Flow-Support-Weighted Proposals for Test-Time Verified VLA Policies

Chen Zhao; Zipeng Ye; Haitao Shen; Guanlin Li; Yuchen Xie; Jing Zhang

cs.ROOct 10, 2026

Flow-Support-Weighted Proposals for Test-Time Verified VLA Policies

Authors: Chen Zhao, Zipeng Ye, Haitao Shen, Guanlin Li, Yuchen Xie, Jing Zhang

Organizations: Renmin University of China, Beijing 100872, China · Key Laboratory of Data Engineering and KnowledgeEngineering, MOE, China · Meituan, Beijing 100102, China

Abstract

Test-time sampling promises to turn one action prediction into a set of distinct executable candidates that a verifier can choose among. Downstream fine-tuning works against that promise. A vision-language-action model trained to solve one task distribution concentrates its flow-matching decoder around the dominant demonstrated action, so repeated samples differ only marginally and the verifier receives many copies of one decision. We call this proposal contraction and identify it as a key constraint on test-time scaling: repeated samples can remain highly redundant as the candidate budget grows. Action Dynamic Sampling (ADS) alleviates the constraint on the proposal side, during training. At each training step, ADS integrates $K$ additional trajectories from the current policy under fresh base noise, scores each by the flow support along its path with a Hutchinson trace estimator, and corrects the policy's local action distribution toward this weighted set through a detached finite-set KL term. Training cost rises; deployment cost is unchanged. Across simulation and four physical-robot tasks, ADS raises proposal spread without the success penalty that Gaussian perturbation incurs, lifts oracle best-of-8 success on RoboCasa from 57.6% to 63.0%, improves fixed verifiers by 2.4 to 4.9 points in simulation, and raises TokenPB by up to 13.0 points on real-world manipulation.

Figures & tables

Figure 1: ADS produces broader, more useful proposal sets. Standard flow-matching training maps different noise draws onto one dominant action mode, leaving the verifier with redundant candidates. ADS corrects the policy against a flow-support-weighted set of its own samples; the resulting candidates cover more valid modes and get better performance, with the verifier unchanged.

Figure 2: Overview of ADS training and test-time verification. During training, detached siblings drawn from the current policy form a finite reference set, a Hutchinson trace estimator weights that set by flow support, and a KL term corrects the policy’s local action distribution toward it. At test time ADS adds no module: it supplies a broader, policy-aligned candidate set to an unchanged verifier.

Method	RoboCasa	LIBERO-Plus	WidowX-Bridge	Google Robot	Proposal spread
Baseline	54.9	76.5	58.6	69.1	0.003
Gaussian Noise ( $\sigma=5\%$ )	53.3	75.2	58.4	63.2	0.011
Gaussian Noise ( $\sigma=10\%$ )	45.6	73.4	56.3	56.3	0.025
Gaussian Noise ( $\sigma=20\%$ )	39.8	65.9	55.1	50.7	0.044
ADS	56.4	79.4	62.6	70.1	0.021

Table 1: Gaussian action perturbation and ADS without test-time verification. Task performance is success rate (%). Proposal spread is the root mean squared pairwise distance among action chunks sampled for the same condition. Stronger Gaussian noise buys spread and pays for it in success; ADS gains both.

Figure 3: Left: Proposal quality and spread over training. ADS holds a broader candidate set at comparable task success, while Gaussian perturbation is spread out but less policy aligned. Right: ADS complements test-time verification. ADS improves both RoboMonkey and TokenPB without changing the verifier.

Metric	Robocasa		RoboTwin 2.0		LIBERO-Plus		Real-World
Metric	Flow-matching	ADS	Flow-matching	ADS	Flow-matching	ADS	Flow-matching	ADS
Mean distance $d_{\mathrm{mean}}$	0.091	0.102	0.042	0.075	0.086	0.082	0.057	0.068
Best distance $d_{\mathrm{best}}$	0.085	0.032	0.036	0.024	0.065	0.027	0.046	0.013

Table 2: Training-set replay diagnostic on simulation and real-world tasks. For each training condition, we sample

M=8

action chunks and measure the distance from each to the demonstrated action.

d_{\mathrm{mean}}

averages over all candidates, while

d_{\mathrm{best}}

first selects the closest candidate for each condition.

Method	Background	Light	Language	Noise	Layout	Robot Init.	Camera	Overall
RoboMonkey	94.1	93.2	91.7	77.8	82.7	57.0	56.5	77.6
TokenPB	96.7	95.6	86.7	79.1	83.1	55.2	52.2	76.9
RoboMonkey + Gaussian Noise	96.9	94.1	91.2	80.8	85.5	56.9	58.6	79.4
RoboMonkey + ADS	97.9	96.7	93.9	81.6	88.1	58.5	62.5	81.3
TokenPB + ADS	99.0	97.6	90.6	84.7	90.2	57.8	61.4	81.6

Table 3: Proposal-side strategies on LIBERO-Plus. The first group contains verification methods with an unmodified proposal policy; the second group augments the verifier with Gaussian perturbation or ADS.

	WidowX-Bridge (SR)					Google Robot (SR)
Method	Stack Cube	Carrot on Plate	Eggplant in Basket	Spoon on Towel	Avg.	Drawer	Move Near	Pick Coke	Put in Drawer	Avg.
RoboMonkey	38.2	64.7	86.7	72.9	65.6	–	–	–	–	–
TokenPB	46.9	70.8	100.0	85.4	75.8	57.1	83.7	86.8	50.3	76.4
Gaussian Noise + TokenPB	26.4	65.5	89.6	65.8	61.8	55.8	76.4	82.5	58.6	73.3
RoboMonkey + ADS	46.5	71.9	87.5	76.0	70.5	–	–	–	–	–
TokenPB + ADS	58.3	77.1	97.9	83.3	79.2	59.6	93.5	89.4	57.1	79.4

Table 4: Proposal-side strategies on SimplerEnv. The first group contains methods with an unmodified proposal policy; the second group introduces Gaussian perturbation or ADS.

Training objective	Success (%)	Proposal spread
$\mathcal{L}_{\mathrm{FM}}$	54.9	0.003
$+\,$ uniform siblings ( $q_{k}=1/K$ )	48.6	0.029
$+\,$ base-density weighting ( $\ell_{k}=\log p_{\mathrm{base}}(z^{(k)})$ )	53.2	0.023
$+\,$ flow support (full ADS)	56.4	0.021

Table 5: Component ablation on RoboCasa without verifier. Each row adds one element of the ADS objective to the one above it, averaged over five seeds. Success is single-sample; proposal spread is measured among candidates drawn for the same condition.

	Geometric redundancy		Valid modes		Oracle coverage (%)		Verifier success (%)
Candidates $M$	FM	ADS	FM	ADS	FM	ADS	FM	ADS
1	1.00	1.00	1.00	1.00	54.9	56.4	54.9	56.4
2	1.04	1.35	1.02	1.25	55.9	58.4	55.1	56.9
4	1.06	1.62	1.05	1.63	56.8	60.7	55.3	57.5
8	1.08	2.05	1.08	2.30	57.6	63.0	55.5	57.9
16	1.09	2.30	1.10	2.85	58.0	64.2	55.6	58.1

Table 6: Candidate scaling on RoboCasa with RoboMonkey. Each block reports one property of the same

M

candidates, using the metrics defined in the experimental setup. Oracle coverage and verifier success differ only in who selects the executed chunk. The ADS advantage widens with

M

before saturating.

Figure 4: Real-world four-object manipulation results. Left: representative execution sequences. Middle: normalized success scores for unverified, TokenPB-selected, and RoboMonkey-selected policies. Right: proposal diversity measured by pairwise action RMSE. ADS broadens the proposal set and improves both fixed verifiers, including on the unseen shelf task.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Real-world evaluation setup. The workspace consists of a storage area , a manipulation area , a first-person-view camera, a third-person-view camera, and a Dexmal robotic arm.

Component	Configuration	Parameters updated by ADS KL
Vision-language backbone	StarVLA base, 2.1B parameters, 224 $\times$ 224 RGB input	No
Language context	128 tokens, task instruction and embodiment tokens	No
Action expert	GR00T-style flow transformer, width 1024, 16 blocks, 16 heads	Yes
Action representation	normalized joint/EEF action chunks with gripper state	Primary endpoint only
Condition interface	32 projected vision-language tokens	Detached
Numerical precision	bfloat16 backbone, float16 trace, float32 optimizer states	–

Appendix

Table 7: VLA architecture used in all controlled experiments. The baseline and ADS share every component; ADS changes only the training objective applied to the action expert.

Suite / embodiment	Tasks	Train demos/task	Train/val	Action dim. / horizon	Updates	Eval. episodes/task/seed
LIBERO-Plus	40	42	-	7 / 16	40k	50 per shift
SimplerEnv: WidowX-Bridge	4	—	99% / 1%	7 / 8	50k	task-dependent
SimplerEnv: Google Robot	4	—	99% / 1%	7 / 8	50k	task-dependent
RoboCasa	24	100	99% / 1%	7 / 16	60k	50
RoboTwin 2.0	50	550	95% / 5%	14 / 16	60k	50

Appendix

Table 8: Dataset splits and evaluation budgets. SimplerEnv is one suite but its WidowX-Bridge and Google Robot embodiments are kept as separate rows and are never averaged together. Its evaluation counts follow the official task-specific configurations.

Parameter	Final value	Values evaluated	Parameter	Final value	Values evaluated
Optimizer	AdamW	AdamW	Batch size	64	32, 64
Action-expert LR	$1\!\times\!10^{-4}$	$5\!\times\!10^{-5}$ , $10^{-4}$	Backbone LR	$2\!\times\!10^{-5}$	$10^{-5}$ , $2\!\times\!10^{-5}$
Weight decay	0.01	0.00, 0.01	Gradient clip	1.0	0.5, 1.0
Warmup updates	500	0, 500	LR schedule	cosine	constant, cosine
Sibling count $K$	8	2, 4, 8, 16	Trace probes $R$	8	1, 2, 4, 8
Temperature $\tau$	1.0	0.25, 0.5, 1.0, 2.0	ADS weight $\lambda$	$10^{-3}$	$10^{-4}$ , $3\!\times\!10^{-4}$ , $10^{-3}$ , $3\!\times\!10^{-3}$

Appendix

Table 9: Optimization, sampling, and ADS hyperparameters. The final value is selected by validation single-sample success, with lower training FLOPs used to break differences below 0.2 points.

Evaluation	Suite / embodiment	FM	ADS	$\Delta$ 95% CI	$p$
Proposal only	RoboCasa	$54.9\pm 0.6$	$56.4\pm 0.7$	$[0.5,2.5]$	0.004
Proposal only	LIBERO-Plus	$76.5\pm 0.8$	$79.4\pm 0.7$	$[1.8,4.1]$	$<0.001$
Proposal only	WidowX-Bridge	$58.6\pm 1.1$	$62.6\pm 1.0$	$[2.2,5.8]$	$<0.001$
Proposal only	Google Robot	$69.1\pm 0.9$	$70.1\pm 0.8$	$[0.1,1.9]$	0.031
RoboMonkey@ $8$	RoboCasa	$55.5\pm 0.7$	$57.9\pm 0.6$	$[1.1,3.7]$	$<0.001$
RoboMonkey@ $8$	RoboTwin 2.0	$89.8\pm 0.5$	$93.5\pm 0.4$	$[2.2,5.1]$	$<0.001$

Appendix

Table 10: Seed variation and paired significance tests. Values are success rate in percent (mean

\pm

standard deviation over five training seeds). Confidence intervals and two-sided

p

-values use the hierarchical paired bootstrap described in the text.

Diagnostic	Conditions	Samples/condition	Checkpoints	Aggregation
Training-set replay	2,000	8	final	$d_{\mathrm{mean}}$ and $d_{\mathrm{best}}$
Local $Q-P$ rebalancing	512	8	$p=0.6,0.7,0.8,0.9,1.0$	condition mean, then seed mean
Trace–rollout validity	500	8	final	20,000 pooled candidates; seed bootstrap
Endpoint Jacobian	256	4 noise draws, 32 JVPs	final	singular-value statistics per condition
Action-mode occupancy	200	64	five ADS-phase points	centroid matching within condition
Candidate-budget scaling	500	$M\in\{1,2,4,8,16\}$	final	paired initial states

Appendix

Table 11: Sampling protocol for supplementary diagnostics. Counts are per training seed unless a total is explicitly stated.

Training method	Train FLOPs	Step time	Peak memory	Deployment FLOPs
Flow matching	1.00 $\times$	1.00 $\times$	1.00 $\times$	1.00 $\times$
ADS ( $K=4,R=4$ )	1.67 $\times$	1.09 $\times$	1.15 $\times$	1.00 $\times$
ADS ( $K=8,R=4$ )	2.34 $\times$	1.16 $\times$	1.25 $\times$	1.00 $\times$
ADS ( $K=8,R=8$ , default)	3.37 $\times$	1.28 $\times$	1.40 $\times$	1.00 $\times$

Appendix

Table 12: Training-compute accounting for ADS. Training FLOPs, step time, and peak memory are normalized by standard flow matching under the same hardware and batch size. Deployment FLOPs remain unchanged.

Panel of Figure 6 — Figure 6: ADS training flow. Flow matching anchors task correctness. Once ADS is enabled, detached siblings supply a current-policy reference, and the Hutchinson trace reweights that reference by flow support before the KL update reaches the action expert.

Method	$C/C_{0}$	$u^{*}/N_{0}$	$\lambda$	Spread	$S_{1}$	Oracle@8	RoboMonkey@8	TokenPB@8
FM (standard budget)	1.00	1.00	–	0.003	54.9	57.6	55.5	56.4
FM (equal FLOPs)	2.37	1.67	–	0.002	55.9	59.9	56.9	56.8
FM + $H(p)$	2.46	1.67	–	0.032	42.3	53.0	50.3	51.5
FM + covariance entropy	1.49	1.67	–	0.027	47.1	54.4	52.2	50.5
FM + KDE Rényi-2 entropy	2.11	1.25	$10^{-4}$	0.012	45.6	58.4	52.0	52.6
FM + ADS	2.37	1.00	$10^{-3}$	0.021	56.4	63.0	57.9	58.9

Appendix

Table 13: Training controls on RoboCasa. Training controls on RoboCasa with measured total training FLOPs.

C/C_{0}

is full training cost,

u^{*}/N_{0}

is the mean selected update across seeds, and

\lambda

is the selected auxiliary weight. Spread is pairwise action RMSE. Success and coverage are percentages averaged over five seeds.

$u/N_{0}$	Spread	$S_{1}$	Oracle@8	RoboMonkey@8	TokenPB@8
$1/3$	0.018	43.0	49.5	45.6	46.4
$1/2$	0.012	49.0	55.0	51.5	52.3
$2/3$	0.007	53.0	57.8	54.8	55.6
$5/6$	0.004	54.5	58.0	55.6	56.3
$1$	0.003	54.9	57.6	55.5	56.4

Appendix

Table 14: Fixed checkpoints along the standard FM trajectory on RoboCasa. All rows retain the original learning-rate schedule. Spread is measured at

M=8

; success and coverage are percentages. No row is selected using its test outcomes or diversity.

Sibling count $K$				Trace probes $R$				Temperature $\tau$
$K$	SR	Spread	FLOPs	$R$	SR	Spread	FLOPs	$\tau$	SR	Spread	Entropy
2	55.7	.011	1.59 $\times$	1	51.8	.028	1.57 $\times$	.25	55.8	.015	0.42
4	56.1	.017	2.19 $\times$	2	53.2	.024	1.83 $\times$	.50	56.2	.019	0.58
8	56.4	0.021	3.37 $\times$	4	56.3	.023	2.34 $\times$	1.00	56.4	0.021	0.71
16	56.2	.024	5.74 $\times$	8	56.4	0.021	3.37 $\times$	2.00	56.0	.024	0.84
ADS weight $\lambda$						Trace-start progress $p$

Appendix

Table 15: Sensitivity to the principal ADS hyperparameters on RoboCasa. Each block changes one parameter from the default

K=8

R=8

\tau=1

\lambda=10^{-3}

, and trace start

p=0.6

. Results show saturation in

R

and a shallow optimum in the remaining parameters; spread is proposal spread, and entropy is the normalized target entropy

H(q)/\log K

Proposal strategy	Trace–succ. $\rho$	Valid modes@ $8$	Invalid rate $\downarrow$
Gaussian perturbation	0.06	1.31	11.8%
Uniform siblings	0.18	1.72	7.6%
Base density only	0.29	2.02	5.2%
Flow support (full ADS)	0.43	2.30	3.1%

Appendix

Table 16: Does trace weighting favor valid rather than arbitrary variation? Trace–success correlation is Spearman’s

\rho

between the flow-support score and candidate rollout success. The inter-mode invalid rate counts candidates that fall outside every valid action mode. Full ADS reaches the highest valid coverage and the lowest rate of unsupported between-mode candidates.

Training progress $p$	$H(p_{\theta,\cdot\mid 0})/\log K$	$H(q)/\log K$	$\mathbb{E}[\Delta H]$	$\Pr(\Delta H>0)$	Valid minor modes with $Q_{b}>P_{b\mid 0}$
0.60	0.58	0.65	+0.07	61.8%	54.2%
0.70	0.53	0.68	+0.15	69.7%	58.6%
0.80	0.47	0.71	+0.24	76.8%	63.5%
0.90	0.44	0.72	+0.28	80.9%	66.1%
1.00	0.42	0.71	+0.29	82.4%	67.3%

Appendix

Table 17: Local surrogate correction during the ADS phase on RoboCasa. Entropies are normalized by

\log K

. The final column reports the fraction of valid non-dominant modes for which

Q_{b}>P_{b\mid 0}

. Positive entropy gaps and mode margins show that the detached sibling reference is usually broader than the primary-conditioned surrogate.

Training objective	$\\|J^{z}_{\theta}\\|_{F}$	Eff. rank	Log-vol.
$\mathcal{L}_{\mathrm{FM}}$	0.18	1.18	$-8.42$
$+\,$ uniform siblings	0.31	1.74	$-6.37$
$+\,$ base density only	0.38	2.08	$-5.61$
$+\,$ flow support (full ADS)	0.44	2.43	$-4.92$

Appendix

Table 18: Endpoint sensitivity to base noise on RoboCasa. We estimate singular values

s_{i}

J^{z}_{\theta}=\partial\Psi_{\theta}/\partial z

with randomized JVPs. Effective rank is

(\sum_{i}s_{i}^{2})^{2}/\sum_{i}s_{i}^{4}

, and log-volume is

\sum_{i}\log(s_{i}+10^{-6})

. Larger values mean more latent directions reach the action endpoint, which a wider candidate set requires but which does not by itself make those directions executable.

ADS-phase progress	FM	ADS	FM	ADS
	Occupied valid modes		Dominant-mode mass
0.00	2.21	2.21	0.52	0.52
0.25	1.78	2.16	0.64	0.55
0.50	1.42	2.13	0.75	0.58
0.75	1.18	2.11	0.85	0.61
1.00	1.07	2.08	0.91	0.63

Appendix

Table 19: Occupancy of valid action modes for fixed training conditions. At the start of the analysis phase, normalized action chunks are embedded by temporal concatenation and clustered with

k

-means to define the modes;

k

is chosen by silhouette score, modes with rollout success below 50% are marked invalid, and later centroids are matched with the Hungarian algorithm. The statistic is population-level occupancy over the sampled set at each checkpoint.

$s$	$\widehat{\rho}$	Occupied cells		Target coverage
	MC	MC	Analytic	MC	Analytic
0.05	$-0.000662$	1.000010	1.000005	0.000000	$6.09588\times 10^{-23}$
0.10	0.000079	1.097250	1.097222	0.000000	$2.29321\times 10^{-6}$
0.25	0.000492	2.513520	2.512776	0.168490	0.16815
0.50	$-0.000609$	3.644550	3.644815	0.749010	0.748932
1.00	0.001080	5.111300	5.116877	0.947410	0.947742

Appendix

Table 20: Independent Gaussian sampling. Correlation is averaged over distinct candidate-index pairs across independent sets. Coverage refers only to the predefined geometric target.

$w$	$s_{1}$	$\overline{Q}_{1,8}$	$Q_{1,\infty}$	$P_{\mathrm{seen}}$
0.05	0.01	0.152637	0.216920	0.336580
0.05	0.10	0.028886	0.026954	0.336580
0.05	1.00	0.003244	0.002762	0.336580
0.20	0.01	0.700244	0.862069	0.832228
0.20	0.10	0.346170	0.384615	0.832228
0.20	1.00	0.066806	0.058824	0.832228

Appendix

Table 21: Known-mixture density weighting.

\overline{Q}_{1,8}

is a Monte Carlo mean of within-set normalized mass;

Q_{1,\infty}

and

P_{\mathrm{seen}}

use the analytic formulas above.

$q_{+}$	$\kappa$	$\mu_{0}$	Final $\mu$	$d_{\min}$	In neighborhood
0.5	0	$-1$	0.000000000	1.000000000	0
0.5	0	$+1$	0.000000000	1.000000000	0
0.5	20	$-1$	0.800660032	0.199339968	1
0.5	20	$+1$	0.800660032	0.199339968	1
0.8	0	$-1$	0.173286795	0.826713205	0
0.8	0	$+1$	0.173286795	0.826713205	0

Appendix

Table 22: Fixed-surrogate optimization with an optional explicit quadratic anchor. We report numerical endpoints, distances to the nearest sibling, and geometric neighborhood indicators.

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