.18
0.084
Flow
0.06
0.093
Chi-Transformer
Flow
3
0.72
0.092
Flow
9
0.68
0.089
Regression
0.20
0.063
Flow
0.62
0.082
Sudeep-DiT
Flow
3
0.76
0.080
Flow
9
0.76
0.080
Table 18: Comparison of validation loss and success rate across different architectures and methods on state- based Too LooggEnng The results show that validation loss is not reliable proxy for policy performance.
G.2 MANIFOLD ADHIERENCE EVALUATION METHOО
To evaluate the manifold adherence, compute the projection error of predicted action onto the space spanned by expert actions at neighboring states. Concretely, given a state, we compute its l2 distance to all states in the training set. Then, we pick & nearest neighbor states and gather their
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corresponding actions A [a(0), aa.) ..... ( Lastly, we compute projection error by projecting to the column space of A: ||a Px(a)]2 = min, ||a AcI12-
H NEAREST NEIGHBOR HYPOTHESIS STUDY
Another popular hypothesis that GCPs are learning a lookup table of observation-to-actioo map- pings (Pari et al., 2021; He et al., 2025). This might be true for relatively simple tasks that do not require high precision and complex generalization, such as Can. However, for tasks that re- quire higher precision and more contact, such Tool-Hang, the nearret-neighhbo//looouu assumption is insufficient to explain the success GCPs. We evaluate the performance of a nearest- neighbor policy (VINN (Pari al., 2021)) on state-based Tool-Hang and find that it achieves success rate of only 12% as shown in Table 19. This is significantly lower than both flow and re- gression methods, indicating that the action manifold Nevertheless, nearest-neighbor can still serve as a proxy for the expert action manifold, as it captures the general trend of actions even though linear combinations of actions in the dataset cannot di- rectly produce the correct action, the expert action manifold should not be too distant. Therefore, in this paper, we use nearest-neighbor as proxy for the linearized expert action manifold rather than directly computing the distance between expert actions in the validation set and predicted actions.
linearly spanned by the expert actions.
Action Chunk Size
Success Rate (%)
16
12
32
2
Table 19: Performance k-nearest neighbor policy on state-based Too L-Hang task. Using same method as VINN with softmax over k=5 nearest neighbors.
H.1 ACTION CHINKK SIZE STUDY
Another equivalent important factor is the action chunk size. Fig. 8 highlights the importance of action chunk size, where regression with larger action chunk can outperform flow with smaller action chunk size. Our ablation also indicates that MIP outperforms flow with smaller action chunk size and matches the performance of flow with larger action chunk size.
Transport (PH, State)
Tool Hang (PH, State)
9.7
1.7
Success Rate
0.0-
E.5
0.4-
Regression
MTP
Flom
Action Chunk Size
Figure Action chunk size ablation. Success rate are averaged across seeds, architectures check- points on Tool-Hang and Transport tasks. Prediction horizon set to powers of to make sure is compatible with Chi-UNet Architecture.
H.2 LOss NORM TYPE ABLATION STUDY
Previous work (Kim et al., 2024) shows that & loss is superior to by loss for regression policy. To test this hypothesis, ablate the loss norm type and compare the performance of different methods with l and l loss. Table 20 shows the performance comparison different methods with é and & loss, where find that & loss generally outperforms l loss, especially for regression policy. However,
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even with L loss, still observe that Regression MIP Flow, highlighting the importance of the stochasticity injection and iterative computation is independent of the loss function.
Architecture
Method
Transport (ph)
Tool Hang (ph)
DiT
Regression L1
0.72/0.64
0.76/0.65
DIT
Regression 1,2
0.50/0..4
0.50/0.37
DIT
MIP L1
0.8 0.81/0.73
0,91/0.84
DIT
MIP L2
0.79/0.69
0,92/0.85
DIT
Flow L.1
0.83/0.76 76
0.93/0.84
DIT
Flow L2
0.81/0.71
0.89/0.75
Transformer
Regression I.1
0.67/0.57
0.44/0.33
Transformer
Regression L2
0.65/0.54
0.31/0.25
Transformer
MIP LI
0.80/0.68
0,85/0.77
Transformer
MIP T.2
0.80/0.69
0,80/0.72
Transformer
Flow L1
0.77/0.69
0.81/0.71
Transformer
Flow L2
0.79/0.65
0.73/0.61
UNet
Regression L1
0.84/0.71
0.71/0.55
UNet
Regression L2
0.66/0.59
0.73/0.59
UNet
MIP L.I
0.85/0.68
0.76/0.67
UNet
MIP L2
0.81/0.72
0.82/0.71
UNet
Flow LI
0.85/0.69 0.83/0.75
0,87/0.71
UNet
Flow 1.2
0.87/0.73
Table 20: Comparison of LI L2 norm different methods and architectures. performance across checkpoints with random seeds.
Report average/best
I DIVERSITY OF GCPS AND RCPs
commonly believed hypothesis is that GCPs can express more diverse behaviors than RCPs by capturing the full distribution of expert actions (Shafiullah al., 2022). We evalute different vari- ants of GCPs and RCPs on FRANKA-KITCIIEN where the expert shows multiple task completion orders. As demonstrated in ?? GCPs with both stochastic and deterministic sampling show similar task completion order diversity. Deterministic policies like regression and MIP (to be introduced in Section 4) also demonstrate similar task completion order diversity. This indicates that, given sparse expert demonstrations, both GCPs and RCP learns high-Lipschhin policies to switch between differ- ent modes given different observations (corresponding to (b.2) case Figure 6). RCPs and GCPs are equally good learning such behaviors (Figure 9) which explain why for both policy parametrizations, even on seemingly multi-modal tasks like FRANKA-KITCHEN.
similar performance
Kitchen Task Comp letion by Method (Top Task Orders per Mothod)
Regression
Flee (Stecheotic)
Flon (Detereinistic)
HIP
Counn
30-
68
Completion
Figure 9: Task completion order in Kitchen environment with different methods. We plot the count of different task completion orders for different methods to evaluate the diversity of the policies. The x-axis shows the task completion order, where each sub-task represented by its initials. For each run, we collect 1000 trajectories with the same seed shared by all methods. For flow, we evaluate both stochastic and deterministic modes.
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J THEORETICAL ANALYSIS OF GCP'S EXPRESSIVITY
J.1 FORMAL STATEMENT OF THEOREM
In this section, we introduce the notation and definition required for the subsequent proofs and provide the formal statement of Theorem 1 from the main text. Throughout, let lle denote any matrix norm satisfying the property | S | ||
|| la. contrast to the notation used in
the main text, define & (a. o) as the solution at time t of the ODE:
dt (a o), with initial condition
(J.1)
Note that Lo (ag 2, o) coincides with the definition of (z o) in the main text. Next, we define the notion of c-loo--oncavity.
Definition J.1 .ol.loo-oncavity...om A distribution with density p V(r) is said to be c-loa-concave if E C22(C (I2) and its Hessian satisfies V²) > xI for all x E R and some x 0.
With this notation in place, we now state the formal version of Theorem 1.
Theorem Suppose that
b = = Le, o),
where I, = (1 -1-11ag + tai, a ~ N(..), ~MPi
(J.2)
where P1 is k-log-concave. Then, we have
(/" V
k(1 t)2 +12
Vo o)l6 $
x(1 - 8)2 + [[Vobb o)]ldds.
(J.3)
In particular, for = 1 we obtain
V 2 (ao, o)ll- S VI+=-I [Va o)llods.
(J.4)
Remark J.1. Theorem 1 follows immediately from the fact that both the operator and the Frobenius norms satisfy |X,X,) X2lle || together with the inequality Eq. (J.4).
J.2 SUPPORTING LEMMAS
We state the supporting lemmas for proving Theorem below and provide their proofs immediately for completeness. As a first step, we analyze the dynamical system satisfied by V.Ra,e(a, o).
Lemma J.1. Define (o, 0) where as is the initial condition, and define the matrices
M, = V Do, o), A, = (V o), Eg = (a, 10)
(J.5)
Then,
d
M, = A.Mr + E,, Mo =
(J.6)
dt
Proof. Since Po.o o) = an. Mo 0. Moreover,
dt V VoTar o) = V. d dt Po, o) = V.. o) [))
(J.7)
= (Vab))aa)) o) V. o) + (Vob)) (0)
(I.8)
Note that, from the previous lemma, we may introduce As,t the solution to the matrix ODE
dt As.t = A.Aa. A = I.
(J.9)
Moreover, it follows that
Ag,t Vola, o).
(J.10)
We now ready state the relation between M, and Agtt
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Lemma
M₁ - L As,
(I.11)
Proof.
Using d de A0, Ar and we consider the time derivative of Ao
d
(at Aot Ao, d
d
de (M
M,)
(J.12)
dt
= - AcM + Ao + Ao, Ex
(J.13)
(J.14)
Note that Ag. is invertible by uniqueness of the ODE solution in Eq. (I.9). Integrating both sides with respect to gives
No = /
(J.15)
Hence, we have
Mc Ao /'" An E,ds.
(I.16)
Note that A = Ae. and Ao, 8,, As, we obtain
M, 1/" As,
(J.17)
An immediate application of the triangle inequality and the property of l|o yields
|Mell 1/" ||AA..llop Es Esllods.
(J.18)
Moreover, ||Ag,ellop admits the bound:
Lemma J.3.
|| S exp (L. || lopdlo
(J.19)
Proof. Define (s, = Naww. We have
d di
|1 [s, (8, 0)11= = [Ifu(s, 1)l2 [w(s
Te (((,))))
(J.20)
(s t)ll2 w A ArA
(J.21)
< | (s,
(J.22)
By Gronwall's theorem and [s, s)ll2 = |اا||| we obtain
(l, t) S cxpl J. ||Ar llopds').
(J.23)
To bound exp (IC ||Ayy llopdss) we use the following result from (Danicls, 2025), included here for completeness.
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Theorem 3 (Restated; Theorem 6 in (Daniels, 2025)). Suppose no N(O,1) and fi k-log- concave distribution with K. > 0. Define
o. BeXi, Xo no:
X1 p1,
(J.24)
and let (z(z) denote the corresponding flow field. Then,
V2Ve(x) & Ka + 3r
(J.25)
kaf + 38
With the result, we can bound exp (Jo' 1| llops)) as follows.
Lemma J.4.
-B I1,0),
where It 1== 1)ug lai, an ~ N(O,I, d1 ~ pl,
(J.26)
where P1 is k-log-concave. Then, we have
f
x(1 - t)2 t2 x(1 - 8)2
| 2a lo)loppsss s log
(J.27)
Proof. By leveraging Theorem 3 for each condition o, we have
Ras + Bar (ab 10)
(J.28)
kaz + 82.
then we have
R + B.BB8 ka? B3.
Vab lo)llpp
(J.29)
Integrating both sides, we obtain
{ Vabz lo)loodsss
Ka + Byd 3r ds
(J.30)
ka? + +3.
2 log(kcaz + (2)
(J.31)
knz + :37
log
(J.32)
kaa + 33
By substitute t and 3r = t, we have
x(1 11-1) 1)2 + f2
(as o) S $log /
(J.33)
x(1 8) + x2
With the preceding components in place, now establish Theorem 2.
J.3 PROOF OF THEOREM 2
By combining Eq. (J.18), Lemma J.3, and Lemma J.4, we have
[V.Po,(ao, o)lb S f/'
x(1 1)2 + t2
n(1 - s)2 + I|Vobi o)llods.
(J.34)
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For I, the function + x(1 s)² + s² attains its minimum at s K+1 inequality then yields
Applying Holder's
L/- V <(1 - s)2 4 82
Vo.0,
(ab o)llads
(J.35)
%
S max sE|0.1|
(V [<(1 [x(1-))2-
[ -)) s2
Vobg(a l))lldds
(J.36)
v + L/" Voobgaa lo)ll.ds.
(J.37)
K REGULARIZATION DOES NOT ACCOUNT FOR MANIFOLD ADHERENCE
In this section, we analyze linear, population-level surrogate for MIP test whether implicir regularization, instantiated via ridge regression and two-step MIP-like like iteration, can explain the observed manifold adherence. We mimic the two passes of MIP with two ridge-regularized linear regressions: (i) regression where the ridge penalty is applied to the Abservation-to-actiio (ii) a regression where the ridge penalty is applied to the action-to-action map. We then compose the two fitted maps to obtain the two-stage inference used by MIP. As shown below, it instead yields smooth spectral shrinkage and does not make the manifold absorbing.
map, and
Throughout we work in expectation (population covariances), so that conclusions reflect model structure rather than finite-sample effects. We assume independence of o, 2, and the additive noise n, and that the inverses write exist (otherwise interpret pseudoinverses the relevant supports).
Setup. Observations o Rdd and actions a E Idd follow the linear model
a= O o n, 1 N(0, E, r²1).
N(0, Σ.) be an auxiliary signal, independent of ao. n), and define == C a + 62E, We consider linear predictors à Bo + Cw with ridge regularization applied either to B (Section K.I) or to C' (Section K.2). This reflects the use of both the observation and the action a, in prediction.
with 1 Let 2
K.1 RIDGE REGRESSION FOR OBSERVATIIONTO-AACIII MAPPING (PENALTY ON B)
We solve
min B.C
E
Bo + Cw aaaal12
all
Define
X
v : BB CJ.
and, at the population level, the second-moment blocks
Σn == 0| = E.,
Σ12 Ejow 1 Elolcia + c22)
Σ21 ΣTz
Σn := 02 = E + +E,) + EZ.,
Σ, : Elao I'E..
E,g E(aw 1 (0"E, Eng),
where, for any vector I, we write x02 := zr
The objective can be written in trace form
f = [a v x (a xX) + BIB
tr(Bi(a TX VX) T) At
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Published conference paper at ICLR 2026
Dropping terms that are constant in (B, C), let
Ex Ex =x==
En
12
== = :
Piaz
Eezl.
Σ21
E22
Then
L= - + T) A
Differentiating gives
VL 2 + + + 2CΣ21 + X2B,
Vol 22 + +BBE1 + 2CE22.
Setting the gradients to zero yields the normal equations:
B + AI) + CΣ²1 Eal
BE + CΣ22 = Lia
Solving the linear system (c.g., by block climination) yields
B = Em Eg E2z Σaz) (E11 + XI) Eng E12
1
(
C= [E -B(E11 + XI)
Using Σ21 -000.. have Egi (0"E)) For (i),
(i) Eql 2-1 En Ena
1
= 0'D - E0' -1
+ +GE -00E0,,0 +2,)
= =c + +,,) + 33, -00 +Ep)
C1
C1
For (ii),
iii
[(En + X1) 222 Σ12|
[[E, + AI)
+ +E,) -E
[[E, O +c - ( - -1 9 T-
TO') + +E, + ( - + $ (0 -1
Isotropic speciaiizoo...,,..... Take E, = I, E. 11 Σ and -- diag(s:) Then
I, (i)
C1
(ii) (O') + (1
(L)
(1 (1 () + c
diag
[
1-1
8t
C1S;
=diag
Act + (1+ X) (c + cg)
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Hence, we obtain
22 Acts, + (1+ ()rr) +22)
1
(1 + 2)
B = diag
C diag
Acts, 2 + (1+ (ern
Define the shrinkage factor for the i-th singular direction by p Ba/ss. 85. Then,
P. Aet + ()
(K.1)
Key implications:
Ridge on B (A > 0) makes shrinkage s; s,-dependent. From (K.1), the factor decreases with 8; (because the denominator has sc)ss). So larger singular directions are shrunk more when À >> 0.
cis, +c ): shrinkage is constant across i.
If no ridge (A 0). B diag(
no ridge and no noise (A = 0, = 0). B = - shrinkage (recovers the standard solution).
no auxiliary z-signal (c2 0). 0.
K.2
RIDGE REGRESSION FOR ACTION-TO-ACO MAPPING (PENALTY C))
We now solve
min: B,C
EB
Bo CC Cw 1oto-da -al2 Al|CI/
The normal equations become
BE + CE21 = Eqa
BE C(E2 + Al) En2.
Solving gives the closed-form estimators:
(Eal C.221 )
= (Eoz Eal Em 112) (Ee2 + AI E21
E E12)
(8)
In the isotropic specialization Vn = -n E. = 1, = diag(s:) (:), one obtains
(c (c)+ c + + +x)
21 I.
B diag
C cी + 2 +
K.3 TWO-PASS LINEAR SURROGATE OF MIP
Let (B1, C1) denote the solution of Section K. with ridge A B, and (B2, C2) denote the solution of $K.2 with ridge X2 on C. To mimic the MIP two-pass inference rule in (4.3), we consider
åg t Bi0,
à d 120 + C2C2g.
We obtain
(B2 CC
Note that e₁ serves the analogue of f, from the main text. From Sections K.1 and K.2 we then obtain
=
(B2 + c1DB1
(c2 + >2)8;
cn
diag
+
e7p² + c +12
en7 + + de A1cts? 8? + (1 + X (c c2))
diag
+ +
(K.2)
c7tp² + c A2
die + (1+ (c c3)
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Moreover, the shrink factor will be
si 1
c \1o3s + (1+ (c +
2
22
(K.3)
$4 G +
A2
Key implications:
no ridge on B (A1 0). No s de-denend shrinkage.
no noise (1) 0). Just same the regular case: & O
Signal-to-noise effect. The quantity in (K.2) rises with and falls with 7, mildly favoring signal over noise by damping noisy directions.
Why this composition is a plausible proxy. The first stage applies ridge penalty to the observation-to-Ban -otiiioontt-atiion parameters B and predicts an interpolant action from observations alone, as in the t 0, 0 pass of MIP. We use ridge here canonical proxy for implicit regularization in the linear setting. The first stage applies ridge penalty to the action-to-action parameters C. It takes the interpolant action input (near 1) together with and produces the final output. Composing the two yields the operator 6 in (K.2), which is the linear analogue of the two-pass prediction of (4.3).
Why shrinkage does not yield manifold adherence. The operator in (K.2) acts as spectral shrinker: because the factors in (K.3) decrease with 8; (for Å1 0). it attenuates the dominant directions more than the weak ones-contrary serve principal directions and damp small and noisy modes. Since these factors lie in (0, 1] and vary smoothly with s,, 77, and Y1 A2, the map lacks any projection like behavior: once a point is off-manifole it is neither returned to nor retained on any low-dimensional subspace. Thus, implicit regularization alone, even with the two-pass composition of MIP, cannot account for the observed manifold adherence.
projection onto a manifold, which would pre-
L Toy EXPERIMENTS: TESTING THE FUNCTION APPROXIMATION CAPABILITIES OF REGRESSION AND FLOW MODELS
L.I OVERVIEW
This appendix summarizes an empirical comparison of training paradigms (regression, flow match- ing, straight flow, MIP) for function approximation with geometric constraints across three tasks: scalar reconstruction, high-dimensional projection with subspace constraints, and Lie algebra rota- tions. Experiments operate low-data regimes (50 training samples) using concatenation and FiLM architectures, with results averaged across multiple random seeds.
L.2 EVALUATION METRICS
Reconstruction: LI and L2 errors measure point-wise approximation quality between predictions fc) and targets f(c).
Three metrics assess geometric constraints in piecewise-constant projection structure: subspace diagonal quantifies predictions outside correct subspace P; for interval off-diagonal tests cross-interval generalization with mismatched projections, and boundary measures smoothness at interval transitions using combined adjacent subspace projections. All metries use normalized form
Projection:
[I(I - Il
Cosine similarity measures angular alignment between predicted and true rotation directions. Projection metric quantifies normalized perpendicular emoo relative to the rotation axis span.
Lie Algebra:
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L.3 KEY FINDINGS
Regression-based approaches achieve lowest L2 reconstruction (0.003197= = 0.000525 with L2 loss and FiLM), consistently outperforming flow-based methods on point-wise approximation tasks.
Task- Mask-Dependent Performance:
Flow Methods Excel Projections: Flow-based training demonstrates superior geometric con- straint satisfaction. Straight flow (flow matching without time conditioning) achieves best boundary projection (0.009769 0.001630) and Lie algebra projection metries (0.063612 0.000952), indi- cating beneficial geometric biases from learning probability transport.
MIP Competitive Performance: MIP combines direct regression with denoising regularization, achieving near-optimal reconstruction while maintaining reasonable geometric constraint satisfac- tion across tasks.
L.4 TRAINING LOSS CONSIDERATIONS
Results focus on 1.2-trained models, providing mathematically grounded objectives for both regres- sion and flow paradigms. While alternative loss functions evaluated empirically, flow-based LI training lacks principled derivation as conditional flow matching is naturally defined for squared error.
L.5 ARCHITECTURAL OBSERVATIONS
Both concatenation and FiL.M architectures demonstrated competitive performance with no consis- tent dominance. FiLM showed marginal advantages on certain geometric metrics for flow-based methods, suggesting affine feature modulation may better capture conditional dependencies prob- ability transport.
L.6 IMPLICATIONS FOR METHOD SELECTION
Tasks prioritizing point-wise reconstruction: regression- based training with L2 loss offers superior accuracy and computational efficiency.
Tasks requiring geometric constraint satisfaction: flow-based training provides significant advantages despite increased evaluation cost.
Straight flow's success suggests time conditioning may be unnecessary, enabling simpler models with competitive performance.
L.7 EXPERIMENTAL DETAILS
Study encompasses 540 runs: modes (regression, flow, straight flow, MIP, MIP one-step) x 2 losses x 2 architectures x 3 tasks x 3 seeds. Configuration: 256 hidden dimensions, 3 layers, ReLU, batch size 32, 50k epochs, Adam with Ir=0.001. Evaluation: 100k test samples; flow methods use Euler integration with ODE steps.
Full
Report:
much- aocch-ado-ane
report. much much-ado-as addo-abouttnoisiiie htt.a///ooootttoiaa/e///a///////
Code:
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M APPENDIX FOR SECTION 2
M.1 MARKOV DECISION PROCESSES CONFIGURATION
We consider a Markov Decision Process M (S. R, P. Po)2 with the state space S, the action space A, the reward R(s,a) a)" obtained by taking action in state s, the transition dynamies p x A + (SS), and the initial-state distribution Po E ASS) To formulate the success rate (i.e.. performance) in this setting, define the reward function
1, R(s, a) = {
if the task successful under (s, a);
(M.1)
otherwise.
Under this definition of rewards, the expected return of policy is - EIΣ, R(s,, at), which reduces to P[success under n.. Hence, (T) exactly equals the success rate of policy n
M.2 INTEGRATED FLOW PREDICTION
For completeness, provide the flow ODE
d dt ba o)
starting from
dg
(M.2)
The associated integrated flow prediction is given by
0 0) + S (a o)dt.
(M.3)
In practice, to approximate the ODE solution for sampling, we employ the following discretized Euler integration.
Definition M.1 (Discretized Euler Integration). We discretize the time interval [0, 1 to N steps with step size h 1/N. The iterates are then updated according to
ag++ 0k + h buk (ak |0), O, ...,, N
(M.4)
The final iterate serves the Euler approximation Φg (z o). We also refer to N as the Number of Function Evaluations (NFEs).
N LLM USAGE
We used LLMs only for minor language polishing (grammar and wording) and to assist with liter- ature search. All technical content, experiments, and conclusions were created and verified by the authors.
FFor simplicity, consider the MDP in this context by identifying the with the observation defined in 2. More generally, one may consider Partially Observable Markov Decision Process (POMDP). where the agent receives observation emitted by an underlying latent state s. Foor ease of exposition, use the same notation for rewards defined random variables and their distri- butions.
40


Reviews:
[{"review_id": "ZEuKfKJdC8", "reviewer": "Reviewer_ddUV", "summary": "This paper seeks to understand the \"secret sauce\" behind the recent success of Generative Control Policies (GCPs) in robotics. While the field has largely attributed this success to the power of generative models (like diffusion) to handle multi-modality or complex representations, this paper systematically analyzes those common assumptions.\n\nThe authors argue that the generative, distributional aspect of these models is their least important feature. Instead, the performance leap comes from a powerful and previously under-appreciated combination of two other design choices: Supervised Iterative Computation (C3) and Stochasticity Injection (C2).\n\nThere're several claims in the paper\n\n1. Analysis on multimodality of GCPs\nThe paper's argument is built in two parts. Firstly, the authors show that on tasks known to be multi-modal, GCPs show no significant performance advantage. Using a set of ablation experiments (e.g., showing that taking the mean action doesn't cause performance to collapse), they provide evidence that the models are not, in fact, learning distinct action modes.\nThe paper challenges the idea that the iterative nature of GCPs makes them deeper or more expressive. Through a theoretical argument (Theorem 1) and an empirical measurement of the Policy Lipschitz Constant, the authors show the exact opposite: successful GCPs learn smoother, less complex functions than their regression counterparts.\n\n2. the authors introduce three core components of any GCP:\nC1: Distributional Learning \nC2: Stochasticity Injection (adding noise during training)\nC3: Supervised Iterative Computation (refining an action over multiple steps)\nThe paper proposes simplified algorithm they call the Minimal Iterative Policy (MIP). MIP is designed to completely discard the complex generative component (C1) and rely only on C2 and C3.\nThe results show that this simple MIP matches the performance of the full, state-of-the-art Flow-based GCPs. This experiment conclusively demonstrates that the C2+C3 combination is the true engine of performance.\n\n3. Finally, the paper provides a powerful explanation for why this C2+C3 combo is so effective.\nThey find that while MIP is no better than regression at simple imitation (i.e., they have the same validation loss), it is dramatically better at producing plausible actions when it encounters unfamiliar states. This iterative refinement process learns to fail gracefully, always steering its actions back toward the \"manifold\" (the space of valid expert actions). The iterative process (C3) fails on its own. The authors find that the injected noise (C2) is the critical ingredient that stabilizes the training, allowing the iterative refinement to learn its task without being derailed by compounding errors.", "questions": "Besides the questions I raised in the weakness part. \n\nMinor Point::\nThe paper's citation on line 159 for the concept of \"straight interpolation\" (or similar flow matching/rectified flow concepts) is incomplete. The authors cite only \"Flow Matching\" (Lipman et al. [3]), but this was one of several foundational works that introduced this concept concurrently.\n\n[1]Albergo, Michael, and Eric Vanden-Eijnden. \"Building Normalizing Flows with Stochastic Interpolants.\" ICLR 2023 Conference. 2023.\n[2] Heitz, Eric, Laurent Belcour, and Thomas Chambon. \"Iterative α-(de) blending: A minimalist deterministic diffusion model.\" ACM SIGGRAPH 2023 Conference Proceedings. 2023.\n[3] Lipman, Yaron, et al. \"Flow Matching for Generative Modeling.\" The Eleventh International Conference on Learning Representations.\n[4] Liu, Xingchao, and Chengyue Gong. \"Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow.\" The Eleventh International Conference on Learning Representations.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. A key finding of this paper is that when using the same network architecture, traditional Regression Control Policies (RCPs) are highly competitive with modern Generative Control Policies (GCPs). This is an important contribution, as it helps correct a potential misconception in the field (stemming from earlier works that used different architectures for baselines) that diffusion policies inherently offer a large performance gain over standard behavior cloning on these tasks.\n\n2. A major strength of this paper is its systematic methodology. The authors first propose a novel taxonomy to deconstruct GCPs into three key components, and then create 'ablation' models (MIP, RR, and SF) to empirically isolate and test the precise performance contribution of each component.", "weaknesses": "1. The paper's use of the Lipschitz constant as a metric for \"expressivity\" is a potential point of confusion. As you noted, the Lipschitz constant measures smoothness or robustness (how much the output can change for a small input change), not necessarily the representational capacity (the class of functions a model can learn). While their point (that GCPs find smoother solutions) is valid, their terminology can be debated.\n\n2. The paper's theoretical argument, presented as Theorem 1 (Informal), feels underdeveloped. It attempts to formally decouple the number of iterative steps from an equivalent network's depth, but the premise of this connection to a diffusion policy's true expressiveness is not rigorously established. As presented, it functions more as a high-level intuition than a solid proof, making it a weak pillar for the paper's claims.\n\n3. A significant limitation is the paper's narrow scope. The strong conclusion that Distributional Learning (C1) is the \"least important factor\" is derived entirely from a single-task, high-precision setting. This overlooks the primary strength of generative models: capturing complex, diverse distributions, which is fundamental for multi-task learning. The success of diffusion models in other domains (e.g., vision) is precisely their ability to model complex, multi-modal data distributions. It is highly plausible that C1 is, in fact, the critical component for scaling to multi-task behavioral distributions, making the paper's central finding a potential artifact of its limited, single-task evaluation.\n\n4. A particularly counter-intuitive finding is the diffusion policy's failure to produce multi-modal behavior, given that diffusion models are expressly designed to capture complex data distributions. This observed uni-modal output may not be a failure of the model itself, but rather an artifact of the DDPM sampling process.\nDDPM functions as a stochastic sampler (an SDE), and its Langevin dynamics term, while beneficial for error correction, could be the mechanism responsible for mode concentration. This stochastic term might be \"correcting\" the trajectory so effectively that it collapses different potential paths into a single, high-density outcome.\nA crucial follow-up experiment would be to test the policy with a deterministic DDIM sampler (an ODE). This would remove the influence of the Langevin term, offering a clearer view of the true learned distribution. It's possible that this would reveal the multi-modality the model did learn, even if it simultaneously lowers task performance by removing the stochastic error correction.\n\nOverall, this is an interesting paper with provocative findings. However, its conclusions feel brittle. They contradict the established behavior of diffusion models in other domains, a discrepancy likely stemming from the paper's narrow, single-task focus. The arguments lack full rigor, relying on conflated terminology (Lipschitz vs. expressivity) and informal theorems. The paper successfully isolates a powerful mechanism (C2+C3) for this specific setting, but it does not provide a generalizable understanding of generative policies in robotics.", "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": "r02fWQoxc7", "reviewer": "Reviewer_QhDz", "summary": "This paper investigates why generative control policies often outperform regression-based ones in imitation learning. The authors aim to show (through controlled evaluations) that this advantage is not due to multimodal behaviors or higher expressiveness of generative models, but rather due to manifold adherence. They argue that, since control policies mainly map observations to actions, such complexity is unnecessary; what matters is incorporating some degree of stochasticity and supervised iterative computation. Based on this insight, they propose a minimal approach that achieves strong results on several continuous control benchmarks.\n\nOverall, I find the contribution valuable for the community. The main evaluation, while not entirely conclusive, is well-designed and offers interesting (and somewhat counterintuitive) insights, showing that generative policies’ strength may not come from multimodality or expressiveness. The second contribution, the MIP framework, also seems practical and empirically validated. I have mixed opinions overall: some parts of the evaluation are not fully rigorous, but the results and reasoning are promising. I would give this paper an initial rating of 6 and look forward to further discussion with the authors and other reviewers.", "questions": "Other than the points above, I also have a more general question. From the perspective of the true action generation process, it’s inherently a distribution. So, in principle, learning this distribution directly (as GCPs do) still seems like the optimal approach. The paper focuses on which components of the objective function work best, but from a first-principles standpoint, modeling the full action distribution still feels like the most appropriate formulation IMO", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. I like the motivation of this work. Decomposing and analyzing the underlying reasons behind the success of generative policies is valuable, especially for RL and imitation learning, where understanding why a model works often matters more than just scaling it up. As often said, RL doesn’t necessarily benefit from ever-larger networks like LLMs or VLMs do  (IMO since the right actions or behaviors come from a subtle, structured distribution that doesn’t simply improve with more data or capacity..).\n\n2. For the most part, the evaluation design is solid, and the conclusions drawn from the hypothesis testing seem meaningful and informative.\n\n3. The proposed minimal algorithm looks sound and provides fresh insights into behavior cloning and how to simplify it (with stochasticity, and iterative computing but not necessarily model the whole distribution) without losing performance.", "weaknesses": "I list both my (tentative) weaknesses and questions together here, since some of them overlap, and a few points might just come from my not fully understanding certain parts of the evaluation.\n\n1. On the conclusion that GCPs cannot really produce multi-modal actions, I have some concerns about the setup of evidence C. Even in deterministic environments and with deterministic policies, the underlying mapping from observation to action can still be stochastic or even multi-modal at the distribution level (e.g., multiple valid actions for the same state). So using this setting to claim the absence of multi-modality might be a bit too strong.\n\n2. Related to that, the theoretical part on policy expressiveness seems to lean heavily on the assumption that there is no multi-modality (so that you can use the k-log-concave unimodal distribution arguments). But this then fully depends on the correctness of the earlier empirical conclusion about multi-modality. In practice, we do observe multimodal behaviors in pretrained GCPs, e.g. in the DP-like setups, so I’d like to see more clarification here.\n\n3, For the claim that C2 supports C3 (i.e., noise injection can replace data augmentation), I’m not fully convinced. The noise here is injected on the action/control side or in the dynamics, while “data augmentation” in imitation learning can also mean visual, semantic, or context-level augmentation, which GCPs can implicitly provide. It might help to define more precisely what kind of “augmentation” you mean.\n\n4. For the pixel-based control part where you show RCP can match GCP, I suspect this may partly come from the architectural alignment you used. If that’s the case, then H1 may not be as strong a takeaway as presented, it could just be an architecture effect. It would be good to clarify whether there is another implication here about the observation-to-action mapping.\n\n5. To make the conclusions more universal, it would help to also test diffusion-based and transformer-based imitation policies. Even if they share the spirit of flow models, it would make the claim solid that the result is not specific to one generative family.\n\n6. About the choice of $t_{*}=0.9$ is this environment-specific or a general setting? If it’s a trade-off hyperparameter, it would be good to say so and show a small sensitivity analysis", "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": "OSASG1ym3X", "reviewer": "Reviewer_Kdio", "summary": "This paper investigates why generative control policies (GCPs)—such as flow- or diffusion-based action models—perform better than standard regression policies in behavior cloning (BC).\nRather than assuming the benefit arises from distributional modeling (learning multimodal \n𝑝\n(\n𝑎\n∣\n𝑜\n)\np(a∣o)), the authors dissect GCPs into three conceptual components:\n\nC1: Distributional learning (learning a stochastic action distribution)\n\nC2: Stochastic training (injecting noise during training)\n\nC3: Supervised iterative computation (multi-step prediction with intermediate supervision)\n\nThrough controlled experiments across 27 BC benchmarks (Robomimic, Adroit, MetaWorld, etc.) with matched architectures, they show that C1 is not important, while C2 and C3 are the true drivers of GCP performance.\nBased on this, they propose a Minimal Iterative Policy (MIP) that keeps only C2 + C3—achieving parity with flow models but with deterministic inference and far lower compute cost.\nThey also introduce manifold adherence as a diagnostic for how closely policy outputs stay near expert data manifolds.", "questions": "Have you tried learning $\\(t^*\\)$ or using a schedule instead of a fixed value (0.9)?\n\nCould MIP’s iterative refinement be stacked (3–4 steps) without re-introducing flow losses—does performance saturate after two steps?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Rigorous experimental isolation of C1, C2, C3 effects.\n\nUnusually broad benchmark coverage (27 tasks).\n\nStrong ablations: flow vs regression vs MIP.\n\nNegative result clearly demonstrated and well-reasoned.\n\nPractical takeaway: simple two-step deterministic policy matches flow performance.\n\n“Manifold adherence” metric offers a novel lens on robustness.", "weaknesses": "All evaluations are offline imitation; no fine-tuning or real-robot rollouts to confirm closed-loop stability.\n\nMIP’s hyperparameters (e.g. noise scale) are fixed; sensitivity analysis would help.", "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": "NdNixPqGyb", "reviewer": "Reviewer_zgJS", "summary": "The paper questions why generative policies, specifically diffusion-based and flow-based imitation learning models, appear to outperform regression-based policies in robot learning. It argues that their success is not due to better distribution modeling or multimodality capture, but rather to algorithmic structure. The paper examined a few hypotheses and then concluded that GCPs are not fundamentally superior distribution learners; their strength arises from how their training and inference processes structure learning.", "questions": "- The paper empirically shows that the combination of stochasticity and iteration works. Is the improvement due to smoother gradient landscapes, implicit ensemble effects, or noise-induced curvature exploration or is there any intuition?\n- Can MIP generalize across different goals or objects without retraining? This would test whether its simplicity generalizes or overfits. Compared to it, flow-based policies often work well in multi-task settings where distributional conditioning is crucial.\n- How does MIP perform under camera noise, lighting changes, or slight object translation? Does MIP maintain robustness to sensor noise or domain shift? Since MIP is deterministic, it may degrade fast.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper conducted comprehensive experiments and had careful comparison with architecture parity and identical backbones, ensuring fairness and validating the hypothesis. It has a large-scale benchmark study across 27 imitation learning datasets that isolates the actual causes of GCP success.\n- It separates generative modeling goals from control objectives and proposes a new non-generative baseline that rivals flow policies MIP.\n- It shows that distribution fitting is largely irrelevant for single-task control, which is a major conceptual correction.\n- The insight that a two-step regression model can rival state-of-the-art flow policies is surprising.", "weaknesses": "- The experiments fix inference to deterministic modes, although guided sampling or diffusion-based control might behave differently.\n- The study focuses on a single-task setting, while multi-task or goal-conditioned GCPs might still benefit from distributional modeling.\n- The paper does not test whether C2+C3 advantages persist under RL fine-tuning or long-horizon planning, which is crucial for robotic applications.", "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)