cs.ROOct 29, 2026

Gaussian Distillation of Flow Policies

Authors: Youssef Saied, Ziyad Sheebaelhamd, Claire Vernade, François Fleuret

Organizations: University of Geneva Geneva, Switzerland · University of Tübingen Tübingen, Germany · University of Technology Nuremberg Nuremberg, Germany · University of Geneva and Meta FAIR Geneva, Switzerland

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Abstract

Flow policies are among the strongest robot controllers and are well suited to modelling multimodal action distributions. Our analysis shows that the continuous-action distributions of the studied flow policies are well approximated by Gaussians. We propose the Gaussian Action Readout (GAR), which replaces a flow policy's final layer with a Gaussian readout trained on just 0.02% of eligible states, while keeping the rest of the network frozen. It samples with one action-network pass, gives closed-form likelihoods, and achieves near-teacher task success at up to twice the inference speed.

Figures & tables

Figure 1: Near-teacher success with one action-transformer evaluation. Diagonal GAR uses 6,000 fitting contexts (0.02% of eligible training states), with teacher networks frozen. Each policy is evaluated on 50 RoboCasa365 ( Nasiriany et al., 2026 ) tasks with 50 episodes per task. Bars show pointwise 95% within-task episode-bootstrap intervals (Appendix F.2 ).
Figure 2: Full-covariance Gaussian approximation across sixteen-action chunks at 300 fixed observations per teacher. Histograms show relative sliced 1-Wasserstein errors (%) for the 48-dimensional translation and rotation groups, normalized by the teacher’s projected command magnitude. Gaussian sampling controls measure finite-sample error. Dashed lines mark medians; logarithmic horizontal axes retain all observations, and lower errors are better.
Figure 3: GAR predicts Gaussian parameters and Bernoulli probabilities from one evaluation of the frozen action transformer. Only small readouts are trained; sampling continuous actions and binary commands produces an action chunk.
TeacherFitting contextsMean error (%)SD error (%)
   DiagonalFull
Xiaomi6,0004.63 [ 3.09 , 7.73 ]5.58 [ 3.61 , 9.03 ]2.91 [ 1.72 , 5.95 ]
Xiaomi24,0004.54 [ 3.00 , 7.37 ]4.35 [ 2.69 , 7.37 ]2.92 [ 1.56 , 5.57 ]
GR00T6,0008.43 [ 5.98 , 12.91 ]9.76 [ 6.51 , 16.42 ]8.09 [ 5.36 , 13.42 ]
GR00T24,0008.29 [ 5.86 , 12.55 ]8.47 [ 5.74 , 14.65 ]7.61 [ 5.08 , 12.21 ]
Table 1: Moment prediction on 600 held-out contexts per teacher, using estimates from teacher samples as references. Entries give median [25th, 75th percentile] errors, with lower values better. Mean and SD errors are 100∥μθ(c)−μ^c∥2/∥μ^c∥2100\|\mu_{\theta}(c)-\widehat{\mu}_{c}\|_{2}/\|\widehat{\mu}_{c}\|_{2} and 100∥σθ(c)−σ^c∥2/∥μ^c∥2100\|\sigma_{\theta}(c)-\widehat{\sigma}_{c}\|_{2}/\|\widehat{\mu}_{c}\|_{2} , respectively, where σ\sigma denotes the vector of marginal SDs. Diagonal and full variants share the mean readout. Errors use normalized coordinates for Xiaomi and decoded coordinates for GR00T, so comparisons are within each teacher. More fitting contexts reduce median diagonal-SD error, with little change in mean error.
TeacherFitting contextsTask groupTeacherRaw one-stepMean onlyDiagonalFull
Xiaomi6,000Overall58.2455.7656.6457.6457.28
  Atomic seen81.1181.2278.4480.5681.67
  Composite seen58.7552.6358.1358.0057.38
  Composite unseen32.0030.2530.6331.5029.75
 24,000Overall58.2455.7656.4457.7256.44
  Atomic seen81.1181.2280.1181.0081.11
Table 2: RoboCasa365 success (%) across 50 tasks and 2,500 unpaired episodes per policy; higher is better. Groups contain 18 seen atomic, 16 seen composite and 16 unseen composite tasks, with 50 episodes per task. Raw one-step uses the teacher without fitted readouts. Mean-only GAR retains stochastic binary commands; Gaussian variants use predicted variance. Context counts refer to mean and covariance fitting; command readouts use 6,000 contexts. Execution, fitting and uncertainty are detailed in Appendices B , C and F.2 , respectively.
 GR00TXiaomi
Continuous-motion policySuccessDifferenceSuccessDifference
Benchmark references from Table 2 : 50 episodes per task
Raw one-step8.13—52.63—
Mean-only GAR, 6,000 contexts7.75—58.13—
Distribution shape: Gaussian versus teacher sampling
Full Gaussian, teacher moments11.72—59.38—
Table 3: Seen-composite success (%) under Table 2 ’s protocol (Appendix F.4 ): 32 episodes per task and model for each intervention, 50 for reused references. Differences are second minus first (pp), with 95% independent within-task bootstrap intervals. All policies retain GAR commands except raw one-step, which uses native decoding.
Figure 4: Gaussian actions produce two routes in a constructed example. All 128 rollouts start at the left dot, pass on either side of the obstacle, and approach the right circle. Appendix G.1 specifies the state-dependent mean and constant covariance.
Figure 5: Xiaomi’s diagonal GAR produces two successful movement orders from the same start. The task is to pack one garlic and one chicken drumstick into each container; A and B distinguish objects of each type. Left: shared initial policy-camera view. Right: recorded simulator states after successive transfers, rendered from a fixed external camera and labeled with elapsed physical time. The two orders account for all three successes among 32 attempts: two follow the upper order and one follows the lower order (Appendix G.2 ).
Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

TeacherFeaturesttCosineΔrel\Delta_{\mathrm{rel}} (%)Norm change (%)Centered cosine
GR00Thh0.010.999931.68+1.160.99958
 uu0.010.999971.29+1.060.99961
 hh0.100.9993711.39+10.760.99684
 uu0.100.9998111.12+10.940.99635
Xiaomihh0.010.999981.03+0.760.99979
 uu0.010.999991.01+0.890.99982
Appendix
Table 4: Feature changes with time at an identical noisy input. Each row averages 32 validation observations and three noise draws, relative to t=0t=0 . Norm change is 100(r−1)100(r-1) ; centered cosine removes the mean feature at each time and noise seed.
Figure 6: One-pass Gaussian control on RoboCasa365. Selected successful executions using GR00T full-covariance readouts (left two) and Xiaomi diagonal readouts (right two). Each action chunk uses one action-transformer evaluation.
Preparation stepResourcesTime
Download, extraction and data checksCPU/network2.71 h wall time
Feature extraction and teacher sampling4 RTX 4090 GPUs3.84 GPU-hours total 1.14 h longest parallel job
Motion and command target estimation12 CPU workers6.9 s wall time
Appendix
Table 5: Preparation cost for Xiaomi’s cache of 7,200 contexts and 3,686,400 teacher chunks. The measurements exclude teacher pretraining and policy evaluation; checkpoint download was not timed.
  Full covarianceDiagonal covariance
TeacherAction groupError (%)Control (%)Error (%)Control (%)
XiaomiTranslation0.80.71.80.7
 Rotation1.81.63.01.5
GR00TTranslation4.63.39.13.4
 Rotation9.56.319.96.4
Appendix
Table 6: Gaussian approximation of translation and rotation across sixteen-action chunks at 300 fixed observations per teacher. Each group has 48 coordinates. Entries are median relative sliced 1-Wasserstein errors (%), normalized by the teacher’s projected command magnitude; lower is better. Controls repeat fitting and evaluation on samples from the corresponding full or diagonal Gaussian (Appendix D ).
Figure 7: Diagonal-covariance Gaussian approximation under the same procedure as Figure 2 , with 300 observations per panel. Approximation errors and Gaussian sampling controls use the same logarithmic bins and axis limits as the full-covariance comparison. Dashed lines mark medians, and all observations are retained.
  Physical errorRelative error, 90th percentile
TeacherCovarianceTranslationRotationTranslationRotation
XiaomiFull0.05900.00717.68.8
 Diagonal0.12070.01197.78.8
GR00TFull0.40540.072614.919.3
 Diagonal0.98670.131629.235.4
Appendix
Table 7: Physical sliced errors and upper-tail relative errors for the sixteen-action comparison on 300 contexts per teacher. Physical errors are medians across contexts, in millimetres for translation and degrees for rotation-vector commands. The last two columns give 90th percentiles of relative error (%); lower values are better. These are descriptive summaries without confidence intervals; Table 6 gives median relative errors and Gaussian controls.
Figure 8: Variance predictions from diagonal GAR fitted on 6,000 contexts, evaluated on 600 held-out contexts per teacher. Predictions track teacher variances but underestimate the largest values, especially for Xiaomi. Curves show medians within logarithmic teacher-variance bins; shading spans the 10th–90th percentiles of predictions, not confidence intervals. The dotted curve compares variance estimates from 128 independent teacher samples against 384-sample targets, providing a finite-sampling reference. The following diagnostics give all predictions and upper-tail errors.
TeacherFitting contextsMean error (%)SD error (%)
   DiagonalFull
Xiaomi6,00013.6316.1910.65
Xiaomi24,00013.4013.9010.34
GR00T6,00020.0829.0725.26
GR00T24,00019.4425.8624.65
Appendix
Table 8: 90th-percentile moment errors on the same 600 evaluation contexts per teacher as Table 1 . All errors are percentages of the teacher mean’s norm; lower is better.
TeacherFitting contextsEstimateTail ratioTail error (%)
Xiaomi—Independent teacher0.97610.9
Xiaomi6,000Diagonal0.57753.0
Xiaomi6,000Full0.50356.5
Xiaomi24,000Diagonal0.66945.5
Xiaomi24,000Full0.54552.4
GR00T—Independent teacher1.0017.4
Appendix
Table 9: Marginal variance accuracy on 600 evaluation contexts per teacher, relative to regularized 384-sample targets v^\widehat{v} . Independent teacher rows use unbiased variances from 128 disjoint teacher chunks per context; other rows use GAR predictions. Tail columns use the largest 1% of target variances pooled across contexts and coordinates, reporting the median individual ratio v/v^v/\widehat{v} (one is exact) and median relative error ∣v/v^−1∣|v/\widehat{v}-1| (lower is better). These targets exceed 0.00784 for Xiaomi and 0.0786 for GR00T, in their respective fitted action coordinates. Here vv denotes the independent estimate or GAR’s predicted marginal variance.
Figure 9: All held-out marginal-variance predictions from the deployed diagonal and full readouts. Gray points include every context–coordinate pair; curves and bands follow Figure 8 . Each teacher uses common axis limits across its four panels, spanning the complete target and prediction ranges without clipping. GR00T’s 16 structurally constant coordinates are excluded. Xiaomi uses normalized action coordinates and GR00T uses decoded coordinates.
Figure 10: Convergence of the 6,000-context diagonal covariance readouts. Lower moment loss is better; vertical dotted lines mark the deployed checkpoints selected by tuning loss. Training curves use a fixed 1,200-context subset and tuning uses 600 contexts from separate demonstrations. Both fits select update 18,000 and stop at 50,000 updates for Xiaomi and 42,000 for GR00T. Each panel shows one fit, with the full recorded update range and a logarithmic loss axis.
ObjectiveSeedSD error (%)Tail ratio
Original175.58 [3.61, 9.03]0.577
 295.29 [3.30, 8.67]0.581
Variance Huber173.16 [1.78, 5.64]0.410
 293.24 [1.76, 5.71]0.411
Log MSE172.81 [1.49, 5.55]0.244
 292.85 [1.48, 5.42]0.248
Appendix
Table 10: Xiaomi diagonal variance-loss ablation on 600 evaluation contexts, with fitting seeds reported separately. SD error is 100∥σθ−σ^∥2/∥μ^∥2100\|\sigma_{\theta}-\widehat{\sigma}\|_{2}/\|\widehat{\mu}\|_{2} , shown as median [25th, 75th percentile]; lower is better. Tail ratio is median predicted/target variance among the largest 1% of targets; one is exact.
Figure 11: Effect of the fitting loss on Xiaomi variance prediction, with checkpoints selected by their respective tuning objectives. Both seeds use all 600 evaluation contexts and 160 coordinates per context. Curves show median predictions in the same 24 logarithmic bins; shading shows the 10th–90th percentiles for each seed, not confidence intervals. Axes are shared across panels and span the full target-variance range. The dotted curve uses 128 independent teacher samples, while targets use 384 samples. Log-space fitting improves typical predictions but increases underestimation of the largest variances.
ModelCovariance contextsTeacher (ms)GAR (ms)Speedup
Threadripper PRO 5955WX: three sessions
Xiaomi6,000220.25115.361.91×\mathbf{1.91}\times [1.909, 1.915]
GR00T6,00068.0143.301.57×\mathbf{1.57}\times [1.566, 1.571]
EPYC 9554: earlier three sessions
Xiaomi24,000202.08111.841.80×1.80\times [1.797, 1.807]
GR00T6,00060.8742.981.42×1.42\times [1.407, 1.421]
Appendix
Table 11: Batch-one latency of diagonal policies on two RTX 4090 systems, including vision-language processing. Mean and command readouts use 6,000 contexts; covariance fitting counts are shown separately. Speedup is teacher/GAR latency; lower latency and higher speedup are better. Three-session measurements use 64 calls per policy per session on 32 matched contexts and report medians of session means and speedups, with observed speedup ranges in brackets. The separate EPYC protocol uses 400 calls per policy on fifty contexts after ten warmups.
ComponentTeacher (ms)GAR (ms)
Vision-language processing65.466.8
Action transformer135.428.3
GAR readout—0.5
Total call latency225.9120.7
Appendix
Table 12: Xiaomi latency breakdown on the Threadripper PRO 5955WX / RTX 4090 system, using the 6,000-context mean and diagonal covariance readouts. Values are means over ten matched observations in a separate profiling session. Module times use CUDA events; total call latency uses synchronized wall-clock timing and also includes preprocessing, transfers and decoding.
TeacherFitting contextsReadoutSuccessesDifference95% interval
Xiaomi6,000Mean only1,416−1.60-1.60[−3.84,0.60][-3.84,0.60]
  Diagonal1,441−0.60-0.60[−2.84,1.60][-2.84,1.60]
  Full1,432−0.96-0.96[−3.20,1.20][-3.20,1.20]
 24,000Mean only1,411−1.80-1.80[−4.08,0.40][-4.08,0.40]
  Diagonal1,443−0.52-0.52[−2.72,1.68][-2.72,1.68]
  Full1,411−1.80-1.80[−4.04,0.40][-4.04,0.40]
Appendix
Table 13: Success differences on fifty tasks with fifty episodes per policy and task (2,500 total), using the settings of Table 2 . Teachers succeed on 1,456 Xiaomi and 577 GR00T episodes. Difference is readout minus teacher success in percentage points; higher favors the readout. Pointwise 95% intervals use 20,000 independent within-task episode-bootstrap resamples (seed 85017), conditional on these tasks and fitted checkpoints.
TeacherFitting contextsDifference95% interval
Xiaomi6,000+1.00+1.00[−1.20,3.24][-1.20,3.24]
 24,000+1.28+1.28[−0.96,3.52][-0.96,3.52]
GR00T6,000+2.20+2.20[0.40,3.96][0.40,3.96]
 24,000+1.24+1.24[−0.56,3.04][-0.56,3.04]
Appendix
Table 14: Diagonal-minus-mean success differences in percentage points, with pointwise 95% intervals from 20,000 independent within-task bootstrap resamples (seed 85017). Positive differences favor diagonal sampling; each policy has fifty episodes per task on the fixed fifty-task suite.
TeacherGARFixedDifference95% interval
Xiaomi57.6455.92+1.72+1.72[−0.44,3.92][-0.44,3.92]
GR00T21.9220.76+1.16+1.16[−0.64,3.00][-0.64,3.00]
Appendix
Table 15: GAR versus Fixed success on RoboCasa365, with 2,500 episodes per policy and teacher. Both policies share their mean and command readouts; only the covariance changes. Success is a percentage, and differences are GAR minus Fixed in percentage points; higher favors GAR. Brackets give pointwise 95% independent within-task episode-bootstrap intervals, conditional on the fitted checkpoints and fixed task suite.
TeacherAction group (unit)GARFixedReduction [95% interval]
XiaomiTranslation (mm)0.5660.9870.420  [0.359,0.483]0.420\;[0.359,0.483]
 Rotation (degrees)0.1450.1960.051  [0.042,0.061]0.051\;[0.042,0.061]
GR00TTranslation (mm)1.5602.6471.088  [0.887,1.299]1.088\;[0.887,1.299]
 Rotation (degrees)0.3360.6360.301  [0.259,0.343]0.301\;[0.259,0.343]
Appendix
Table 16: Mean sliced 1-Wasserstein distance to teacher action chunks on 143 held-out contexts per teacher; lower is better. GAR and Fixed share their means and command predictors. Distances and reductions use the action group’s stated unit; reduction is Fixed minus GAR, so positive values favor GAR. Brackets give pointwise 95% paired context-bootstrap intervals from 10,000 resamples, conditional on the fitted readouts and cached samples.
Continuous-motion policyGR00TXiaomi
GAR: predicted diagonal moments94 / 800464 / 800
Gaussian: teacher-estimated diagonal moments67 / 512300 / 512
Gaussian: teacher-estimated full moments60 / 512304 / 512
Teacher sampling, GAR commands61 / 512296 / 512
Teacher-estimated mean, no motion noise58 / 512283 / 512
Teacher-estimated mean, GAR variances75 / 512308 / 512
Appendix
Table 17: Native-protocol diagnostic outcomes across sixteen seen composite tasks, in successes/episodes. All policies use GAR’s binary-command rule. The reference reuses 50 episodes per task from Table 2 ; each intervention uses 32 per task.
TaskTeacherRaw one-stepGAR
Soup and tomato sauce into basket494748
Cream cheese and butter into basket484747
Turn on stove and place moka pot505050
Bowl into drawer and close it484648
Two mugs onto plates505050
Book into caddy505050
Appendix
Table 18: LIBERO-Long successes on fifty common initial states per task. The teacher is Xiaomi-Robotics-0 ; diagonal GAR uses 6,000 fitting contexts. Overall entries are successes out of 500 episodes.
Figure 12: Behavioral diversity of diagonal GAR and the Xiaomi teacher from identical starts. Each scene has 32 executions per policy, including all successes and failures. Bars show first-object and full movement-order entropy at 5 cm; lines show 95% within-policy, within-scene bootstrap intervals. Zero observed entropy does not exclude rare alternatives, and equal entropy does not establish matching choices.
SceneTeacher countsDiagonal counts
Dishwasher 1cup: 19; bowl: 13cup: 15; bowl: 17
Dishwasher 2dish A: 32dish A: 32
Lunches 1vegetable A: 2; vegetable B: 30vegetable A: 1; vegetable B: 31
Lunches 2chicken A: 2; garlic A: 30garlic A: 32
Leftovers 1chicken drumstick: 4; vegetable: 28vegetable: 32
Leftovers 2vegetable: 32vegetable: 32
Appendix
Table 19: First-object counts for diagonal GAR and the Xiaomi teacher across all 32 attempts per policy and scene, using the 5 cm threshold. These are movement choices rather than successful completions; no ties or no-movement outcomes occur in these scenes. A and B distinguish objects of the same type within a scene.
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