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arxiv: 2602.16169 · v2 · pith:FJRF2FEFnew · submitted 2026-02-18 · 💻 cs.LG · cs.CL

Discrete Stochastic Localization for Non-autoregressive Generation

Yunshu Wu , Jiayi Cheng , Longxuan Yu , Partha Thakuria , Rob Brekelmans , Evangelos E. Papalexakis , Greg Ver Steeg This is my paper

Pith reviewed 2026-05-22 11:32 UTC · model grok-4.3

classification 💻 cs.LG cs.CL
keywords discrete diffusionnon-autoregressive generationstochastic localizationmasked diffusion modelstext generationsignal-to-noise ratio invarianceunit sphere embeddings
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The pith

A single trained network supports many per-token noise schedules for non-autoregressive discrete generation by making the denoiser invariant to nominal SNR.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies the dependence of denoising on timestep-specific noise levels as the main reason continuous diffusion trails masked discrete models on sequences. It introduces Discrete Stochastic Localization, which places token embeddings on the unit sphere so that the Bayes-optimal denoiser no longer varies with the chosen signal-to-noise ratio. Because the same network therefore works for any collection of per-token SNR paths, masked diffusion emerges as one endpoint of the family. Fine-tuning a pretrained masked model with this framework raises MAUVE scores on OpenWebText for every step budget tested and also unlocks random-order autoregressive and hybrid sampling without further training.

Core claim

Discrete Stochastic Localization embeds discrete tokens as continuous points on the unit sphere and defines a localization channel under which the Bayes-optimal denoiser is invariant to the nominal signal-to-noise ratio. As a direct result, one network parameterizes an entire family of valid per-token SNR trajectories, with the standard masked-diffusion trajectory recovered exactly when the schedule reaches its endpoint.

What carries the argument

Unit-sphere token embeddings under the localization channel, which enforce invariance of the Bayes-optimal denoiser to nominal SNR and thereby decouple the trained network from any single noise schedule.

If this is right

  • One network supports an arbitrary family of per-token SNR paths rather than a single fixed schedule.
  • Masked diffusion appears as the endpoint case of the same family.
  • Fine-tuning raises MAUVE on OpenWebText for every tested step budget from 128 to 1024.
  • The same checkpoint enables random-order autoregressive sampling and hybrid continuous-then-discrete sampling at T=48 steps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The SNR invariance may simplify training pipelines for other discrete domains such as protein sequences or source code.
  • Hybrid schedules could be optimized on the fly for different quality-speed trade-offs without retraining.
  • The continuous embedding view may allow direct transfer of continuous-diffusion techniques like guidance to discrete settings.

Load-bearing premise

The Bayes-optimal denoiser is invariant to the nominal signal-to-noise ratio under the localization channel.

What would settle it

An experiment in which the learned denoiser produces measurably different outputs for identical inputs under two different nominal SNR values would directly contradict the claimed invariance.

Figures

Figures reproduced from arXiv: 2602.16169 by Evangelos E. Papalexakis, Greg Ver Steeg, Jiayi Cheng, Longxuan Yu, Partha Thakuria, Rob Brekelmans, Yunshu Wu.

Figure 1
Figure 1. Figure 1: Discrete Stochastic Localization (DSL). A single SNR-invariant denoiser supports arbitrary per-token SNR paths, including remasking-induced “backtracking”, motivating mixed￾corruption training to better match refinement-time drafts and improve step-efficiency. A recurring challenge in NAR generation is the training– sampling mismatch. Models are trained to predict tokens under partially corrupted ground-tr… view at source ↗
Figure 2
Figure 2. Figure 2: Sampling diagnostics under a fixed step budget. (a) Masking and reveal schedule. (b) Remasking intensity and realized rewrites per token. (c) Posterior sharpening measured by mean max-probability and top-p nucleus size. _ MASK A _ MASK B C OK C D OK D B WRONG E F OK F D WRONG G pos 1 2 3 4 5 6 7 init true init=ABCDBFD true=ABCDEFG t=0 t=2 t=4 t=6 t=8 t=9 t=10 A A C D B F D A A A D A F D A A A D A F G A A A… view at source ↗
Figure 3
Figure 3. Figure 3: ReMDM-style discrete correction on a cyclic toy. (a) We corrupt the ground-truth sequence ABCDEFG by masking two positions and inserting two visible-but-wrong tokens, yielding CDBFF. (b) Confidence-driven remasking enables subsequent steps to rewrite low-confidence visible tokens; the refinement trajectory corrects both masked and wrong tokens and can recover ABCDEFG within 10 refinement steps in this exam… view at source ↗
Figure 4
Figure 4. Figure 4: Endpoint smoothing improves near-clean calibration. We compare atomic ROAR endpoints (γ∈ {0, γmax}) to smoothed endpoint ranges (γ ∼ Unif(0, γmin) or γ ∼ Unif(cγmax, γmax)). On held-out data, we measure calibration under teacher forcing on corrupted inputs; smoothing reduces ECE at large SNR and yields reliability closer to the diagonal at SNR=100. 128 256 512 1024 Steps (T) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 MAU… view at source ↗
Figure 5
Figure 5. Figure 5: Endpoint smoothing improves the step–quality trade￾off under fixed decoding. Using the same ReMDM-style sam￾pler with principled η-cap (eta cap on; no uncertainty-guided remasking) and identical schedules, smoothed-endpoint check￾points achieve higher MAUVE across step budgets (and compara￾ble/better GenPPL), while atomic endpoints show weak gains as T increases. 7.5. Ablation: smoothing ROAR endpoints imp… view at source ↗
Figure 6
Figure 6. Figure 6: Log-normal Distribution Choice. Optimization and batching. We train for a maximum of 100,000 optimizer steps with no learning-rate warmup (num warmup steps=0). Everything else in training setting is the same as MDLM training setting. SNR sampling distribution (mixed ROAR/lognormal). We denote by γ the signal-to-noise ratio (SNR) used in the training corruption process. We use a mixed SNR path: with probabi… view at source ↗
read the original abstract

Continuous diffusion is a natural framework for non-autoregressive generation but has generally lagged behind masked discrete diffusion models (MDMs) on discrete sequence generation. We argue that the bottleneck is not continuity itself, but a representation in which denoising depends on timestep-indexed noise regimes. We introduce \emph{Discrete Stochastic Localization} (DSL), a continuous-state framework with unit-sphere token embeddings whose Bayes-optimal denoiser is invariant to the nominal signal-to-noise ratio (SNR) under the localization channel. One trained network then supports an entire family of per-token SNR paths, with endpoint masked-diffusion paths as a special case. Fine-tuning a pretrained MDLM checkpoint with DSL substantially improves distributional faithfulness (MAUVE) on OpenWebText across all step budgets from $T{=}128$ to $T{=}1024$, and the same checkpoint supports random-order autoregressive sampling, as well as a hybrid continuous-then-discrete sampler using as few as T=48 total steps -- without distillation or retraining.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Discrete Stochastic Localization (DSL), a continuous-state framework for non-autoregressive discrete sequence generation that employs unit-sphere token embeddings. It claims that the Bayes-optimal denoiser under the localization channel is invariant to nominal signal-to-noise ratio (SNR), so that a single trained network can support an arbitrary family of per-token SNR paths (with masked-diffusion paths as a special case). Fine-tuning a pretrained Masked Discrete Language Model (MDLM) checkpoint with DSL is reported to improve distributional faithfulness (MAUVE) on OpenWebText across step budgets T=128 to T=1024 and to enable random-order autoregressive sampling plus hybrid continuous-then-discrete sampling with as few as T=48 steps, all without distillation or retraining.

Significance. If the invariance property transfers from the Bayes-optimal denoiser to a trained network and the reported MAUVE gains prove robust, the work would offer a practical unification of continuous diffusion and masked discrete models, reducing the need for schedule-specific retraining and enabling flexible sampling strategies from one checkpoint. The parameter-free character of the invariance (when it holds) and the reuse of a single network across paths are genuine strengths that could influence future non-autoregressive generation research.

major comments (2)
  1. [§3 (DSL definition and invariance derivation)] The central claim that one trained network supports the entire family of SNR paths rests on the Bayes-optimal denoiser being invariant to nominal SNR under the localization channel with unit-sphere embeddings. The manuscript derives this invariance for the optimal denoiser but provides no explicit verification (e.g., via consistency checks or ablation across SNR schedules) that the learned neural-network approximation preserves the same invariance for fixed embeddings. If approximation error is SNR-dependent, the single-network property fails for unseen paths; this is load-bearing for the practical contribution.
  2. [§5 (Experiments)] Table 1 (or equivalent experimental table) and the accompanying text report MAUVE improvements after DSL fine-tuning but supply no error bars, number of runs, or ablation isolating the effect of the localization channel versus other fine-tuning choices. Without these controls it is impossible to assess whether the gains are statistically reliable or merely reflect training variance.
minor comments (2)
  1. [Abstract] The abstract states improvements “across all step budgets from T=128 to T=1024” but does not list the exact baselines (standard MDLM, continuous diffusion, etc.) or the precise MAUVE values; these should be added for reproducibility.
  2. [§2–3] Notation for the localization channel and the precise mapping from unit-sphere embeddings to the forward process should be introduced earlier and used consistently; current presentation leaves the channel definition somewhat implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major comment below and are prepared to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The central claim that one trained network supports the entire family of SNR paths rests on the Bayes-optimal denoiser being invariant to nominal SNR under the localization channel with unit-sphere embeddings. The manuscript derives this invariance for the optimal denoiser but provides no explicit verification (e.g., via consistency checks or ablation across SNR schedules) that the learned neural-network approximation preserves the same invariance for fixed embeddings. If approximation error is SNR-dependent, the single-network property fails for unseen paths; this is load-bearing for the practical contribution.

    Authors: We agree that an explicit empirical verification of SNR-invariance for the trained network would strengthen the central claim. The manuscript derives the invariance rigorously for the Bayes-optimal denoiser under the localization channel and unit-sphere embeddings, and the reported experiments already demonstrate that a single fine-tuned checkpoint supports multiple distinct sampling paths (including masked-diffusion endpoints, random-order autoregressive sampling, and hybrid continuous-discrete sampling) without retraining. Nevertheless, we did not include dedicated consistency checks or ablations that directly test whether approximation error remains independent of nominal SNR. In the revised manuscript we will add such verification, for example by evaluating the same trained model on several held-out SNR schedules and reporting generation metrics to confirm practical invariance. revision: yes

  2. Referee: Table 1 (or equivalent experimental table) and the accompanying text report MAUVE improvements after DSL fine-tuning but supply no error bars, number of runs, or ablation isolating the effect of the localization channel versus other fine-tuning choices. Without these controls it is impossible to assess whether the gains are statistically reliable or merely reflect training variance.

    Authors: We acknowledge that the current experimental reporting lacks error bars, the number of independent runs, and an explicit ablation isolating the localization channel. The manuscript reports MAUVE gains across step budgets T=128 to T=1024 after DSL fine-tuning of a pretrained MDLM checkpoint, but does not quantify run-to-run variance or compare against standard fine-tuning without the localization objective. In the revised version we will include standard deviations from multiple runs and add an ablation that compares DSL fine-tuning against conventional fine-tuning of the same checkpoint to better isolate the contribution of the localization channel. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces DSL as a continuous-state framework with unit-sphere embeddings and states that its Bayes-optimal denoiser is invariant to nominal SNR under the localization channel. This invariance is presented as a direct mathematical property of the defined channel, enabling the single-network support for multiple SNR paths (including masked-diffusion endpoints) as a consequence rather than an independent prediction. The central empirical results—MAUVE improvements on OpenWebText after fine-tuning a pretrained MDLM checkpoint across T=128 to T=1024, plus support for random-order AR and hybrid sampling—are separate validations that do not reduce to the invariance claim by construction. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the provided claims. The derivation chain remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the invariance of the Bayes-optimal denoiser under the newly defined localization channel; this is treated as a domain assumption rather than a derived result.

axioms (1)
  • domain assumption The Bayes-optimal denoiser is invariant to the nominal signal-to-noise ratio (SNR) under the localization channel
    Invoked to justify that one network supports an entire family of SNR paths.
invented entities (1)
  • Discrete Stochastic Localization channel with unit-sphere token embeddings no independent evidence
    purpose: To achieve SNR invariance for the denoiser
    Newly introduced construction that enables the multi-path property.

pith-pipeline@v0.9.0 · 5727 in / 1316 out tokens · 34147 ms · 2026-05-22T11:32:01.808085+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 7 internal anchors

  1. [1]

    Block Diffusion: Interpolating Between Autoregressive and Diffusion Language Models

    Arriola, M., Gokaslan, A., Chiu, J. T., Yang, Z., Qi, Z., Han, J., Sahoo, S. S., and Kuleshov, V . Block diffusion: Inter- polating between autoregressive and diffusion language models.arXiv preprint arXiv:2503.09573,

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Nearly d-linear convergence bounds for diffu- sion models via stochastic localization.arXiv preprint arXiv:2308.03686,

    Benton, J., De Bortoli, V ., Doucet, A., and Deligianni- dis, G. Nearly d-linear convergence bounds for diffu- sion models via stochastic localization.arXiv preprint arXiv:2308.03686,

    work page arXiv

  3. [3]

    M., Du, Y ., Simchowitz, M., Tedrake, R., and Sitzmann, V

    Chen, B., Monso, D. M., Du, Y ., Simchowitz, M., Tedrake, R., and Sitzmann, V . Diffusion forcing: Next-token pre- diction meets full-sequence diffusion.arXiv preprint arXiv:2407.01392,

    work page arXiv

  4. [4]

    Continuous diffusion for categorical data

    Dieleman, S., Sartran, L., Roshannai, A., Savinov, N., Ganin, Y ., Richemond, P. H., Doucet, A., Strudel, R., Dyer, C., Durkan, C., et al. Continuous diffusion for categorical data.arXiv preprint arXiv:2211.15089,

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    Mask-predict: Parallel decoding of conditional masked language models.arXiv preprint arXiv:1904.09324,

    Ghazvininejad, M., Levy, O., Liu, Y ., and Zettlemoyer, L. Mask-predict: Parallel decoding of conditional masked language models.arXiv preprint arXiv:1904.09324,

    work page arXiv 1904

  6. [6]

    Non-Autoregressive Neural Machine Translation

    Gu, J., Bradbury, J., Xiong, C., Li, V . O., and Socher, R. Non-autoregressive neural machine translation.arXiv preprint arXiv:1711.02281,

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    Dependency Networks for Collaborative Filtering and Data Visualization

    Heckerman, D., Chickering, D. M., Meek, C., Rounthwaite, R., and Kadie, C. Dependency networks for collabo- rative filtering and data visualization.arXiv preprint arXiv:1301.3862,

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    Autoregressive diffusion models,

    Hoogeboom, E., Gritsenko, A. A., Bastings, J., Poole, B., Berg, R. v. d., and Salimans, T. Autoregressive diffusion models.arXiv preprint arXiv:2110.02037, 2021a. Hoogeboom, E., Nielsen, D., Jaini, P., Forr´e, P., and Welling, M. Argmax flows and multinomial diffusion: Learning categorical distributions.Advances in neural information processing systems, 3...

    work page arXiv

  9. [9]

    Z., Kim, H., Kakade, S., and Chen, S

    Kim, J., Kim, S., Lee, T., Pan, D. Z., Kim, H., Kakade, S., and Chen, S. Fine-tuning masked diffusion for provable self-correction.arXiv preprint arXiv:2510.01384,

    work page internal anchor Pith review arXiv

  10. [10]

    Information-theoretic diffusion

    URL https://arxiv. org/abs/2302.03792. Lee, J., Mansimov, E., and Cho, K. Deterministic non- autoregressive neural sequence modeling by iterative re- finement.arXiv preprint arXiv:1802.06901,

    work page arXiv

  11. [11]

    Lovelace, J., Kishore, V ., Chen, Y ., and Weinberger, K. Q. Diffusion guided language modeling.arXiv preprint arXiv:2408.04220,

    work page arXiv

  12. [12]

    Montanari, A

    Accessed: 2025-05-11. Montanari, A. Sampling, diffusions, and stochastic localiza- tion.arXiv preprint arXiv:2305.10690,

    work page arXiv 2025

  13. [13]

    Hopfield Networks is All You Need

    Ramsauer, H., Sch ¨afl, B., Lehner, J., Seidl, P., Widrich, M., Adler, T., Gruber, L., Holzleitner, M., Pavlovi´c, M., Sandve, G. K., et al. Hopfield networks is all you need. arXiv preprint arXiv:2008.02217,

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  14. [14]

    Anchored diffu- sion language model.arXiv preprint arXiv:2505.18456,

    Rout, L., Caramanis, C., and Shakkottai, S. Anchored diffu- sion language model.arXiv preprint arXiv:2505.18456,

    work page arXiv

  15. [15]

    Is noise condition- ing necessary for denoising generative models?arXiv preprint arXiv:2502.13129,

    Sun, Q., Jiang, Z., Zhao, H., and He, K. Is noise condition- ing necessary for denoising generative models?arXiv preprint arXiv:2502.13129,

    work page arXiv

  16. [16]

    Hart: Efficient visual generation with hybrid autoregressive transformer

    Tang, H., Wu, Y ., Yang, S., Xie, E., Chen, J., Chen, J., Zhang, Z., Cai, H., Lu, Y ., and Han, S. Hart: Efficient visual generation with hybrid autoregressive transformer. arXiv preprint arXiv:2410.10812,

    work page arXiv

  17. [17]

    BERT has a Mouth, and It Must Speak: BERT as a Markov Random Field Language Model

    Wang, A. and Cho, K. Bert has a mouth, and it must speak: Bert as a markov random field language model.arXiv preprint arXiv:1902.04094,

    work page internal anchor Pith review Pith/arXiv arXiv 1902

  18. [18]

    S., and Kuleshov, V

    Wang, G., Schiff, Y ., Sahoo, S. S., and Kuleshov, V . Re- masking discrete diffusion models with inference-time scaling.arXiv preprint arXiv:2503.00307,

    work page arXiv

  19. [19]

    Energy-based diffusion language models for text generation.arXiv preprint arXiv:2410.21357,

    Xu, M., Geffner, T., Kreis, K., Nie, W., Xu, Y ., Leskovec, J., Ermon, S., and Vahdat, A. Energy-based diffusion language models for text generation.arXiv preprint arXiv:2410.21357,

    work page arXiv

  20. [20]

    −logP(s) =−logP(x) = 1/2 Z ∞ 0 dtE z(t)|x[∥x− ˆx(z)∥2] Probability relates to MMSE, for any one-to-one embedding (Guo et al., 2005; Kong et al.,

    work page 2005

  21. [21]

    Table 2.Summary of notation and key relations. A.1. Optimal Denoiser is SNR invariant We now derive the optimal denoiser for the noise channel with per token SNR described in the main text. The denoiser is as follows, where we first re-write with Bayes rule, then expand the Gaussian noise channel. ˆx(z,γ)≡E pγ(x|z)[x] = P x pγ(z|x)P(x)x pγ(z) = P x pγ(z|x...

    work page 2025

  22. [22]

    However, the distributional error bound is derived from a practical perspective when the limit can never be achieved

    proves diffusion models and stochastic localization are equivalent under a time change, it is in the limit setting where T→ ∞ . However, the distributional error bound is derived from a practical perspective when the limit can never be achieved. Therefore, our analysis does not conflict with the result in (Montanari, 2023). A.6. Prior Mismatch Scaling vs....

    work page 2023

  23. [23]

    Rewrites-per-token

    Training is conducted in full precision (FP32). 3Our implementation usestorch.roll. 16 Discrete Stochastic Localization for Non-autoregressive Generation Figure 6.Log-normal Distribution Choice. Optimization and batching.We train for a maximum of 100,000 optimizer steps with no learning-rate warmup (num warmup steps=0). Everything else in training setting...

    work page 2006

  24. [24]

    Continuous diffusion baselines include Plaid (Gulrajani & Hashimoto, 2023), CDCD (Dieleman et al., 2022)

    BaselinesWe compare DSL against state-of-the-art continuous and discrete diffusion models, and autoregressive models (Vaswani et al., 2017). Continuous diffusion baselines include Plaid (Gulrajani & Hashimoto, 2023), CDCD (Dieleman et al., 2022). Discrete diffusion baselines include Discrete Diffusion Model (D3PM) (Austin et al., 2021), Score Entropy Disc...

    work page 2017

  25. [25]

    and MD4 (Shi et al., 2024). For autoregressive models, we choose Any-order Autoregressive Models ARDM (Hoogeboom et al., 2021a) and MAC (Shih et al., 2022), and flow-based methods IAF/SCF (Ziegler & Rush, 2019), AR Argmax Flow (Hoogeboom et al., 2021b), Discrete Flow (Tran et al., 2019), and Multinomial Diffusion (Hoogeboom et al., 2021b), according to th...

    work page 2024

  26. [26]

    You don’t have time for this

    BIP that only need the hard press. Now I think it works very well against your blindness. At RT11 Here’s how it works. I’ve adjusted the fast slider to go above the 60’s. Eventually, myshoot rate is now 4 percent. It works all on the T-Mobile bands. If I’m too low, I can cross it up the corner, but if it’s with a stick hand (Carl Rasmus could cut), I infr...

    work page 2014