Quantile Head for Vision-Language-Action Models

1 INTRODUCTION

• Regression provides a single-pass point estimate, while standard flow-matching samplers model action distributions through iterative integration and repeated evaluations.The paper frames these approaches as complementary: direct deterministic control versus distributional sampling.Source: §1 · p. 1
• Quantile Head extends a single action prediction to a median and lower and upper quantiles that characterize each action coordinate's conditional distribution.The quantiles are jointly predicted to combine single-pass prediction with distributional modeling.Source: §1 · p. 1, Figure 1 · p. 2
• With calibrated nearby quantiles, fixed gaps, and matched correction speed, joint supervision lowers direct median-update variance relative to median-only supervision.This is a local result under the stated conditions.Source: §1 · p. 2
• The head uses a frozen pretrained VLM, learnable prompts, and positive gaps around the median to generate ordered quantiles in one forward pass.A masked pinball loss jointly supervises the quantiles; inference defaults to the median.Source: §1 · p. 2
• Quantile Head achieves state-of-the-art performance on LIBERO, LIBERO-Plus, and LIBERO-Pro, and the best performance among compared methods on two physical robot tasks.The paper also reports lower median-update noise under stated conditions.Source: §1 · p. 2

2 RELATED WORK

• Regression-style heads predict actions in one forward pass, but point predictions do not explicitly represent conditional action distributions.Squared and absolute losses target the conditional mean and median, respectively, for unrestricted predictors (Koenker & Bassett Jr, 1978).Source: §2 · p. 2
• Standard diffusion and flow-matching samplers require successive denoising or integration steps and repeated model evaluations, although recent VLA methods also support one-step generation.SnapFlow uses progressive self-distillation; Let It Be Simple uses high-noise training (Luan et al., 2026; Chen et al., 2026b).Source: §2 · p. 3
• Quantile Head jointly supervises ordered action quantiles for VLA control, with median decoding by default and optional quantile sampling.Prior work includes advantage-weighted quantile regression (Richter & Wattenhofer, 2019), ordered median-anchored outputs, and calibrated error quantiles for candidate ranking (Chen et al., 2026a).Source: §2 · p. 3

3.1 THEORETICAL ANALYSIS

• The shared reference-path objective unifies squared action regression and conditional flow matching, while integrated pinball loss provides equivalent CDF supervision up to a factor of two.Under the stated local conditions, auxiliary calibrated quantiles reduce direct median-update variance relative to median-only updates.Source: §3.1 · p. 4

3.2 QUANTILE HEAD

• A frozen VLM and learnable prompts provide context to an action expert, whose head predicts a median and positive gaps that accumulate into ordered quantiles.Positive gaps prevent quantile crossing, and separately learned left and right gaps permit asymmetric spacing; the median is the default decoded action.Source: §3.2 · pp. 4–5

3.3 MASKED PINBALL LOSS

• Masked pinball loss supervises the predicted quantiles, randomly withholding selected valid action-coordinate labels while sharing each coordinate's mask across its quantiles.The mask is redrawn on each training forward pass; evaluation uses all valid coordinates without random dropping.Source: §3.3 · p. 6

4.1 EXPERIMENTAL SETTING

• Evaluation covers LIBERO, LIBERO-Plus, LIBERO-Pro, and two real-robot manipulation tasks, with LIBERO-Plus perturbing 40 tasks into 10,030 instances.The LIBERO suites each contain ten tasks; LIBERO-Plus varies viewpoints, robot starts, instructions, lighting, backgrounds, sensor noise, and object layouts.Source: §4.1 · p. 6

4.2 TASK PERFORMANCE WITH DEFAULT MEDIAN DECODING

• Quantile Head reaches 99.3% average success across four LIBERO suites, leads LIBERO-Plus and LIBERO-Pro comparisons, and averages 75.0% across two real-robot tasks.On LIBERO-Plus, it reaches 87.1% zero-shot and 89.1% fine-tuned success; on LIBERO-Pro, it reaches 60.2% without further training. The real-robot average compares with 58.5% for π0.5, a 16.5 percentage-point gain.Source: §4.2 · p. 7

4.3 ABLATION STUDIES

• Across four LIBERO suites, median-decoded Quantile Head reaches 99.3% average success versus 96.0% for L2, 97.3% for L1, and 96.3% for sampled Flow Matching.Mean episode time including failures falls from 8.13 s for L1 to 5.88 s, a 27.7% reduction.Source: §4.3 · p. 7, Table 4 · p. 8
• Median-Only reaches 97.5% mean success versus 99.3% for the full head, while Detached Median Anchor reaches 98.7% and Symmetric Quantiles 97.9%.Task success does not measure error to the true conditional median; the reported rates are consistent with potential benefits from shared supervision and the direct gradient path.Source: §4.3 · p. 8
• Joint quantile supervision raises mean evaluation success in all four prompt and label-mask configurations: 83.5% to 91.8%, 95.5% to 98.8%, 86.0% to 97.0%, and 97.5% to 99.3%.These matched comparisons use median inference without quantile sampling.Source: §4.3 · p. 8

4.4 EXPLORING ALTERNATIVE SAMPLING STRATEGIES

• On LIBERO-Long, narrowing the sampling window from [0.1, 0.9] to [0.4, 0.6] raises success from 93.0% to 99.6%, 1.0 percentage point above median decoding.The comparison uses cross-chunk correlation ρ = 0.7; median decoding remains the default and requires no sampling choices.Source: §4.4 · p. 8, Figure 4 · p. 9
• In one LIBERO-Long rollout, predicted marginal action ranges are narrower during selected transport moments and wider during grasping and placing.This pattern suggests a possible role for local adjustments around object contact but does not establish distributional accuracy.Source: §4.4 · p. 8, Figure 5 · p. 9
• In two controlled scenes, uniform and density-weighted quantile sampling reach 92% and 95% success without an obstacle, but only 6% and 13% with one.With the obstacle, median decoding, both regression heads, and Flow Matching reach 0%; density weighting requires no retraining.Source: §4.4 · p. 8

• Quantile Head jointly supervises ordered marginal action quantiles for single-pass median control.Source: §5 · p. 9
• The local analysis finds lower median-update variance than median-only supervision with nearby calibrated quantiles, fixed gaps, smooth densities, and matched correction rates.Source: §5 · p. 9
• Joint supervision improves mean success across four matched LIBERO configurations, while the median policy reaches 99.3% on LIBERO and 75.0% across two real-robot tasks.Sampling yields gains and losses; systematic strategy selection remains future work.Source: §5 · p. 9

A.1 PROOF OF THEOREM 1

• Threshold integration uses all cutoffs to respect action-axis distance, but this weighting is a supervision choice rather than the only possible one.The proof permits flat segments and jumps, without requiring strict ordering or a continuous density.Source: §A.1 · pp. 14–15
• First moments suffice for the CDF-loss argument, while squared-risk decomposition for flow matching requires finite second moments.Bounded actions satisfy the label condition, but affine normalization without clipping does not itself bound targets.Source: §A.1 · pp. 14–15
• Integrated threshold-event supervision is equivalent to quantile regression, while the shared objective recovers squared action regression and conditional flow matching under their specified choices.The pairwise term in the CDF loss follows from the integrated threshold loss rather than being added to the regression or CFM objective.Source: §A.1 · pp. 14–15
• When representable, the true conditional CDF is the unique population optimum, and its quantile representation minimizes the integrated quantile loss.The result allows atoms and requires neither a density nor symmetry; representability is needed only to attain the lower bound.Source: §A.1 · p. 16
• A finite grid identifies only its selected quantiles for each action marginal, not intervening quantiles, omitted tails, or cross-coordinate dependence.For the paper’s K = 21 levels on [0.025, 0.975], the finite pinball objective need not equal the continuous integrated loss.Source: §A.1 · p. 16
• The output-level approximation argument does not guarantee that a shared finite network represents the functions or that training finds them.Random label dropping preserves the finite objective in expectation, but the masking analysis does not establish finite-sample convergence or closed-loop performance.Source: §A.1 · pp. 16–17

Assumptions and comparison

• For calibrated, distinct symmetric quantile levels sufficiently close to 1/2, fixed gaps and matched local correction speeds yield lower variance in direct median updates.The comparison assumes a fixed input and a fixed finite number of quantiles; it does not require a symmetric conditional density.Source: §A.2 · pp. 17–18

Gradient variance

• Nested quantile indicators have dependent scores, so the proof uses their composite-quantile covariance rather than assuming independent quantile scores.The covariance is min(τk, τℓ) − τkτℓ.Source: §A.2 · p. 17

Local mean correction

• Symmetric probability levels cancel first-order terms, leaving the local mean-correction slope changed only at second order.This cancellation does not require symmetry of the density or equal left and right gaps.Source: §A.2 · p. 18

Noise after matching correction speeds

• For sufficiently close symmetric levels, the normalized joint-update noise ratio is below one, uniformly over symmetric grid shapes for fixed K.The argument permits varying relative spacing and requires only sufficiently small positive ε.Source: §A.2 · p. 18

Scope of the conclusion

• The theorem establishes a local center-update mechanism, not a guarantee of joint-training generalization or task success.It does not cover inaccurate gaps, simultaneous gap and shared-network updates, or certify the wider experimental quantile grid.Source: §A.2 · p. 18

Simulation training and evaluation

• Models use a frozen VLM, 16 prompt tokens, and 21 quantile levels; LIBERO and LIBERO-Plus training uses one epoch from the same pretrained checkpoint.LIBERO evaluations use three seeds and 1,000 episodes per suite; LIBERO-Plus evaluates all 10,030 instances, while LIBERO-Pro evaluates 100 episodes per task for each seed-trained model.Source: B EXPERIMENTAL DETAILS AND ADDITIONAL… · p. 19

Aggregation and comparisons

• External baseline results retain their source protocols, while Table 4 provides matched action-head comparisons; LIBERO-Plus Total is instance-weighted across 10,030 instances.LIBERO-Pro averages perturbation and suite rates equally; the standard LIBERO comparison averages its four suite rates equally.Source: B EXPERIMENTAL DETAILS AND ADDITIONAL… · p. 19

Number of quantiles

• 98.3–98.6% success occurs for K = 15–25, versus 95.7–96.5% for K = 5–11; default K = 21 reaches 98.6%.Rates average three training seeds, with 1,000 evaluation episodes per seed and setting; K = 31 falls to 97.6%.Source: Figure 7 · p. 20, B EXPERIMENTAL DETAILS AND ADDITIONAL… · p. 20

Ablations and flow-matching inference

• The component ablations vary pinball supervision, label masking, and prompts; the matched flow-matching baseline uses ten Euler integration steps per replanning call.Median-Only retains the quantile head but supervises only its median, and disabling joint supervision leaves gap branches without supervised gradients.Source: B EXPERIMENTAL DETAILS AND ADDITIONAL… · p. 20

Additional benchmark results

• Table 5 reports Quantile Head success rates by LIBERO-Pro suite and perturbation type, while Table 6 reports LIBERO-Plus rates by suite and perturbation.Table 6 specifies instance-weighted aggregation across the four suites for Overall.Source: Table 5 · p. 21, Table 6 · p. 21

Label mask ratio

• 98.6% success occurs at masking ratios r = 0.1 and r = 0.3, compared with 98.0% without masking.Success falls to 87.2% at r = 0.9 and 0.0% at r = 1.0, where all valid action labels are dropped.Source: Figure 8 · p. 20, B EXPERIMENTAL DETAILS AND ADDITIONAL… · p. 21

Sampling analysis

• Sampling analyses compare quantile decoders with a Gaussian surrogate that matches the predicted median and 10–90% width while changing the marginal inverse CDF.The study also varies rank windows and temporal rank structure; the gripper uses its median.Source: B EXPERIMENTAL DETAILS AND ADDITIONAL… · p. 21

Per-dimension action distributions

• Figure 9 presents predicted action spread across six motion dimensions as median-centered ranges in environment-command units from the Figure 5 rollout, with rotation panels vertically magnified fivefold.Source: Figure 9 · p. 22

Per-dimension ranges

• The ranges average median-centered endpoints across six coordinates, excluding the gripper, and describe marginal ranges rather than the range of an averaged action.They use the first predicted step; outer quantile knots are linearly extrapolated.Source: §B.1 · p. 22

Figure 9

• Figure 9 plots ranges in environment-command units, with separate translation and rotation scales and rotation magnified fivefold.The plot draws on 26 replanning observations from one successful LIBERO-Long episode.Source: §B.1 · p. 22

Rollout example

• The example rollout places alphabet soup and tomato sauce in a basket; “Place” marks lowering into the basket, not completed release.Video frames are matched to observations by recorded indices.Source: §B.1 · p. 22

Platform

• The platform uses an AgileX Piper arm, a two-finger gripper, a wrist RGB camera, and a fixed external RGB camera.The setup and arm assembly are labeled in Figure 10.Source: §B.2 · p. 22, Figure 10 · p. 23

Tasks

• The two tasks are placing an apple on a yellow plate and removing a cuboid from a blue plate.Figure 11 frames show task sequences, with keyframes ordered in time and arrows indicating motion direction.Source: §B.2 · p. 22, Figure 11 · p. 23

Training and evaluation

• Each task has 100 demonstrations; Quantile Head and π0.5 share the training budget and evaluation conditions, but π0.5 fully fine-tunes its VLM.Each method and task is evaluated for 100 trials per trained model, with timeouts counted as failures.Source: §B.2 · pp. 22–24

Task and demonstrations

• The two successful 315-action demonstrations take opposite 11 cm lateral detours from an identical initial observation, with no left/right instruction cue.Training reserves 25% of batch slots for the ambiguous shared start and samples both trajectories equally for the remainder.Source: §B.3 · p. 24

Training

• All variants train for 6,000 updates with the VLM frozen; Quantile Head additionally trains a gap projection, so trainable parameter counts differ.The variants share batch sequences and action-side initialization.Source: §B.3 · p. 24

Fixed-observation sampling

• At one fixed observation, Quantile Head draws 1,000 samples without another network forward pass per draw; Flow Matching uses ten integration steps.Quantile sampling shares one rank across six motion coordinates and ten prediction steps, with the gripper at its median; these draws are not closed-loop episodes.Source: §B.3 · p. 24, Figure 13 · p. 26

Closed-loop results

• At the shared training initial state, three sampled Quantile decoders achieve 87–94% success versus Flow Matching’s 3%; median, L1, and L2 each achieve 100% with identical repeated action sequences within each trained model.Source: Table 7 · p. 26, §B.3 · p. 27

Evaluation protocol

• Each closed-loop setting receives 100 episodes from the shared initial state, executing ten actions before replanning and stopping by 400 actions.No weights are updated during evaluation.Source: §B.3 · pp. 25–26

Density-weighted sampling

• 95% success with density-weighted sampling compares with 92% for uniform shared ranks and 100% for median decoding, without retraining.This supplemental evaluation uses the same no-obstacle Quantile models.Source: §B.3 · p. 28

Task and demonstrations

• The added visible, collidable box measures 6 × 4.4 × 19 cm; two new 430-action demonstrations take opposite detours and finish without recorded obstacle contact.A straight-approach control fails and records contact on 179 of 475 executed steps.Source: §B.4 · p. 28, Figure 15 · p. 29

Training

• All four variants are retrained for 6,000 updates on only the two obstacle demonstrations, with the VLM frozen and shared action-side initialization.Each variant uses the same training batch sequence within this experiment.Source: §B.4 · p. 28

Action distributions

• Quantile samples peak near both demonstrated dx and dy targets, while L1 and L2 lie between them and Flow Matching spreads over wider action ranges.The plotted marginal distributions are evaluated at one shared initial observation; task-completion accuracy is evaluated separately.Source: §B.4 · pp. 28–29, Figure 16 · p. 30

Task success

• 6.0% success for shared-rank Quantile sampling is highest among the seven obstacle-setting decoders; independent ranks reach 4.0%, native ranks 1.0%, and the other four 0.0%.The comparison uses 100 episodes at the fixed training initial state; it does not establish reliable obstacle avoidance.Source: §B.4 · p. 30, Table 8 · p. 31

Density-weighted sampling

• 13% success with density-weighted shared ranks compares with 6% for uniform shared ranks at the same obstacle initial state.The six-coordinate density-weighted decoder uses squared mean marginal density; its 13% rate does not establish reliable obstacle avoidance.Source: Table 9 · p. 32, §B.4 · p. 34

Trajectory interpretation

• In the illustrated run, successful shared-rank trajectories first cross the obstacle-center plane on both sides, three on the left and three on the right.This describes the first crossing only; it does not mean every sampled route succeeds or avoids contact throughout.Source: §B.4 · p. 32

Sampling interval

• 15% success with shared ranks over [0.2, 0.8] exceeds full-range [0, 1] sampling by 9 percentage points, but success does not rise monotonically as the interval narrows.Median decoding achieves 0% in this interval comparison; no model is retrained when changing intervals.Source: §B.4 · p. 32, Figure 18 · p. 33