When One Step Is Enough: An Empirical Study of Euler Integration in Flow-Matching Policies

Shivakanth Sujit; Luca Nunziante; Rousslan Fernand Julien Dossa; Kai Arulkumaran

cs.ROOct 22, 2026

When One Step Is Enough: An Empirical Study of Euler Integration in Flow-Matching Policies

Authors: Shivakanth Sujit, Luca Nunziante, Rousslan Fernand Julien Dossa, Kai Arulkumaran

Organizations: Araya Inc., Tokyo

Abstract

Flow matching is now the dominant approach for vision-language action (VLA) robot policies, thanks to its expressiveness and simple training objective. However, inference for standard flow matching requires iteratively solving an ODE, limiting the control frequency at deployment. While several works have proposed novel methods to allow for single-shot action generation, we ask whether such additional objectives are actually necessary. We find that across six flow matching VLAs, one-step action generation largely matches multi-step sampling at scale: multi-step integration confers little to no consistent advantage at inference. We empirically find that this holds as long as models are sufficiently trained and have enough capacity. Practically, the implication is that for large VLAs, the single-step policy comes essentially for free from the multi-step model, enabling low-latency deployment without retraining, distillation, or auxiliary objectives.

Figures & tables

Figure 1: 1-step vs. reference-step success rates (SR) across all (model, benchmark) cells in our study;

\Delta=\text{SR}_{\text{ref}}-\text{SR}_{1}

is the performance gap in percentage points (pp). Points on or near the x=y diagonal indicate a negligible performance gap between the two conditions. For the three larger models (

\pi_{0.5}

, LAP, XR-0) all reported gaps are small and either statistically equivalent under two one-sided t-tests (TOST) or inconclusive with one-step having a higher aggregate SR. In the real world pooled deltas

\Delta

are at most

3.7\,

pp, although with some task-level differences.

Model	Total params	VLM backbone	VLM params	Reference steps	Latency (ms)
					Reference	1 Step
DiT [ 13 ]	$211$ M	CLIP ViT-B/16 + CLIP text [ 28 ]	$149.2$ M	$10$	$30$	$15\,{\scriptscriptstyle(2.0\times)}$
SmolVLA [ 32 ]	$450$ M	SmolVLM2 [ 21 ]	$\sim 350$ M	$10$	$436$	$91\,{\scriptscriptstyle(4.8\times)}$
X-VLA [ 43 ]	$900$ M	Florence-2-L (pruned) [ 37 ]	$570$ M	$10$	$215$	$70\,{\scriptscriptstyle(3.1\times)}$
FLOWER [ 29 ]	$947$ M	Florence-2-L (pruned) [ 37 ]	$570$ M	$4$ – $5$ 1	$140$	$60\,{\scriptscriptstyle(2.3\times)}$
$\pi_{0.5}$ [ 4 ]	$3.3$ B	PaliGemma VLM [ 3 ]	$3.0$ B	$10$	$80$	$50\,{\scriptscriptstyle(1.6\times)}$

Table 1: Model overview. We report the total parameter count, VLM backbone details, reference number of Euler integration steps as reported by authors, and action generation latency on a single A100 GPU, with the speedup of 1 step in parentheses.

Panel of Figure 2 — Figure 2: Real world Franka suite: spoon drawer , open drawer , open oven , close oven (left to right).

Model	LIBERO SGO		LIBERO-10		LIBERO+ SGO		LIBERO+ 10		CALVIN SR@5		CALVIN Avg. Len.		RoboTwin Clean		RoboTwin Rand.
Model	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step
DiT	89.9%	91.9%	68.4%	53.8%	30.8%	28.6%	22.0%	14.5%	–	–	–	–	–	–	–	–
SmolVLA	72.2%	82.0%	34.6%	44.2%	4.8%	5.0%	1.2%	0.7%	–	–	–	–	–	–	–	–
X-VLA	97.6%	97.3%	96.2%	96.0%	70.3%	70.4%	67.4%	66.9%	71.0%	70.0%	4.18	4.14	65.5%	65.1%	24.3%	23.3%
FLOWER	97.3%	96.5%	89.6%	83.6%	77.6%	74.4%	72.0%	59.3%	78.5%	75.5%	4.51	4.43	–	–	–	–
$\pi_{0.5}$	98.5%	98.9%	94.4%	94.4%	86.4%	86.3%	77.6%	76.1%	–	–	–	–	–	–	–	–

Table 2: Simulation metrics with TOST equivalence colouring (Sec. 3.3 ). SGO groups Spatial, Goal and Object suites. – indicates that no public checkpoint was available for evaluation.

Model	Spoon $\to$ Drawer		Open Drawer		Open Oven		Close Oven		Pooled
	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step
DiT	33.0	32.0	59.0	64.0	0.0	1.0	22.0	23.0	28.5	30.0
	$[-8.2,\,+10.2]$		$[-13.5,\,+3.6]$		$[-4.4,\,+1.7]$		$[-7.2,\,+5.2]$		$[-5.0,\,+2.0]$
X-VLA	0.0	0.0	68.0	72.0	0.0	0.0	49.0	50.0	29.2	30.5
	$[-2.6,\,+2.6]$		$[-9.5,\,+1.4]$		$[-2.6,\,+2.6]$		$[-7.5,\,+5.5]$		$[-3.4,\,+0.9]$
FLOWER	46.0	50.0	81.0	97.0 *	20.0	31.0	92.0	75.0 *	59.8	63.2

Table 3: Real-world success rates (%) per task and pooled (n=100 per task, 400 pooled). In grey the 90% CI for

\Delta=p_{\mathrm{ref}}-p_{\mathrm{1}}

. Cells with * marks a significant McNemar difference (

\alpha=0.05

; per-task decisions are after Holm correction within each model; no correction for pooled analysis).

Figure 3: Paired jerk-RMS comparison of real world rollouts. Violins are kernel density estimates; boxes give quartiles with 5–95% whiskers; the coloured line is the median.

n

is the number of samples (pairs with both runs successful).

Panel of Figure 4 — Figure 4: D3IL Avoiding results. In (a), ∗ indicates a significant McNemar difference ( ${p<0.05}$ ).

Figure 5: Step-count gap over training on LIBERO. Each panel reports

\Delta=\text{SR}_{\text{ref}}-\text{SR}_{1}

with shaded area as 90% CIs. Multi-step gains, when present, are concentrated on LIBERO-10 and fade with training. The short-horizon suites are close to step-invariant throughout.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A1: Comparison between the 1-step and 10-step integration approaches: L2 distance in the action space against the absolute value of the success rate gap. Each marker is a checkpoint (labeled 5k–29k), and color/shape of the marker identify the initialization variant. Dashed lines capture per-variant trend.

Model	Params	$k$	Excess (%)	Error (%)
DiT	211M	10	0.59	9.87
SmolVLA	450M	10	2.35	29.29
X-VLA	900M	10	$<$ 0.01	0.52
FLOWER	947M	4	0.24	4.56
$\pi_{0.5}$	3.3B	10	0.25	4.85
LAP	3.3B	10	0.27	5.00

Appendix

Table A1: Flow geometry and normalised endpoint error across released VLA checkpoints, evaluated on

4000

LIBERO demonstration observations (

1000

per suite), with eight fixed noise draws per observation. Excess measures the relative length of the integration path beyond the straight-line distance between its endpoints; Error measures normalised disagreement between one-step and reference-step actions. Smaller values indicate straighter trajectories and closer action agreement, respectively.

	reference-step Success	reference-step Failure
1-step Success	$a$	$b$
1-step Failure	$c$	$d$
	$N=a+b+c+d$

Appendix

Table B1: Contingency table. Each of the

N

matched rollouts is classified by the joint outcome of the two inference settings:

a

when both methods succeed,

b

when 1-step succeeds but reference-step fails,

c

when 1-step fails but reference-step succeeds, and

d

when both fail.

Model	$h$	LIBERO SGO		LIBERO-10		CALVIN SR@5		CALVIN Avg. Len.
Model	$h$	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step
X-VLA	1	61.6%	62.9%	10.6%	13.6%	0.0%	0.0%	0.31	0.30
	10	96.7%	97.2%	94.4%	94.4%	55.2%	56.4%	3.64	3.64
	20 $\dagger$	–	–	–	–	71.0%	70.0%	4.18	4.14
	30 $\dagger$	97.6%	97.3%	96.2%	96.0%	–	–	–	–
FLOWER	1	97.1%	96.7%	90.0%	85.0%	46.7%	4.8%	3.48	1.44

Appendix

Table C1: Replanning horizon

H

analysis on LIBERO and CALVIN ABC

\rightarrow

D. Success rates and average chain length are reported along with TOST equivalence colouring (see Sec. B ). SGO pools Spatial, Goal and Object suites; LIBERO-10 is its own suite.

\dagger

marks that model’s deployment default for that benchmark. LIBERO and CALVIN defaults differ for X-VLA (30 vs. 20), so those rows are benchmark-specific.

Task prompt	Median length in frames (seconds)
Close the drawer	106 (8.83)
Close the oven	113.5 (9.46)
Open the drawer	177.5 (14.79)
Open the oven	119 (9.92)
Put the green plate in the oven	197 (16.42)
Put the mug in the white basket	195 (16.25)

Appendix

Table D1: Median episode length per task in the Real World dataset (410 total episodes). Lengths are reported in frames, with the corresponding duration in seconds in parentheses.

Model	Benchmark	Budget	Checkpoint	Batch	Chunk	Replan
DiT	Real World	$30\,000$	$30\,000$	256	25	15
	LIBERO	$30\,000$	$30\,000$	256	25	10
	D3IL Avoiding	$50\,000$	$50\,000$	256	10	10
X-VLA	Real World	$50\,000$	$50\,000$	128	25	25
X-VLA	D3IL Avoiding	$10\,000$	$10\,000$	128	10	10
FLOWER †	Real World	$40\,000$	$23\,000$	32	25	15

Appendix

Table D2: Training and deployment hyperparameters. Budget denotes the total number of fine-tuning gradient steps, while checkpoint denotes the checkpoint used for evaluation. We also report the action chunk size and the replanning horizon.

Model	Gradient Step	Success Rate (%) $\uparrow$		Normalised Entropy $\uparrow$
Model	Gradient Step	1 Step	Reference	1 Step	Reference
DiT	$10\,000$	4.0	76.5	0.000	0.045
	$20\,000$	19.5	79.5	0.000	0.000
	$30\,000$	78.5	98.0	0.218	0.037
	$40\,000$	86.5	99.5	0.101	0.000
	$50\,000$	100.0	70.0	0.000	0.148

Appendix

Table D3: D3IL Avoiding success rate and normalised entropy across evaluated training checkpoints. The checkpoints that were used for the main results are highlighted in grey.

Benchmark	Splits	Data Samples	Tasks	Eval. Budget
LIBERO [ 18 ]	Spatial, Object, Goal (SGO); LIBERO 10	$273{,}465$	$40$	$50$
LIBERO+ [ 10 ]	Spatial, Object, Goal (SGO); LIBERO 10	$2{,}238{,}036$	$40$	$\sim 250$
CALVIN [ 23 ]	ABC $\to$ D [ 23 ]	$1{,}071{,}743$	$389$	$1000$ 1
RoboTwin [ 6 ]	Clean, Randomised	$248{,}105$	$50$	$100$
Real world [ 42 ]	-	$68{,}872$	$9$	$100$
D3IL Avoiding [ 9 ]	-	$96$	$1$	$200$

Appendix

Table D4: Benchmark Overview. Eval. Budget refers to the number of per-task rollouts.

Model	SR@1		SR@2		SR@3		SR@4		SR@5		Avg. Len.
Model	Reference	1-step	Reference	1-step	Reference	1-step	Reference	1-step	Reference	1-step	Reference	1-step
X-VLA	95.8%	95.1%	89.8%	88.9%	83.8%	83.1%	77.9%	77.3%	71.0%	70.0%	4.18	4.14
FLOWER	99.5%	99.3%	96.1%	95.2%	91.6%	89.9%	85.8%	83.2%	78.5%	75.5%	4.51	4.43 ${\color[rgb]{1,0,0.5}\blacktriangledown}$
XR-0	99.9%	99.9%	98.1%	97.8%	95.6%	95.4%	92.2%	92.2%	87.2%	88.1%	4.73	4.73

Appendix

Table E1: CALVIN ABC-D success rates for 1-step and reference-step variants. Glyphs on SR@5 and Avg. Len. encode the 1-step vs reference-step comparison:

{\color[rgb]{0,0.6,0}\blacktriangle}

1-step significantly better,

{\color[rgb]{1,0,0.5}\blacktriangledown}

1-step significantly worse. Avg. Len. uses a Welch t-test at

{p<0.05}

Model	LIBERO SGO		LIBERO-10		LIBERO+ SGO		LIBERO+ 10
Model	Reference	1 Step	Reference	1 Step	Reference	1 Step	Reference	1 Step
DiT	89.9%	91.9% ${\color[rgb]{0,0.6,0}\blacktriangle}{\color[rgb]{1,0,0.5}\blacktriangledown}$	68.4%	53.8% ${\color[rgb]{1,0,0.5}\blacktriangledown}$	30.8%	28.6%	22.0%	14.5% ${\color[rgb]{1,0,0.5}\blacktriangledown}$
SmolVLA	72.2%	82.0% ${\color[rgb]{0,0.6,0}\blacktriangle}{\color[rgb]{1,0,0.5}\blacktriangledown}$	34.6%	44.2% ${\color[rgb]{0,0.6,0}\blacktriangle}$	4.8%	5.0% ${\color[rgb]{1,0,0.5}\blacktriangledown}$	1.2%	0.7%
X-VLA	97.6%	97.3%	96.2%	96.0%	70.3%	70.4%	67.4%	66.9%
FLOWER	97.3%	96.5%	89.6%	83.6%	77.6%	74.4% ${\color[rgb]{1,0,0.5}\blacktriangledown}$	72.0%	59.3% ${\color[rgb]{1,0,0.5}\blacktriangledown}$
$\pi_{0.5}$	98.5%	98.9%	94.4%	94.4%	86.3%	86.2%	77.7%	76.3% ${\color[rgb]{1,0,0.5}\blacktriangledown}$

Appendix

Table E2: LIBERO and LIBERO+ success rates. SGO is an average over Spatial, Goal, and Object suites.

Figure E1: DiT per-task analysis on LIBERO. Violins sharing the same letter but with a single

\star

, are separable at

\alpha_{\textrm{global}}=0.05

but are not after Bonferroni correction.

Figure E2: SmolVLA per-task analysis on LIBERO. Violins sharing the same letter but with a single

\star

, are separable at

\alpha_{\textrm{global}}=0.05

but are not after Bonferroni correction.

Figure E3: X-VLA per-task analysis on LIBERO. Violins sharing the same letter but with a single

\star

, are separable at

\alpha_{\textrm{global}}=0.05

but are not after Bonferroni correction.

Figure E4: FLOWER per-task analysis on LIBERO. Violins sharing the same letter but with a single

\star

, are separable at

\alpha_{\textrm{global}}=0.05

but are not after Bonferroni correction.

Figure E5:

\pi_{0.5}

per-task analysis on LIBERO. Violins sharing the same letter but with a single

\star

, are separable at

\alpha_{\textrm{global}}=0.05

but are not after Bonferroni correction.

Figure E6: LAP per-task analysis on LIBERO. Violins sharing the same letter but with a single

\star

, are separable at

\alpha_{\textrm{global}}=0.05

but are not after Bonferroni correction.

Figure E7: XR-0 per-task analysis on LIBERO. Violins sharing the same letter but with a single

\star

, are separable at

\alpha_{\textrm{global}}=0.05

but are not after Bonferroni correction.

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Vision-language-action (VLA) policies based on flow matching generate action chunks through repeated evaluations of an action expert. Increasing the number of integration steps raises inference cost, but does not necessarily improve closed-loop success. We propose Coda, which reallocates part of this integration budget to a single learned endpoint correction. A frozen policy first completes a few-step noise-to-action trajectory; a lightweight Transformer then predicts a demonstration-supervised residual using the candidate action, source noise, and shared observation-prefix cache. Only the corrector is trained. On 50 RoboTwin Easy tasks, five-step Coda improves success from 71.64% to 74.68% over the matched five-step baseline, while reducing forward latency by 30.2% relative to the default ten-step policy. A two-step configuration achieves 71.88% success with a 2.12

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