Geometry
How far is the action mismatch?
Measure exact one-dimensional Wasserstein-1 distance on sorted, decoded physical values. Larger probability displacements incur larger transport costs.
Physical action distance Vision–Language–Action Models / ICLR 2027 · Under review
PGOT: Physically Grounded
Optimal Transport
for Distilling Vision-Language-Action Models
with Discrete Autoregressive Decoding
01 / CoNav-UAV
Coordinate-conditioned
00:47 ↗Navigate the world.
02 / LIBERO-10
Instruction-conditioned
01:31 ↗Act with precision.
Physical structure. Compact policies. Consistent completion.7B / 9B teachers 0.8B students
01 / The idea
Action tokens have
physical meaning.
Moving probability to a nearby control value and moving it to a distant one produce different physical errors. Conventional probability matching does not explicitly encode this distance. PGOT brings action geometry into distillation.
Abstract in brief
We introduce Physically Grounded Optimal Transport (PGOT) for distilling vision-language-action policies with discrete autoregressive action decoding.
Geometry aligns teacher and student action distributions using decoded physical distances. Criticality emphasizes transitions in expert actions. Reliability conditions teacher transfer on agreement with demonstrations.
With Qwen2.5-VL-7B and Qwen3.5-9B teachers, domain-specific Qwen3.5-0.8B students improve average success and repeated completion on UAV navigation and robotic manipulation.
Same probability mismatch.
Different physical distance.
InteractiveTeacherStudent
Forward KL1.367Unchanged
Physical W₁0.206Scales with distance
Illustrative 9-bin distributions on [0, 1], each with 0.60 peak probability and 0.05 in every other bin. PGOT uses 256 action tokens.
02 / The method
Three components.
One grounded objective.
PGOT uses teacher output distributions and offline expert demonstrations. All three components act during training; deployment retains the student’s architecture and autoregressive action decoder.
Criticality
Where does control change?
Increase distillation weight around large changes in neighboring expert actions, concentrating learning on navigation maneuvers and gripper transitions.
Action-transition weightingReliability
Which teacher guidance should transfer?
Attenuate teacher bins far from the demonstrated action and gate transport by teacher–expert agreement in the decoded physical action space.
Expert-conditioned transfer
03 / The evidence
Small students.
More consistent execution.
PGOT achieves the highest success rate and three-out-of-three completion among the evaluated policies in both domains, using the same distillation coefficients.
150 start–goal configurations · 3 executions each · 0–500 m · 200-step horizon
| Policy | SR (%) ↑ | Pass³ (%) ↑ | CR (%) ↓ | TR (%) ↓ | MMBC (m) ↑ |
|---|---|---|---|---|---|
| Teacher SFT 7B | 87.56 ± 1.39 | 78.67 | 9.78 | 2.67 | 5.00 |
| Student SFT | 53.33 ± 1.15 | 38.67 | 44.00 | 2.67 | 2.78 |
| FKL | 85.33 ± 1.33 | 74.67 | 11.78 | 2.89 | 5.27 |
| RKL | 78.00 ± 2.31 | 68.00 | 16.00 | 6.00 | 5.00 |
| ABKD | 80.89 ± 1.02 | 67.33 | 15.56 | 3.56 | 4.98 |
| ToDi | 75.56 ± 3.42 | 62.00 | 17.78 | 6.67 | 5.00 |
| Shallow-π | 85.33 ± 0.67 | 74.00 | 10.89 | 3.78 | 5.09 |
| VLA-AD | 74.89 ± 0.38 | 58.00 | 20.89 | 4.22 | 4.20 |
| PGOT Ours | 88.00 ± 2.91 | 82.00 | 8.00 | 4.00 | 5.72 |
↗
82.00% of navigation configurations succeed in all three executions. PGOT also has the lowest collision rate and largest mean minimum building clearance among the evaluated student policies.
10 tasks × 50 initial states · 3 executions each · 520-step horizon
| Policy | SR (%) ↑ | Pass³ (%) ↑ | MES ↓ |
|---|---|---|---|
| Teacher SFT 9B | 49.13 ± 0.61 | 37.20 | 408.02 |
| Student SFT | 45.47 ± 0.95 | 31.40 | 409.95 |
| FKL | 44.93 ± 1.33 | 39.40 | 413.11 |
| RKL | 37.73 ± 0.83 | 22.20 | 433.07 |
| ABKD | 41.93 ± 1.03 | 27.40 | 416.14 |
| ToDi | 43.73 ± 0.81 | 27.40 | 416.21 |
| Shallow-π | 47.07 ± 1.62 | 32.40 | 407.13 |
| VLA-AD | 39.73 ± 0.90 | 25.20 | 424.94 |
| PGOT Ours | 49.47 ± 0.64 | 45.40 | 400.47 |
↗
45.40% three-out-of-three completion on LIBERO-10. PGOT also uses the fewest mean execution steps among the evaluated policies, counting all episodes including failures.
SR: mean success rate ± sample standard deviation across three evaluation rounds. Pass³: fraction of initial configurations completed in all three executions. CR / TR: collision / timeout rate. MMBC: mean minimum building clearance. MES: mean execution steps. Bold marks the best student value. All students use Qwen3.5-0.8B; action sampling temperature is 0.3. Source: paper, Table 1.
What does each component contribute?Component ablations
Removing any component lowers SR and Pass³ in both domains. Removing Geometry produces the largest decrease in both metrics.
| Objective | UAV SR (%) ↑ | UAV Pass³ (%) ↑ | LIBERO SR (%) ↑ | LIBERO Pass³ (%) ↑ |
|---|---|---|---|---|
| PGOT | 88.00 ± 2.91 | 82.00 | 49.47 ± 0.64 | 45.40 |
| w/o Geometry | 82.22 ± 2.14 | 72.67 | 44.60 ± 0.87 | 29.80 |
| w/o Criticality | 83.78 ± 1.54 | 74.67 | 48.67 ± 0.90 | 42.40 |
| w/o Reliability | 84.89 ± 3.29 | 75.33 | 47.67 ± 1.62 | 34.20 |
Source: paper, Table 2. Removing Geometry sets w₁ = 0; Criticality sets γ = 0; Reliability sets α = βᵣ = 0.
04 / In motion
From action tokens
to behavior.
Side-by-side replays compare PGOT, FKL, and Shallow-π on matched initial configurations. These are selected qualitative examples; the tables report the full benchmark evaluation.
01 / CoNav-UAV
4 selected cases · 1080p / 60 fpsUAV navigation
Jump to
Rows: PGOT, FKL, Shallow-π. Columns: first-person, third-person, and bird’s-eye views. In these four selected replays, PGOT succeeds while both baselines collide. Silent video on a shared policy-step timeline, normalized within each case; terminal frames are held.
Download video02 / LIBERO-10
3 selected cases · 1080p / 60 fpsRobotic manipulation
Jump to
Columns: PGOT, FKL, Shallow-π. Rows: agent and wrist camera views. In these three selected replays, PGOT succeeds while both baselines time out. Silent video at 1× simulated time; terminal frames are held.
Download video05 / A closer look
Understanding
the improvement.
Training and paired-rollout diagnostics connect the components to complementary gradients, control transitions, and fewer corrective actions.
Complementary transport gradients
Geometry contributes a distinct gradient direction: mean angles to filtered FKL are 73.94° on CoNav-UAV and 72.05° on LIBERO-10.
More consistent UAV navigation
On CoNav-UAV, PGOT reduces validation physical W₁ by 10.42% relative to w/o Geometry. On jointly successful attempts, reversals and backtracking decrease by 62.73% and 32.83% relative to w/o Reliability.
Explore the research
Ground the objective.
Keep the student compact.
The anonymous repository includes training, data preparation, evaluation, and metrics.