Residual Modeling Closes the Regression and Generative Policy Gap in Robot Learning

Yuchen Zhou; Jiacheng You; Weikang Wan; Weijun Dong; Yang Gao; Jiayuan Mao

cs.ROOct 8, 2026

Residual Modeling Closes the Regression and Generative Policy Gap in Robot Learning

Authors: Yuchen Zhou, Jiacheng You, Weikang Wan, Weijun Dong, Yang Gao, Jiayuan Mao

Organizations: University of Pennsylvania · Tsinghua University · UC San Diego

Abstract

Learning from demonstration has enabled impressive robot behaviors. A common choice for policy learning is to use diffusion or flow matching (Flow-Policies), which often outperforms direct action regression trained with mean squared error (MSE-Policies). This gap is commonly attributed to multimodal demonstrations. We revisit this gap from the perspective of statistical modeling: how action-prediction residuals shape policy optimization. Our analysis of real-world robot demonstration data reveals substantial state-dependent variation in residual scales and heavier-than-Gaussian tails. While both MSE-Policies and Flow-Policies exhibit heavy-tailed action residuals, their training gradients behave differently: MSE allocates more gradient magnitude to observations with large action residuals, which hurts optimization. Motivated by these findings, we introduce heteroscedastic Student-t action regression (HT-Policies), which learns input-dependent residual scales and reduces the influence of heavy tails. HT-Policies predict action chunks with a single feed-forward pass and can reuse pretrained flow-matching-based policy networks as the backbone. Across four simulation benchmarks and real-robot evaluations, HT-Policies achieves success rates competitive with generative policy baselines, both when trained from scratch and from pretrained vision-language-action and world-action models, despite being faster in training and inference. Together, these findings shed light on the practical advantages of generative objectives in robot learning from demonstrations and offer an efficient direct-regression alternative for a range of architectures and tasks. Project page: https://the-labone.github.io/regression-policy-project/

Figures & tables

Figure 1: (a) We attribute the performance gap between MSE- and Flow-Policies to different statistical modeling of prediction residuals. (b) We found observation-dependent and heavy-tailed residuals for MSE-Policies on multiple benchmarks. (c) Benchmark success rates (%) for different policy classes. We use flow-matching for RoboCasa-GR1 and SIMPLER, and diffusion for RoboMimic.

Figure 2: Example tasks. We show 4 tasks from RoboCasa-GR1, Bridge, Fractal, and Tool-Hang.

Figure 3: Heavy-tailed residuals. Both MSE-Policies (a) and Flow-Policies (b) exhibit heavy-tailed action-prediction residuals. Percentages report scalar residuals exceeding three times their coordinate RMS in absolute value.

Figure 4: Gradient distribution for different objectives and noise levels. We show the relative gradient contribution (%) of each action-residual decile. For Flow, gradient RMS is computed separately for high-noise (

t<0.5

) and low-noise (

t>0.5

) regimes. See also the Appendix Figure 12 .

Figure 5: Input-dependent residual scale. We evaluate a GR00T N1.7 model trained with the MSE regression loss. (a) Residual vs. action RMS. (b) Residual RMS vs. episode progress on three Bridge tasks. Shaded bands indicate 95% confidence intervals..

Figure 6: Heavy tails after normalization. (a) HG-regression residuals normalized by predicted

\sigma(o)

; (b) flow-matching residuals normalized by per-observation action sampling.

Backbone	DP	MSE	HT
U-Net	93.8/84.7	81.7/72.3	95.3 / 85.6
Transformer	94.3 / 84.1	82.6/74.6	91.9/82.9

Table 1: Success rates (%) for policies trained from scratch. (a) RoboMimic. Best / mean over the last ten evaluated checkpoints ( Chi et al., 2023 ) , averaged over nine tasks and three seeds per task, with state and image results weighted equally. (b) RoboCasa-GR1. A randomly initialized GR00T N1.7 action head trained with the official fine-tuning recipe.

Figure 7: Real-robot success rates (%). We evaluate on 3 tasks: Push-T, Insert-T, and Peel-Note.

Model	Benchmark	Flow (Official)	Our Evaluations (10 seeds)
Model	Benchmark	Flow (Official)	Flow	MSE	HT
$\pi_{0.5}$	LIBERO (4-suite avg.)	96.9	$\mathbf{97.09\pm 0.04}$	$96.72\pm 0.09$	$96.97\pm 0.07$
GR00T N1.7	RoboCasa-GR1	44.5	$42.34\pm 0.68$	$35.55\pm 0.67$	$\mathbf{50.01\pm 0.51}$
	SIMPLER / Bridge	62.3	$62.06\pm 0.40$	$58.51\pm 0.37$	$\mathbf{62.31\pm 0.65}$
	SIMPLER / Fractal	72.5	$\mathbf{67.83\pm 0.39}$	$60.60\pm 0.43$	$65.25\pm 0.44$
Cosmos 3	LIBERO-10	95.2	$95.60\pm 0.90$	$95.10\pm 0.26$	$\mathbf{96.50\pm 0.45}$

Table 2: Success rate (%) for fine-tuning from pretrained policies. We report the mean success rate and its standard error of a single checkpoint across 10 evaluation seeds. We include official results for the flow-matching–based fine-tuning for reference. For

\pi_{0.5}

, we report the average performance in four LIBERO suites. Bold marks highlight the best-performing model in each row.

Figure 8: Residual modeling and optimization. Top: Policy success rates across four settings. Bottom: Each action-residual decile’s share of summed per-sample parameter-gradient RMS under MSE, HT, and low-noise Flow (

t>0.5

). Each decile contains approximately

10\%

of the samples.

Figure 9: Mean log-likelihood vs. policy success rate.

Figure 10: Training curves for different models on different benchmarks. HT-Policies reach strong task success rates with fewer training steps than Flow-Policies.

	GR00T N1.7			$\pi_{0.5}$			Cosmos3-Nano
Metric	Flow	HT	Speed-up	Flow	HT	Speed-up	Flow	HT	Speed-up
Head latency $\downarrow$	29.30	7.92	$3.70\times$	62.50	6.42	$9.74\times$	1266.10	65.96	$19.19\times$
Total latency $\downarrow$	46.20	24.52	$1.88\times$	123.58	67.94	$1.82\times$	1271.76	71.34	$17.83\times$
FPS $\uparrow$	21.65	40.78	$1.88\times$	8.09	14.72	$1.82\times$	0.79	14.02	$17.83\times$

Table 3: Inference efficiency benchmarking. We evaluate all policies on a single NVIDIA RTX 5090 GPU with batch size 1. Latencies are in milliseconds.

Figure 11: Selection of

\nu

. (a) Success rate vs.

\nu

on RoboCasa-GR1 and (b) Success rate vs.

c=\nu/d

on three datasets, where

d

is the action-chunk dimension.

\nu\to\infty

gives heteroscedastic Gaussian.

Appendix figures & tables30 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Episodes	Observations	$d$	MSE/HG coordinates
RoboCasa-GR1	240	960	232	222,720
Bridge	400	1,600	48	76,800
Fractal	400	1,600	48	76,800
Tool-Hang	200	1,600	144	230,400

Appendix

Table 4: Residual-tail samples. Bridge, Fractal, and Tool-Hang exclude the gripper channel from the plotted residuals. Each Flow-Policy contributes 64 independently evaluated action draws per observation.

Figure 12: Gradient imbalance across objectives and noise levels. Curves show the relative gradient contribution (%) of each action-residual decile. For flow matching, gradient RMS is averaged over noise draws and timesteps separately for high noise (

t<0.5

) and low noise (

t>0.5

Setting	MSE	Flow	HT
RoboCasa-GR1	1,536	1,536	1,536
Bridge	2,048	2,048	2,048
Fractal	2,048	2,048	2,048
Tool-Hang	4,096	4,096	4,096

Appendix

Table 5: Observation counts for the parameter-gradient diagnostics. GR00T is trained for 60k updates on GR1 and 20k updates on Bridge and Fractal. Tool-Hang uses state observations and Chi-UNet.

Dataset	Quantile function $Q(u)$	High noise ( $t<0.5$ )	Low noise ( $t>0.5$ )
RoboCasa-GR1, Bridge, Fractal	$0.999[1-(1-u)^{2/3}]$	Points 1–10 ( $62.5\%$ ) $[0,\,0.4795)$	Points 11–16 ( $37.5\%$ ) $[0.4795,\,0.999]$
Tool-Hang	$u$	Points 1–8 ( $50\%$ ) $[0,\,0.5)$	Points 9–16 ( $50\%$ ) $[0.5,\,1]$

Appendix

Table 6: Flow-matching timestep groups for gradient diagnostics. Ranges span the full quantile intervals represented by each group; percentages denote their training probability mass.

Figure 13: Gaussian tail-bound comparison across datasets. GR00T uses 60k updates on RoboCasa-GR1 and 20k on Bridge and Fractal; LIBERO results use

\pi_{0.5}

. Light curves show individual timesteps, the thick blue curve their pointwise maximum, and the dashed curve the Gaussian norm reference.

Figure 14: Gaussian tail-bound comparison across training stages. GR00T on RoboCasa-GR1 after 5k, 10k, 20k, 40k, and 60k updates, with

d=232

. Curves follow the same convention as Figure 13 .

Figure 15: Gradient allocation and policy success across training stages on RoboCasa-GR1. Columns show results after 5k, 10k, 20k, 40k, and 60k updates. Top: Each action-residual decile’s share of summed per-sample parameter-gradient RMS, with flow-matching gradients evaluated at low-noise timesteps (

t>0.5

). The dashed line marks the

10\%

sample share of each decile. Bottom: Success rates of MSE-Policies, Flow-Policies, and HT-Policies at the corresponding training stages; HT-Policies have the highest success rate at all five stages.

Model / benchmark	$K=1$	$K=16$ mean	$\Delta$ (pp)
GR00T N1.7 / RoboCasa-GR1	40.63	43.23	$+2.61$
$\pi_{0.5}$ / LIBERO	97.250	97.325	$+0.075$
Cosmos3-Nano / LIBERO-10	96.00	98.00	$+2.00$

Appendix

Table 7: Action averaging in fixed Flow-Policies. Success rates (%) from one evaluation run per condition.

\Delta

is the change in percentage points, computed before rounding.

Task	$K=1$	$K=16$ mean	$\Delta$ (pp)
PnPBottle $\to$ CabinetClose	$8/20$ (40.00)	$12/20$ (60.00)	$+20.00$
PnPCan $\to$ DrawerClose	$2/20$ (10.00)	$6/20$ (30.00)	$+20.00$
PnPCup $\to$ DrawerClose	$0/20$ (0.00)	$4/20$ (20.00)	$+20.00$
PnPMilk $\to$ MicrowaveClose	$3/20$ (15.00)	$1/20$ (5.00)	$-10.00$
PnPPotato $\to$ MicrowaveClose	$4/20$ (20.00)	$1/20$ (5.00)	$-15.00$
PnPWine $\to$ CabinetClose	$3/20$ (15.00)	$3/21$ (14.29)	$-0.71$

Appendix

Table 8: Action averaging on RoboCasa-GR1. Entries give successes / completed rollouts, with success rates (%) in parentheses. Task names follow Table 22 . The mean weights all 24 tasks equally;

\Delta

is computed before rounding.

	LIBERO-10		LIBERO-Goal		LIBERO-Object		LIBERO-Spatial
ID	$K=1$	$K=16$	$K=1$	$K=16$	$K=1$	$K=16$	$K=1$	$K=16$
0	$96/100$	$93/100$	$99/100$	$99/100$	$100/100$	$100/100$	$100/100$	$100/100$
1	$98/100$	$98/100$	$98/100$	$100/100$	$100/100$	$100/100$	$100/100$	$99/100$
2	$99/100$	$96/100$	$89/100$	$91/100$	$100/100$	$100/100$	$100/100$	$100/100$
3	$100/100$	$100/100$	$94/100$	$94/100$	$97/100$	$98/100$	$97/100$	$100/100$
4	$99/100$	$98/100$	$100/100$	$100/100$	$100/100$	$99/100$	$98/100$	$98/100$

Appendix

Table 9: Action averaging for

\pi_{0.5}

on all four LIBERO suites. Entries give successes / completed rollouts. Task IDs are listed in Tables 20 and 21 .

ID	Task	$K=1$	$K=16$ mean
0	Put both the alphabet soup and the tomato sauce in the basket.	$8/10$	$9/10$
1	Put both the cream cheese box and the butter in the basket.	$10/10$	$10/10$
2	Turn on the stove and put the moka pot on it.	$9/10$	$10/10$
3	Put the black bowl in the bottom drawer of the cabinet and close it.	$10/10$	$10/10$
4	Put the white mug on the left plate and the yellow and white mug on the right plate.	$10/10$	$10/10$
5	Pick up the book and place it in the back compartment of the caddy.	$10/10$	$10/10$

Appendix

Table 10: Action averaging for Cosmos3-Nano on LIBERO-10. Entries give successes / completed rollouts. Task IDs follow the LIBERO-10 ordering in Table 21 .

Method	$\alpha$	pass@1	pass@2	pass@4	pass@8
Flow-Policy (native)	—	76.25	93.68	99.67	100.00
HT-Policy (no noise)	0	81.00	—	—	—
HT-Policy ( $\sigma$ -scaled)	0.125	79.38	95.61	99.76	100.00
HT-Policy (fixed scale)	0.125	78.88	95.39	99.81	100.00
HT-Policy ( $\sigma$ -scaled)	0.25	78.63	95.82	99.90	100.00
HT-Policy (fixed scale)	0.25	76.88	94.25	99.56	100.00

Appendix

Table 11: Exploration pass@

k

(%) on state-based Tool-Hang with Chi-Transformer. Estimates are computed per initial state from eight rollouts and averaged over 100 states.

\alpha

controls HT-Policy sampling noise. Dashes indicate unreported values or an inapplicable noise multiplier.

Model / dataset	Updates	Batch	Base LR	Warmup	$\nu$
$\pi_{0.5}$ / LIBERO	30,000	32	$2.5\!\times\!10^{-5}$	1,000	1,400
GR00T / RoboCasa-GR1	60,000	512	$10^{-4}$	3,000	928
GR00T / Bridge	20,000	1,024	$10^{-4}$	1,000	224
GR00T / Fractal	20,000	1,024	$10^{-4}$	1,000	224
Cosmos3-Nano / LIBERO-10	2,000	$\leq 2,048$	$5\!\times\!10^{-5}$	500	640

Appendix

Table 12: Training budgets for the large-model experiments. Batch size is global. Cosmos3-Nano batch size depends on packing.

Dataset	Episodes	Frames	$H$	$D_{a}$	$d$
LIBERO, four suites ( $\pi_{0.5}$ )	1,693	273,465	50	7	350
RoboCasa-GR1	24,000	5,820,277	8	29	232
Bridge	53,192	1,893,026	8	7	56
Fractal	87,212	3,786,400	8	7	56
LIBERO-10 (Cosmos3-Nano)	379	101,469	16	10	160

Appendix

Table 13: Training data and effective action dimensions. Episode and frame counts refer to the datasets used for training.

Model / benchmark	Tasks	Rollout budget/ task/seed	Execute	Flow steps	Step limit
$\pi_{0.5}$ / Spatial and Object	$2\times 10$	100	10	10	280
$\pi_{0.5}$ / Goal	10	100	10	10	300
$\pi_{0.5}$ / LIBERO-10	10	100	10	10	520
GR00T / RoboCasa-GR1	24	20	8	4	720
GR00T / Bridge–WidowX	7	50	4	4	300
GR00T / Fractal–Google Robot	6	100	1	4	300

Appendix

Table 14: Evaluation settings for the 10-seed comparison in Table 2 . Rollout budgets are per task and evaluation seed; actual GR00T rollout counts can slightly exceed these budgets. “Execute” is the number of actions applied before replanning. MSE-Policies and HT-Policies each use a single forward pass.

	U-Net		Transformer
Task	DP-C	HT	DP-T	HT
	best/last10	best/last10	best/last10	best/last10
Lift-PH	1.00 /0.98	1.00 / 1.00	1.00 / 1.00	1.00 /0.93
Lift-MH	1.00 /0.97	1.00 / 1.00	1.00 / 1.00	1.00 /0.96
Can-PH	1.00 /0.96	1.00 / 1.00	1.00 / 1.00	1.00 /0.94
Can-MH	1.00 /0.96	1.00 / 0.98	1.00 /0.94	1.00 / 0.98

Appendix

Table 15: DP and HT with state observations. Bold marks the highest value per row and metric across both backbones, before rounding.

	U-Net		Transformer
Task	DP-C	HT	DP-T	HT
	best/last10	best/last10	best/last10	best/last10
Lift-PH	1.00 / 1.00	1.00 /0.95	1.00 / 1.00	1.00 / 1.00
Lift-MH	1.00 / 1.00	1.00 /0.95	1.00 /0.99	1.00 /0.99
Can-PH	1.00 /0.97	1.00 /0.94	1.00 /0.98	1.00 / 1.00
Can-MH	1.00 /0.96	1.00 /0.92	1.00 / 0.98	1.00 /0.95

Appendix

Table 16: DP and HT with image observations. Bold marks the highest value per row and metric across both backbones, before rounding.

Method	Lift		Can		Square		Transport		Tool-Hang	Push-T	Mean
	PH	MH	PH	MH	PH	MH	PH	MH
Sudeep-DiT
Flow	1.00 / 1.00	1.00 /0.99	1.00 / 1.00	1.00 /0.94	1.00 / 0.94	0.88/0.75	0.80/0.70	0.40/0.27	0.86/0.75	0.98 /0.95	0.89/0.83
MSE	1.00 / 1.00	1.00 /0.99	1.00 /0.98	0.92/0.90	0.94/0.86	0.72/0.53	0.50/0.44	0.12/0.06	0.52/0.39	0.92/0.83	0.76/0.70
Straight Flow	1.00 / 1.00	1.00 /0.98	1.00 /0.99	0.96/0.90	0.96/0.93	0.72/0.66	0.56/0.48	0.20/0.14	0.70/0.59	0.90/0.86	0.80/0.75
MIP	1.00 / 1.00	1.00 /0.99	1.00 / 1.00	0.98/0.95	0.98/ 0.94	0.90/0.81	0.76/0.68	0.44/0.38	0.92 / 0.88	0.95/0.92	0.89/0.86

Appendix

Table 17: Scores on state-based tasks. Entries report best/last5, averaged over three seeds. RoboMimic reports success; Push-T reports normalized coverage. Baselines follow Table 11 of Pan et al. (2026) . Bold marks the best value per task and metric across all methods and backbones, before rounding.

Method	Lift		Can		Square		Transport		Tool-Hang	Push-T	Mean
	PH	MH	PH	MH	PH	MH	PH	MH
Sudeep-DiT
Flow	1.00 / 1.00	1.00 / 1.00	1.00 /0.99	0.96/0.94	0.96/ 0.94	0.82/0.76	0.84/0.83	0.32/0.20	0.78 /0.57	0.92 / 0.89	0.86/0.81
MSE	1.00 / 1.00	1.00 /0.99	1.00 / 1.00	0.92/0.81	0.94/0.84	0.74/0.67	0.74/0.56	0.14/0.08	0.28/0.18	0.83/0.77	0.76/0.69
Straight Flow	1.00 /0.99	1.00 /0.99	1.00 /0.98	0.98/0.95	1.00 /0.93	0.82/0.72	0.86/0.83	0.26/0.19	0.46/0.40	0.85/0.79	0.82/0.78
MIP	1.00 / 1.00	1.00 /0.99	1.00 /0.98	1.00 /0.96	1.00 /0.92	0.90/0.83	0.90/0.84	0.50/0.31	0.76/ 0.66	0.91/0.87	0.90/0.84

Appendix

Table 18: Scores on image-based tasks. Entries report best/last5, averaged over three seeds. RoboMimic reports success; Push-T reports normalized coverage. Baselines follow Table 12 of Pan et al. (2026) . Bold marks the best value per task and metric across all methods and backbones, before rounding.

Suite	Flow (OpenPI)	Flow	HT
LIBERO-Spatial	98.80	$97.13\pm 0.11$	$\mathbf{97.71\pm 0.13}$
LIBERO-Object	98.20	$99.44\pm 0.06$	$\mathbf{99.80\pm 0.00}$
LIBERO-Goal	98.00	$96.55\pm 0.19$	$\mathbf{96.94\pm 0.23}$
LIBERO-10	92.40	$\mathbf{95.23\pm 0.14}$	$93.44\pm 0.06$
Mean	96.85	$\mathbf{97.09\pm 0.04}$	$96.97\pm 0.07$

Appendix

Table 19: Success rates (%) of

\pi_{0.5}

across LIBERO suites. Flow and HT report mean

\pm

standard error across 10 evaluation seeds for each trained model. Bold marks the higher mean of Flow and HT.

ID	Task	Flow	HT
LIBERO-Spatial
0	Pick up the black bowl between the plate and the ramekin and place it on the plate.	$\mathbf{99.90\pm 0.10}$	$\mathbf{99.90\pm 0.10}$
1	Pick up the black bowl next to the ramekin and place it on the plate.	$\mathbf{99.60\pm 0.22}$	$99.30\pm 0.21$
2	Pick up the black bowl from table center and place it on the plate.	$\mathbf{99.70\pm 0.21}$	$\mathbf{99.70\pm 0.15}$
3	Pick up the black bowl on the cookie box and place it on the plate.	$\mathbf{99.40\pm 0.22}$	$98.80\pm 0.36$
4	Pick up the black bowl in the top drawer of the wooden cabinet and place it on the plate.	$\mathbf{94.20\pm 0.39}$	$94.10\pm 0.64$

Appendix

Table 20: Task-level success rates (%) of

\pi_{0.5}

on LIBERO-Spatial and LIBERO-Object. Task scores are mean

\pm

standard error over 10 evaluation seeds for each trained model. Bold marks the higher mean for each task.

ID	Task	Flow	HT
LIBERO-Goal
0	Open the middle drawer of the cabinet.	$96.10\pm 0.46$	$\mathbf{99.40\pm 0.16}$
1	Put the bowl on the stove.	$98.20\pm 0.53$	$\mathbf{98.50\pm 0.48}$
2	Put the wine bottle on top of the cabinet.	$89.90\pm 0.60$	$\mathbf{90.30\pm 1.16}$
3	Open the top drawer and put the bowl inside.	$92.50\pm 0.48$	$\mathbf{95.40\pm 0.85}$
4	Put the bowl on top of the cabinet.	$99.40\pm 0.16$	$\mathbf{99.50\pm 0.27}$

Appendix

Table 21: Task-level success rates (%) of

\pi_{0.5}

on LIBERO-Goal and LIBERO-10. Task scores are mean

\pm

standard error over 10 evaluation seeds for each trained model. Bold marks the higher mean for each task.

Task	Flow (official)	Flow	HT
PnPBottle $\to$ CabinetClose	70.00	$42.00\pm 4.23$	$\mathbf{58.26\pm 5.53}$
PnPCan $\to$ DrawerClose	70.00	$23.90\pm 2.68$	$\mathbf{50.64\pm 4.51}$
PnPCup $\to$ DrawerClose	35.00	$8.40\pm 1.83$	$\mathbf{23.58\pm 3.32}$
PnPMilk $\to$ MicrowaveClose	45.00	$11.00\pm 2.56$	$\mathbf{25.00\pm 4.22}$
PnPPotato $\to$ MicrowaveClose	40.00	$6.00\pm 1.25$	$\mathbf{26.31\pm 3.84}$
PnPWine $\to$ CabinetClose	65.00	$24.88\pm 2.98$	$\mathbf{49.74\pm 1.60}$

Appendix

Table 22: Task-level success rates (%) on RoboCasa-GR1. Flow and HT report mean

\pm

standard error across 10 evaluation seeds for each trained model. Bold marks the higher mean of Flow and HT.

Task	Flow (official)	Flow	HT
carrot on plate	58.00	$\mathbf{63.00\pm 1.94}$	$29.80\pm 1.17$
close drawer	97.00	$98.40\pm 0.50$	$\mathbf{98.80\pm 0.53}$
put eggplant in basket	53.00	$50.80\pm 2.33$	$\mathbf{93.03\pm 1.24}$
put eggplant in sink	2.00	$2.00\pm 0.84$	$\mathbf{65.00\pm 1.82}$
open drawer	100.00	$\mathbf{100.00\pm 0.00}$	$72.60\pm 1.19$
spoon on towel	78.00	$\mathbf{87.02\pm 1.13}$	$61.40\pm 1.98$

Appendix

Table 23: Task-level success rates (%) on SIMPLER / Bridge. Flow and HT: mean

\pm

standard error over 10 evaluation seeds. Mean averages seven tasks; bold marks the higher mean of Flow and HT.

Task	Flow (official)	Flow	HT
pick coke can	100.00	$\mathbf{97.90\pm 0.41}$	$97.80\pm 0.57$
pick object	94.00	$\mathbf{90.11\pm 0.98}$	$72.30\pm 1.01$
move near	100.00	$\mathbf{98.50\pm 0.50}$	$93.10\pm 0.87$
open drawer	65.00	$\mathbf{52.10\pm 1.21}$	$46.10\pm 1.33$
close drawer	69.00	$58.84\pm 0.97$	$\mathbf{66.20\pm 1.19}$
place in closed drawer	7.00	$9.50\pm 1.01$	$\mathbf{16.00\pm 0.54}$

Appendix

Table 24: Task-level success rates (%) on SIMPLER / Fractal. Flow and HT: mean

\pm

standard error over 10 evaluation seeds. Mean averages six tasks; bold marks the higher mean of Flow and HT.

ID	Task	Flow	HT
0	Put both the alphabet soup and the tomato sauce in the basket.	$92.00\pm 4.16$	$\mathbf{95.00\pm 2.24}$
1	Put both the cream cheese box and the butter in the basket.	$\mathbf{100.00\pm 0.00}$	$\mathbf{100.00\pm 0.00}$
2	Turn on the stove and put the moka pot on it.	$94.00\pm 2.21$	$\mathbf{99.00\pm 1.00}$
3	Put the black bowl in the bottom drawer of the cabinet and close it.	$94.00\pm 2.21$	$\mathbf{97.00\pm 1.53}$
4	Put the white mug on the left plate and put the yellow and white mug on the right plate.	$95.00\pm 1.67$	$\mathbf{96.00\pm 2.21}$
5	Pick up the book and place it in the back compartment of the caddy.	$\mathbf{99.00\pm 1.00}$	$98.00\pm 1.33$

Appendix

Table 25: Task-level success rates (%) of Cosmos3-Nano on LIBERO-10. Mean

\pm

standard error over 10 evaluation seeds; bold marks the higher mean per row.

		Action head		Whole model
Model	Method	NFE $\downarrow$	Latency $\downarrow$	Latency $\downarrow$	FPS $\uparrow$	Speedup $\uparrow$
GR00T N1.7	Flow	4	89.2	139.1	7.19	1.94 $\times$
GR00T N1.7	HT (ours)	1	24.3	71.7	13.95	1.94 $\times$
$\pi_{0.5}$	Flow	10	169.7	260.8	3.83	2.42 $\times$
$\pi_{0.5}$	HT (ours)	1	17.1	107.6	9.29	2.42 $\times$
Cosmos3-Nano	Flow	30	1368.2	1377.5	0.73	16.74 $\times$

Appendix

Table 26: Inference efficiency on an NVIDIA A800. Batch size one. All latency values are reported in milliseconds. NFE denotes the number of action-head evaluations per action chunk. Speedup is the Flow-to-HT whole-model latency ratio.

Objective	SR (%) $\uparrow$
MSE	37.80
$L_{1}$	27.00
RMSE	46.00
Huber Huber (1964)	38.10
Student- $t$	39.50
Heteroscedastic Gaussian	42.50

Appendix

Table 27: Regression loss ablation. GR00T N1.7 on RoboCasa-GR1, 60k steps, 24 tasks. Upper/lower blocks omit/use input-dependent scales. Both Student-

t

variants use

\nu=928

Family	Fixed-scale penalty	Learned-scale penalty
Gaussian	$S$	$S/(2\sigma^{2})+d\log\sigma$
$L_{1}$	$Q$	$Q/\sigma+d\log\sigma$
Radial norm (RMSE)	$R$	$R/\sigma+d\log\sigma$
Huber	$h_{\delta}(R)$	$h_{\delta}(R/\sigma)+d\log\sigma$
Student- $t$	$T_{\nu}(R/\sigma_{0})+d\log\sigma_{0}$	$T_{\nu}(R/\sigma)+d\log\sigma$

Appendix

Table 28: Implemented regression penalties. Penalties are shown before division by the number of valid action coordinates. Let

S=\|r\|_{2}^{2}

R=\|r\|_{2}

Q=\|r\|_{1}

, and

T_{\nu}(z)=\frac{\nu+d}{2}\log(1+z^{2}/\nu)

. The learned scale

\sigma=\sigma(o)

is scalar.

Setting	Updates	$d$	Reported $\nu$
RoboCasa-GR1	60,000	232	$2,116,464,\mathbf{928},1450,\infty$ (HG)
Bridge	20,000	56	$56,112,\mathbf{224},350,448$
$\pi_{0.5}$ / LIBERO	30,000	350	$700,\mathbf{1400},2187.5$

Appendix

Table 29: Degrees-of-freedom settings for Figure 11 . Bold marks the choice

\nu=4d

(

c=4

), which gives the highest tested success in each setting. HG denotes the Gaussian limit.

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