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When Do Discrete Actions Make Good Continuous Controllers?

Zeji Yi, Soham Dasgupta, Xinyang Zhang, Koushil Sreenath, Guannan Qu, Xi Chen, Max Simchowitz, Guanya Shi

arXiv:2610.20429v1cs.RO

TL;DR

Robot foundation models commonly predict continuous actions, raising the question of whether robot control requires a different representation from other AI modalities. The paper compares discrete and continuous action prediction and finds that discrete actions can be effective for robot control, while coordination among token choices remains a key challenge.

  • Problem

    Robot models commonly predict continuous actions for precision, while other AI models use discrete tokens, prompting the question of whether robot actions require continuous prediction.

    Source: §1 · p. 2

  • Method

    The paper systematically compares continuous and discrete action parameterizations, including regression and flow heads, across single-task and multi-task settings.

    Source: §1.1 · p. 2

  • Results

    Discrete action prediction can support precise control in simulation and real-world tasks, and the experiments identify coordination of token choices as a key difficulty.

    Source: §6 · p. 12

  • Takeaways & Limitations

    The findings support treating discrete action prediction as an effective robot-control approach and a central component of future robot foundation models.

    Source: §6 · p. 12

    The tested reconstruction-based tokenizers do not improve on the best uniform-bin DF result, and the value of alternatives may depend on the prediction head and available refinement steps.

    Source: §6 · p. 12

Abstract

from arXiv · show

Though modern AI models like LLMs predict discrete tokens, today's robotic foundation models represent actions as continuous vectors. In this paper, we challenge this common consensus by asking if continuous action parametrizations are superior to their discrete counterparts. We conduct an extensive comparison of single-step regression and multi-step flow variants of both continuous and discrete policies in both from-scratch single-task and multi-task VLA training, in both simulation and on real hardware. In all regimes, we find that discrete action prediction, even the most naive, uniform-bin quantization, can match or even outperform continuous-action policies in task success rate. Moreover, we find that the challenge to discrete action policies is \emph{not} action precision, but rather factorization error introduced by product-distribution predictions across time and action dimensions. We show that, when paired with the correct interventions---iterative computation through discrete flows, or distribution sharpening through classifier-free guidance---discrete flow policies can turn this weakness into a strength, enabling more responsive action changes during a predicted action chunk. Lastly, we show that attempts to explicitly inject geometric information into discrete policies, e.g. through corruption processes that emulate continuous flow matching, do \emph{not} yield consistent improvements. Taken together, our findings question existing assumptions that robotic action prediction requires distinct design decisions compared to other AI domains, and motivates the possibility of discrete action prediction as a compelling alternative to today's continuous action parametrizations. Link for the webpage \href{https://iscoyizj.github.io/discrete-action-website/}{https://iscoyizj.github.io/discrete-action-website/}

1 Introduction

Robot learning commonly favors continuous action prediction for precision, while other AI models often use discrete tokens. This paper compares continuous and discrete robot action heads and reports that discrete heads can match or outperform continuous ones when appropriately designed.

1 Introduction

  • The paper asks whether robot actions must be continuous or whether discrete representations can compete with or outperform them.The prevailing view favors continuous actions as lossless and potentially more precise.Source: §1 · p. 2

1.1 Contributions

  • The study compares four continuous and discrete regression and flow heads across single-task and multi-task VLA settings, simulation, and real-world experiments.It evaluates closed-loop success, action prediction accuracy, and within-chunk temporal variation.Source: §1.1 · pp. 2–3
  • Discrete heads, including simple discrete regression, can show greater control precision than continuous counterparts despite potentially lossy quantization.The authors attribute this to greater reactivity to both action-chunk time and conditioning observations.Source: §1.1 · p. 3
  • Iterative discrete flow and classifier-free guidance mitigate factorization error from independent token predictions, improving control.Independently predicted tokens can combine incompatible choices across time and action dimensions.Source: §1.1 · p. 3
  • Explicit geometric encoding and alternative tokenizers provide limited additional gains; none exceeds uniform-bin DF with the standard mixture path.Adjacent-bin output embeddings are similar, suggesting discrete heads can learn local geometry without explicitly encoding it.Source: §1.1 · p. 3

2 Preliminaries: Action Representations and Prediction Heads

The paper represents action chunks with either continuous outputs or quantized tokens, then compares one-pass regression heads with flow-based generation. Discrete flow iteratively updates tokens, while classifier-free guidance combines conditional and unconditional predictions.

2 Preliminaries: Action Representations and Prediction Heads

  • An action chunk contains H actions with M coordinates per timestep; the policies use relative end-effector actions, including translation, rotation, and gripper coordinates.The chunk is conditioned on policy inputs such as images, language, and proprioception.Source: §2 · p. 3
  • Continuous regression predicts a chunk in one pass by squared-error regression, whereas continuous flow generates it by integrating a learned velocity field.The regression population minimizer is the conditional mean, which need not be a demonstrated action when multiple behaviors are plausible.Source: §2 · p. 4
  • Uniform quantization maps each normalized and clipped action coordinate to one of B bins, flattens the chunk into tokens, and decodes tokens to bin centers.Unless stated otherwise, the vocabulary size is B = 512.Source: §2 · p. 4
  • Discrete regression predicts quantized tokens independently by position and decodes their greedy choices, while discrete flow iteratively updates a token chunk conditioned on its current state and flow time.The multi-stage computation introduces position-wise correlation in the final tokens.Source: §2 · pp. 4–5
  • Classifier-free guidance combines conditional and unconditional predictions for discrete heads and applies the combination during each refinement or integration step for flow heads.The guidance weight controls the combination; continuous regression is not guided because its single output chunk is not a distribution to sharpen.Source: §2 · p. 5

3 Discrete Action Prediction Can Match Continuous Parametrizations

Across the evaluated settings, discrete action prediction matches or often outperforms continuous prediction. Its advantage is associated with stronger temporal variation and responsiveness to changing observations, rather than consistently lower reconstruction error.

3 Discrete Action Prediction Can Match Continuous Parametrizations

  • The evaluation spans from-scratch RoboMimic tasks, multi-task VLA simulation tasks, and real-world tasks, using closed-loop success and held-out action prediction error.Tool Hang and Transport are identified as tasks demanding high control precision.Source: §3 · pp. 5–6
  • Discrete flow is competitive with or often slightly surpasses continuous flow, while discrete regression consistently outperforms continuous regression across VLA and from-scratch settings.Discrete policies also remain competitive in end-effector reconstruction error; DF has lower error early in the action chunk and in finer-motion regimes.Source: §3 · p. 6, Figure 2 · p. 7

3.1 What Enables the Performance of Discrete Policies?

  • Across parametrizations, reconstruction errors are roughly similar and show no consistent correlation with task success; DR can have higher or lower error than CR while achieving greater success.This argues against reconstruction error alone accounting for discrete policies’ control performance.Source: §3.1 · p. 6
  • Continuous regression has a jerk ratio substantially below 1, while discrete regression is near or sometimes above 1; the ratio measures predicted-to-target temporal variation.The passage suggests discrete policies represent larger within-chunk variation more readily in a single pass, with the difference shrinking for flow heads.Source: §3.1 · pp. 6–7
  • Discrete policies better fit subtle small-motion variations, and DF has the lowest RoboCasa end-effector error in the chunk’s first steps.Continuous heads underestimate expert chunk changes; DR tracks expert responsiveness most closely, while DF is closer than CF at every tested state displacement.Source: §3.1 · pp. 7–8, Figure 2 · p. 7, Figure 3 · p. 7

4 Factorization Error and Iterative Computation in Discrete Policies

Discrete regression’s reactivity comes with factorization error: independent token choices can form inconsistent action chunks. Guidance and iterative discrete-flow refinement reduce this error, bringing guided DR close to DF performance.

4 Factorization Error and Iterative Computation in Discrete Policies

  • Factorization error occurs when independently predicted token choices combine incompatible alternatives across time or action dimensions, even if each token marginal is learned correctly.The product of token marginals need not recover the demonstrated joint distribution of action chunks.Source: §4 · p. 8
  • In Push-T, changing relative probabilities of separated movement modes across the prediction horizon illustrates how factorized decoding introduces spurious temporal variation.The examples illustrate the quantitative finding that DR has more mode-mixture than DF, especially at zero CFG.Source: §4 · pp. 9–10
  • CFG raises RoboCasa success from 27.1% to 49.2%, while DF cuts inconsistent-mode predictions 5.28-fold versus DR and adds 4.7 success points.DF brings the average below one inconsistent-mode prediction per chunk at NFE ≥10.Source: §4 · p. 10
  • DF retains greater action diversity than DR, but performance gains do not track diversity, suggesting preserving full action diversity is unnecessary for task success.Performance improves up to NFE 10 while action diversity remains relatively constant.Source: §4 · p. 10
  • Temporal autoregression reaches 42.8% RoboCasa success without CFG, exceeding DR’s 27.1% but remaining below DF.The AR policy predicts each subsequent action conditioned on the observation and previously generated actions.Source: §4 · p. 10
  • Factorization error helps explain DR’s relative performance gap; CFG sharpening and DF’s iterative computation are both effective ways to address it.The paper characterizes factorization error as temporal inconsistency and reports that guided DR approaches DF performance.Source: §4 · p. 11

5 Additional Structure Provides Limited Gains for Discrete Policies

The section tests whether adding action geometry or changing tokenization improves discrete policies beyond uniform-bin categorical prediction. These interventions provide limited gains and do not surpass the best flat-bin DF result.

5 Additional Structure Provides Limited Gains for Discrete Policies

  • Metric-path DF reaches 52.8% pooled RoboCasa success, below the 53.9% for mixture-path DF; tested explicit geometry adds 2.1 points for DR.Label smoothing changes DR’s targets, whereas the metric path changes DF’s corrupted inputs.Source: §5 · p. 11
  • At their best recorded complete operating points, DF reaches 52.4% RoboCasa success with nFSQ, 46.2% with FAST, and 38.2% with RVQ.None exceeds the broader campaign’s best flat-bin DF result of 53.9%.Source: §5 · p. 11
  • Quantile bins and nFSQ improve unguided DR over flat-bin DR, but their gains are limited against guided flat-bin DR and do not raise best recorded performance.Quantile bins and nFSQ reach 49.8% and 41.6%, respectively, versus 27.1% for unguided flat-bin DR; guided flat-bin DR reaches 49.2%.Source: §5 · p. 11
  • Adjacent action-bin embeddings have high cosine similarity, suggesting discrete policies can learn local geometry and possibly explaining why imposing it explicitly adds little.The paper presents learned local geometry as a possible explanation for the limited gains from explicit structure.Source: §5 · pp. 11–12

6 Discussion and Future Work

The discussion argues that discrete action prediction can support precise robot control, while identifying coordination among token choices as a central challenge. It points to iterative refinement and guidance as ways to address that challenge, and proposes testing shared token-based action generation across modalities.

6 Discussion and Future Work

  • Simple action tokens can support precise control in simulation and real-world tasks, making discrete prediction an effective approach to robot control.The discussion frames this as challenging the distinction between robot actions and other AI modalities.Source: §6 · p. 12
  • Independent token predictions can combine incompatible behaviors; iterative refinement and guidance help resolve conflicts while retaining responsiveness.The discussion connects this coordination challenge to diffusion language models, where token predictions must form a coherent joint output.Source: §6 · p. 12
  • The tested reconstruction-based tokenizers do not improve on uniform-bin DF, motivating tokenizations designed to predict coupled action choices.Their value may depend on the prediction head and the number of generation steps available; evaluation should include consistency and control performance alongside reconstruction quality.Source: §6 · p. 12
  • The authors propose testing whether shared representations and prediction machinery across modalities improve co-training and transfer while preserving action consistency for closed-loop control.They describe this as a natural next step for learning action generation within a token-based framework.Source: §6 · p. 12

AI Use Statement

  • The authors report using generative AI tools for implementation, analysis, figures, and editing, but not for hypothesis formation or experiment design.They state that they reviewed the AI-assisted work and take responsibility for the final paper content.Source: AI Use Statement · p. 12

A Related Work

  • Prior head comparisons often changed representation together with decoding or architecture, motivating comparisons that vary representation and generation computation independently.The related-work discussion notes mixed results across existing studies and describes the paper’s controlled comparison under one backbone and training recipe.Source: A Related Work · p. 28

B Environment Details; B.1 Evaluation Measures and Comparison Protocol

  • The study spans pretrained VLAs on RoboCasa and LIBERO, offline diagnostics on DROID, and from-scratch policies on RoboMimic and Push-T, using prediction error, temporal variation, and closed-loop success measures.The evaluation protocol distinguishes training fit from held-out error; robot translation errors are measured in millimeters and Push-T errors in pixels.Source: §B.1 · p. 30

B.2 From-Scratch Training Experiments

  • The single-task RoboMimic policies are state-based and trained from scratch without a pretrained vision-language backbone.They are evaluated on the tasks described in Appendix D.1.Source: §B.2 · p. 30

B.3 VLA-Scale Experiments

The appendices specify the VLA-scale evaluation, shared data mixture, action-head objectives, and inference procedures. They also describe the architectures and training settings used for both VLA-scale and from-scratch policies.

B.3 VLA-Scale Experiments

  • VLA-scale heads share a PaliGemma backbone, 24-action horizon, quantile normalization statistics, and seed; each head is evaluated on five checkpoints and five-seed closed-loop episodes.The reported success rate is the mean over seeds, and the best of the five checkpoints is selected.Source: §B.3 · p. 31

B.4 VLA Training Data Mixture

  • The VLA training mixture combines six RoboCasa365 datasets, LIBERO-plus, and DROID, with source sampling weighted by the square root of filtered frame counts.MimicGen’s frame count is capped at DROID’s for computing its sampling weight; 2% of each source’s episodes are held out for offline diagnostics.Source: §B.4 · p. 31

C.1 Continuous Heads

  • Continuous regression predicts actions in one pass, while continuous flow learns a velocity field and generates actions through Euler updates from Gaussian noise.The supplied passages describe the regression population minimizer as the conditional mean, which need not be a demonstrated action when multiple actions are plausible.Source: §C.1 · p. 32

C.2 Scalar Action Quantization

  • The scalar codec quantizes each normalized action coordinate into bins and decodes bin centers, with 512 bins per coordinate unless stated otherwise.The bin width is determined by each coordinate’s quantization range, and decoding restores the action-chunk shape before normalization is inverted for execution.Source: §C.2 · pp. 32–33

C.3 Discrete Training Objectives

  • Discrete regression uses tokenwise cross-entropy and per-position argmax, while discrete flow predicts clean tokens from corrupted chunks using clean-token cross-entropy.On RoboCasa, DR uses distance-kernel soft targets; on Push-T, it uses hard-label cross-entropy.Source: §C.3 · p. 33

C.4 Discrete-Flow Source Distribution and Sampler

  • Discrete flow starts from a factorized source, such as uniform bins or a mask, and iteratively updates tokens until the final chunk contains no mask tokens.With a uniform source, previously updated tokens may be revised, and subsequent predictions condition on the updated chunk, allowing dependencies across positions to emerge through staged computation.Source: §C.4 · p. 34

C.5 Classifier-Free Guidance

  • Classifier-free guidance combines conditional and null-condition predictions to reweight discrete token distributions or adjust continuous-flow velocities.For discrete flow, guidance is applied at each refinement step; it adds an action-head pass per evaluation, while the backbone is encoded once.Source: §C.5 · pp. 34–35

C.6 Omitted Details for Policy Architecture; C.6.1 Single-Task From-Scratch Architectures

  • The VLA action head is a separate transformer over 24-step chunks, with real-valued elements for continuous heads and binned tokens for discrete heads.For from-scratch RoboMimic, policies predict 10 actions and execute eight before replanning; discrete heads use 512 bins per coordinate.Source: §C.6.1 · p. 35

C.6.2 VLA-Scale Architectures

The VLA-scale policies use a fine-tuned PaliGemma backbone and a separate action transformer that processes action coordinates as a sequence. Continuous and discrete heads differ in whether those elements are real values or bin tokens.

C.6.2 VLA-Scale Architectures

  • The backbone is PaliGemma-3B initialized from π0 vision-language weights and fine-tuned; a 64-query resampler combines backbone states with proprioception into condition tokens.The backbone processes up to three camera images and 48 text tokens.Source: §C.6.2 · pp. 35–36
  • A separate transformer processes the 24-step action chunk as a sequence of seven action coordinates, represented as real values for continuous heads or tokens for discrete heads.The head uses self-attention over actions, cross-attention to condition tokens, and learned position and coordinate embeddings plus flow-time features.Source: §C.6.2 · p. 36
  • The reported DF policy uses uniform bins, a uniform source, and schedule α(t) = t^3; condition tokens are dropped with probability 0.1 to enable guidance at inference.The evaluation protocol and training settings are specified in the appendices.Source: §C.6.2 · p. 36

D.1 RoboMimic Results

Across the supplied appendix results, discrete and continuous flow heads perform strongly in both compared action spaces, while the held-out chunk diagnostics distinguish their temporal behavior and errors. Action parameterization also affects success, with absolute actions performing worst in the reported comparison.

D.1 RoboMimic Results

  • RoboMimic results report each task and head family’s best success across tokenizer, flow-variant, and guidance settings, using 32 evaluation episodes.The supplied passages also describe Figure 6 as training-sample EEF position error over each run’s final 10 evaluations.Source: §D.1 · p. 36, Figure 6 · p. 38

D.2 Action Parameterization and Action Space

  • Absolute actions give the lowest RoboCasa success for every head; the drop from relative to absolute actions is smaller for DR and DF than for CF and CR.The reported drops are 2.4 points for DR, 6.9 for DF, 13.7 for CF, and 21.4 for CR.Source: Table 6 · p. 38, §D.2 · p. 39
  • The action-space comparison selects CFG and, for flow heads, NFE separately for each head and action space.The appendix defines delta actions as per-step pose increments, relative actions from the prediction-time EEF pose, and absolute actions in the robot base frame.Source: §D.2 · pp. 38–39
  • Relative joint-space and end-effector-space actions give nearly the same success for every head, with a largest difference of 0.8 points for CF.DF and CF achieve the highest success in both spaces, DR remains competitive, and CR performs less well.Source: Table 7 · p. 39, §D.2 · p. 39

D.3 Backbone Variants

  • The backbone-variant comparison reports each head’s highest pooled success over checkpoints, guidance weights 0 or 2, and flow-head NFE values 5, 10, or 20.It includes a fine-tuned π0.5-initialized backbone and a frozen π0-initialized backbone.Source: §D.3 · p. 40, Table 8 · p. 40

E.1 Temporal Variation on Held-Out Chunks

  • DF’s EEF-error 95th percentile is below DR’s on seven held-out sources, with LIBERO-plus the exception.Across sources, median jerk ratios range from 0.294–1.313 for DR and 0.343–0.589 for DF; Figure 8 provides per-source views of the Figure 2 diagnostics.Source: §E.1 · p. 40, Figure 8 · p. 40

E.2 Prediction Error Under Varying CFG

Guidance can raise DF’s action-prediction error even as it improves control; responsiveness and disturbance stability also differ across policy heads. The diagnostics examine how predictions change with observations and small simulator disturbances, alongside ambiguity under iterative refinement.

E.2 Prediction Error Under Varying CFG

  • DF’s pooled error is lowest at λ = 0.5 (14.86 mm), then rises to 17.97 mm at λ = 3, passing DR at about λ = 2.5.Unguided DF has 15.16 mm error versus DR’s 18.06 mm; DR stays nearly flat at 17.44–17.53 mm for λ ≥1.Source: §E.2 · p. 41, Figure 9 · p. 41

E.3.1 Disturbance Stability

  • At ε = 0.1, disturbance excess is 19% of the sampling floor for DF and 6% for CF, versus 213% for DR.Flow heads have the highest disturbance stability; DF proposals are broader than CF’s (σ = 1.217 versus 0.764).Source: §E.3.1 · p. 42, Table 9 · p. 42

E.3.2 State Responsiveness

  • At lag 4, responsiveness is 0.94 for DR, 0.70 for DF, 0.63 for CF, and 0.58 for CR; both flow heads under-respond to the expert.DR tracks expert response most closely across lags, while CF is more under-responsive than DF; the passage distinguishes responsiveness from disturbance stability.Source: §E.3.2 · p. 42, Table 10 · p. 43

E.4.1 Measuring a Distribution’s Response to Disturbance

  • The disturbance test intervenes on the simulator because adding noise only to proprioceptive input produced no measurable response.It uses 144 snapshots from DF rollouts across 18 RoboCasa atomic tasks, and queries heads at CFG 0 (NFE 10 for flows).Source: §E.4.1 · pp. 43–44
  • The disturbance is small: it changes normalized proprioception by about 0.003–0.021, below the reference data’s median neighbor spacing of 0.32.These diagnostics use CFG 0 and NFE 10, whereas reported control success uses separately selected settings.Source: §E.4.1 · p. 44

F.1 Experimental Methodology

  • DF’s mean ambiguous-token count falls from 4.94 at NFE 2 to 1.50, 0.57, and 0.22 at NFE 5, 10, and 20, compared with DR’s 3.01.The sweep changes both refinement steps and the final prediction time; ambiguity is detected on each final conditional token prediction before averaging DF draws.Source: §F.1 · p. 44, Figure 10 · p. 45

F.2 Factorization Error and L2 Loss Diagnostic Results

Across six sources, DF has lower action-prediction error than CF, while DR and CR lead on different individual sources. Decoding-budget effects depend on guidance: coarse decoding leaves unguided flows relatively stable but sharply degrades guided CF.

F.2 Factorization Error and L2 Loss Diagnostic Results

  • DF has the lowest L2 prediction error on four of six sources and lower error than CF on all six; DR leads on LIBERO-plus and CR on RoboCasa composite-seen.On LIBERO-plus, DR’s error is 2.53 mm against CR’s 9.88 mm.Source: §F.2 · p. 45, Table 11 · p. 45

F.3.1 Task Performance Under Varying NFE

  • Without guidance, both flow heads lose about five success-rate points at NFE 2, then change little from NFE 5 to 100.At NFE 2, DF reaches 47.0 ± 2.5% and CF 45.6 ± 1.3%; with λ = 2, DF ranges from 52.3% to 53.6% across budgets.Source: §F.3.1 · p. 45
  • Guided CF falls to 0.3 ± 0.3% success at NFE 2 and 41.9 ± 1.7% at NFE 5, recovering to 52.4 ± 2.4% at NFE 10.The reported task-performance comparisons use fixed checkpoints while varying decoding steps and guidance.Source: §F.3.1 · p. 45, §F.3 · p. 45
  • From NFE 5 to 20, DF’s translation-token entropy decreases while across-draw action diversity persists.The figure uses 4,000 validation observations and 16 draws per observation; diversity is RMS pairwise translation-command distance.Source: Figure 12 · p. 46

F.3.2 Task Performance Under Varying CFG

Guidance has a much larger effect on DR than on flow heads, while the best setting depends on the head and decoding budget. Alternative tokenizers and metric-path variants do not surpass the main uniform-bin mixture-path DF result; FAST DR also fails validity checks.

F.3.2 Task Performance Under Varying CFG

  • DR rises from 27.1 ± 1.2% success without guidance to 48.9 ± 3.6% at λ = 2, with its highest mean, 50.8 ± 2.8%, at λ = 3.5.Moderate guidance adds only 1.1 points for DF and 0.9 for CF; strong guidance at λ = 4 lowers CF to 43.1 ± 1.6% and DF to 49.8 ± 1.4%.Source: §F.3.2 · pp. 46–47, Figure 13 · p. 47

F.4 Visualization of Mode Inconsistency in RoboCasa

  • The RoboCasa example depicts two action modes—move consistently or move then hold—and illustrates how independent token sampling can switch between motions across timesteps.The move thumbnails come from steps 12 and 24 of a DR-argmax replay; the hold thumbnail repeats the initial image.Source: Figure 14 · p. 47, §F.4 · p. 47

G.1 Performance Under Different Tokenization

  • The nFSQ result is 52.4%, slightly above the frequency-bin metric-path result of 51.8%, but neither exceeds the main mixture-path DF result of 53.9%.The 53.9% comparison uses a different recipe and horizon.Source: §G.1 · p. 48, Table 12 · p. 48
  • FAST DR reaches only 25–40% valid decodes during training and ends at 0%, so it has no closed-loop evaluation; FAST AR reaches 48.2% and FAST DF 46.2%.Flat scalar bins do not have this sequence-validity failure because every token sequence decodes.Source: §G.1 · p. 48
  • With frequency bins, DR success is nearly unchanged by guidance at 49.8% versus 49.0%, and unguided DR is well above flat-bin DR’s 27.1%.The authors offer no concrete explanation, but suggest action encoding matters more when inference computation is limited.Source: §G.1 · p. 48

G.2.1 From a Mixture Path to a Metric-Induced Path

Metric-induced corruption concentrates errors near the clean action bin, unlike the uniform-background mixture path, but explicit-geometry variants do not consistently improve success. The section also reports learned local bin ordering in discrete output representations.

G.2.1 From a Mixture Path to a Metric-Induced Path

  • The mixture path raises the clean-token probability while leaving every incorrect bin equally likely; the Gibbs metric path instead favors nearby errors as β increases.The path and its clock are separate choices: the clock controls how quickly the model moves through the kernel family.Source: §G.2.1 · p. 49
  • At matched expected bin error, the mixture path can be often exactly correct but spread remaining errors across the vocabulary, whereas the metric path allocates more errors to nearby bins.Thus the denoising problem changes even when average corruption magnitude is held constant.Source: §G.2.1 · p. 49, Figure 15 · p. 49
  • Training retains clean-token cross-entropy, while geometry enters through the corruption path and decoding uses kinetic-optimal discrete flow matching rates.The decoder gives positive rates only to moves closer to the sampled clean endpoint and caps each token’s rate-vector norm at 50.Source: §G.2.1 · pp. 49–50

G.2.2 Evaluated Metric-Path Variants

  • The bin-progress clock with p = 1 reaches 52.8% pooled success, while the entropy-progress clock with p = 2 reaches 4.6%; the former remains below mixture-path DF’s 53.9%.The equal-frequency bin-progress variant reaches 51.8%, and the effect of explicit geometry depends strongly on path construction.Source: §G.2.2 · p. 50

G.3 Bin Representation and Input-Sensitivity Study

  • Discrete output embeddings show local value ordering: output adjacency is 0.3724 for DR and 0.3953 for DF, with embedding-distance/value-distance correlations of 0.5835 and 0.5579.The analysis supports local order, while DF input distances saturate at larger separations.Source: §G.3 · pp. 50–51, Figure 16 · p. 50
  • Input-token interventions change decoded outputs: at DF time 0.75, offsets of 1, 16, and 64 bins change normalized output by 0.0093, 0.1368, and 0.5121.DR’s total-variation response remains about 0.0125–0.0141 across offsets of 1–256 bins.Source: Figure 18 · p. 51, §G.3 · p. 52

G.3.1 Combined Output-Representation Visualization

  • Output biases track occupied-bin frequency, with Pearson r = 0.809 for DR and 0.720 for DF.The output PCA projections also organize bins by value, though their first two components explain about 11% and 9% of weight variance.Source: Figure 17 · p. 51, §G.3.1 · p. 52

H.1 Which Fine Structure Does Categorical Prediction Preserve?

Synthetic targets with controlled fine structure reveal that which policy performs best depends on the target: continuous regression excels on smooth detail, while categorical prediction excels on steep detail.

H.1 Which Fine Structure Does Categorical Prediction Preserve?

  • The controlled targets combine a broad sine with either a fast-sine fine term or a square wave, allowing fine structure to vary independently.Robot data mixes too many factors to isolate this question.Source: §H.1 · p. 52
  • On the smooth target, CR’s held-out error is about 250 times lower than DR’s; both continuous heads fall below the hard-bin codec floor, unlike discrete heads.The codec floor limits argmax-decoded outputs but not continuous outputs, so bin resolution alone does not explain the discrete heads’ larger error.Source: §H.1 · pp. 52–53, Figure 19 · p. 52
  • On the square-wave target, DR has 3.2 times lower MSE and 4.5 times lower MAE than CR; DF has 31 percent lower error than CF.Categorical predictions jump at the edge, while regression rounds off the square-wave corners.Source: Figure 19 · p. 52, §H.1 · p. 53
  • Quantized Regression behaves like CR on both targets, while CF tracks CR and DF trails DR; the reversal tracks categorical formulation, with argmax a likely contributor.The authors suggest that smoothly changing logits can produce decoded-bin jumps, representing sharp transitions.Source: §H.1 · p. 53

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