Gaussian Action Readout

Gaussian Distillation
of Flow Policies

Replace a robot flow policy’s last layer with a Gaussian readout.
Similar task success. One action-network pass.

Youssef Saied1Ziyad Sheebaelhamd2Claire Vernade3François Fleuret1,4

1University of Geneva2University of Tübingen3University of Technology Nuremberg4Meta FAIR

Read the paper PDFCode release soon

One Gaussian. Different paths.

Interactive illustration
128 trajectories from the same starting point branch above and below a circular obstacle, then reunite at the goal.
Gaussian actions can produce diverse behavior through diffusion in physical time. Change the noise to see how paths emerge.
How this illustration works

This is a constructed 2D example, not a robot experiment. At each step, the action is drawn from one Gaussian whose mean depends on the current position. All 128 paths share the same rule and starting point; no route is assigned in advance. The slider scales the action’s standard deviation while keeping the mean field and random draws fixed. The circle marks a region to avoid; this illustration has no collision physics.

1 passthrough the action network
0.02%of eligible states for fitting
Up to 2×faster full policy calls
Closed formaction likelihoods

The observation

A complex sampler.
A surprisingly simple distribution.

Flow policies generate actions through several network evaluations. But how complex are the distributions they produce?

For the same robot observation, we sample many action chunks and compare their motion to a Gaussian.
A Gaussian closely approximates both models’ motion distributions, especially Xiaomi’s.

For Xiaomi, relative errors are just 0.8% for translation and 1.8% for rotation across 16-action chunks.

Median relative sliced Wasserstein errors over 300 observations, using full covariance. The orange control measures the error expected when the samples themselves come from a Gaussian.

Xiaomi translation and rotation Gaussian approximation errors closely track the Gaussian sampling control.
The fitted Gaussian captures Xiaomi’s motion distribution closely, including correlations across the action chunk.

The method

Keep the network.
Replace the last layer.

Our Gaussian Action Readout (GAR) turns a pretrained flow policy into a policy that samples directly. We freeze the network and train small readouts to predict the distribution’s parameters from its features.

Robot observations enter a frozen vision-language backbone and action transformer. Trainable GAR readouts predict Gaussian motion and Bernoulli commands, then sample an action chunk.
One action-network evaluation gives the parameters for Gaussian motion and binary commands.
01

Collect the moments

Sample the teacher at 6,000 contexts to estimate means, covariances and command frequencies.

02

Fit the readouts

Use the teacher’s frozen features. Only the new readouts are trained, on 0.02% of eligible states.

03

Sample and score directly

Draw actions and compute their likelihoods without integrating a flow or making extra network calls.

With features and targets cached, fitting Xiaomi’s readouts takes about two minutes.

On the robot tasks

Similar task success.
Fewer network evaluations.

We evaluate Xiaomi-Robotics-1 and GR00T N1.5 on 50 RoboCasa365 tasks, with 2,500 episodes per policy.

Task success with diagonal GAR
ModelTeacherGARSpeedup
Xiaomi58.24%57.64%1.91×
GR00T23.08%21.92%1.57×

GAR uses one action-network pass, replacing five flow steps for Xiaomi and four for GR00T.

Speedups measure the complete policy call, including the vision-language backbone, at batch size one.

Does Gaussian shape cost task success?

We also compare teacher motion samples with a Gaussian fitted to the teacher’s mean and full covariance, keeping binary-command handling fixed. Both models achieve similar success. These controls support the Gaussian approximation in practice, while separate mean-replacement tests show that mean-estimation errors can limit GR00T’s mean-only control.

See the controlled comparisons in the paper →
Selected successful GAR rollouts. GR00T above, Xiaomi below; excerpts shown at 4× speed.

Explore the intuition

What’s hiding in the flow?

A Gaussian has a center and a spread. Move them below to see what the flow reveals about each.

The center

Mean

The mean is hiding in the initial velocity. Add the input noise back: every starting point lands on the same mean.

At t = 0ε + v(ε, 0) = μ
Different noise inputs plus their initial velocities all give the same mean action.
Different noise inputs. One answer.

The noise is known. One velocity evaluation gives the mean. For a Gaussian, that is also its most likely action.

The spread

Variance

Different spreads give the same initial velocity. Move forward in flow time to see their velocity fields separate.

At t = ½ σ² = 2 + α2 − α α = v(x, ½) − μx − μ / 2 α is the velocity line’s slope; x ≠ μ/2.
Narrow and wide Gaussians with the same mean have identical velocity at time zero and different slopes at time one half.
Different spreads. Identical velocity at t = 0.

GAR learns to predict variance from frozen features, using teacher samples as training targets.

These illustrations use ideal linear flow matching from independent Gaussian noise. The mean identity also holds for non-Gaussian targets. GAR is fitted to the deployed teacher’s finite-step samples. Here, t is flow time, not physical time.

Beyond a single decision

One Gaussian does not mean
one behavior.

Sampled actions change the state, which changes the next Gaussian. Through this feedback, a simple distribution at each decision can produce different behaviors over time: diffusion in physical time.

Two successful Xiaomi GAR executions from the same lunch-packing scene move the objects in different orders.
Same initial scene, different packing orders. Two selected successful Xiaomi GAR executions in RoboCasa.

The toy above shows the idea. The robot executions show that a Gaussian action policy can also support different task-solving behaviors.

The paper

Gaussian Distillation
of Flow Policies

Youssef Saied, Ziyad Sheebaelhamd,
Claire Vernade and François Fleuret

arXiv:2610.40271

Code release soon.

BibTeX
@misc{saied2026gaussian,
  title={Gaussian Distillation of Flow Policies},
  author={Youssef Saied and Ziyad Sheebaelhamd
    and Claire Vernade and Fran\c{c}ois Fleuret},
  year={2026},
  eprint={2610.40271},
  archivePrefix={arXiv},
  url={https://arxiv.org/abs/2610.40271}
}