Early-Bird Decoding: Accelerating Diffusion LLMs with Learnable Block Sizes and Parallel Sampling

# Early-Bird Decoding: Accelerating Diffusion LLMs with Learnable Block Sizes and Parallel Sampling
Source: [https://arxiv.org/html/2609.16450](https://arxiv.org/html/2609.16450)
Lixuan Wei††thanks:Equal contribution\. Work done while interning at Purdue EcoAI Lab\.Wei Zhou11footnotemark:1Affiliation:Georgia Institute of TechnologyJianwen WuAffiliation:Purdue UniversityYipeng ShenAffiliation:Purdue UniversityMeiling WangAffiliation:Purdue UniversityHaoran You††thanks:Corresponding author\. Email:[haoran@purdue\.edu](mailto:[email protected])Affiliation:Purdue University

###### Abstract

Diffusion large language models \(dLLMs\) offer a promising parallel decoding paradigm as an alternative to autoregressive generation through iterative unmasking\. However, dLLMs typically require many steps before token confidence reaches the decoding threshold, resulting in inefficient inference even with block\-wise KV caching\. To accelerate dLLM inference, wefor the first timepropose an “early\-bird \(EB\)” decoding framework, motivated by the observation that tokens with similarly low entropy tend to cluster and can be jointly decoded earlier, before reaching the confidence threshold\. In particular, our EB\-Decode framework integrates two key enablers: \(1\) a learnable network that adaptively groups tokens with similar uncertainty into variable\-length blocks, rather than relying on fixed block sizes; \(2\) a position\-aware sampler that learns to unmask tokens in parallel using fewer decoding steps within predicted variable\-length blocks\. Both components are developed without modifying pretrained dLLM weights and can therefore be directly deployed as plug\-ins during serving, with negligible training and inference overhead\. Extensive experiments across three models and four benchmarks consistently validate our observation and the effectiveness of EB\-Decode, achieving3\.53–18\.76×\\timeshigher throughput than the vanilla decoding method and up to1\.58×\\timeshigher throughput over the strongest baseline, Fast\-dLLM, with comparable accuracy\.

## 1Introduction

Diffusion large language models \(dLLMs\) have emerged as a compelling alternative to autoregressive \(AR\) models, breaking the sequential bottleneck of token\-by\-token decoding by generating text in parallel through iterative denoising over masked positions\[[2](https://arxiv.org/html/2609.16450#bib.bib2),[42](https://arxiv.org/html/2609.16450#bib.bib42),[40](https://arxiv.org/html/2609.16450#bib.bib40),[56](https://arxiv.org/html/2609.16450#bib.bib56)\]\. This paradigm enables bidirectional context utilization and makes dLLMs a practical and increasingly scalable approach for fast, high\-quality text generation, as demonstrated by large\-scale systems such as Gemini Diffusion\[[15](https://arxiv.org/html/2609.16450#bib.bib15)\], Seed\-Diffusion\[[45](https://arxiv.org/html/2609.16450#bib.bib45)\], and Mercury\[[25](https://arxiv.org/html/2609.16450#bib.bib25)\]\. However, current open\-source dLLMs such as LLaDA\[[40](https://arxiv.org/html/2609.16450#bib.bib40),[64](https://arxiv.org/html/2609.16450#bib.bib64),[5](https://arxiv.org/html/2609.16450#bib.bib5)\]and Dream\[[56](https://arxiv.org/html/2609.16450#bib.bib56)\]still struggle to consistently deliver better accuracy\-efficiency tradeoffs compared to their AR counterparts of similar size; for example, the AR modelLLaMA3\-8B\-Instructachieves 48\.0 tokens/s, whereas the dLLMLLaDA\-8B\-Instructreaches only 3\.5 tokens/s under an NVIDIA A100\-PCIe 40GB GPU setup\. Its accuracy also drops from 83\.5% at 1,024 denoising steps to 54\.1% at 256 denoising steps\[[49](https://arxiv.org/html/2609.16450#bib.bib49),[41](https://arxiv.org/html/2609.16450#bib.bib41),[24](https://arxiv.org/html/2609.16450#bib.bib24),[29](https://arxiv.org/html/2609.16450#bib.bib29)\], which limits the widespread adoption of dLLMs in real\-world deployments\.

Recently, several works have attempted to mitigate the inference efficiency challenges of dLLMs, including semi\-AR variants for block\-wise sequential generation\[[1](https://arxiv.org/html/2609.16450#bib.bib1),[10](https://arxiv.org/html/2609.16450#bib.bib10)\], Key\-Value \(KV\) and activation caching\[[53](https://arxiv.org/html/2609.16450#bib.bib53),[33](https://arxiv.org/html/2609.16450#bib.bib33),[38](https://arxiv.org/html/2609.16450#bib.bib38),[21](https://arxiv.org/html/2609.16450#bib.bib21),[8](https://arxiv.org/html/2609.16450#bib.bib8)\], efficient sampling algorithms\[[22](https://arxiv.org/html/2609.16450#bib.bib22),[4](https://arxiv.org/html/2609.16450#bib.bib4),[51](https://arxiv.org/html/2609.16450#bib.bib51),[35](https://arxiv.org/html/2609.16450#bib.bib35),[23](https://arxiv.org/html/2609.16450#bib.bib23),[48](https://arxiv.org/html/2609.16450#bib.bib48),[60](https://arxiv.org/html/2609.16450#bib.bib60),[36](https://arxiv.org/html/2609.16450#bib.bib36)\], and step distillation\[[19](https://arxiv.org/html/2609.16450#bib.bib19),[13](https://arxiv.org/html/2609.16450#bib.bib13),[9](https://arxiv.org/html/2609.16450#bib.bib9),[30](https://arxiv.org/html/2609.16450#bib.bib30),[59](https://arxiv.org/html/2609.16450#bib.bib59)\]\. However, most of them rely on fixed block selection and confidence thresholds during decoding, implicitly assuming uniform token difficulty along the sequence, which wastes parallelism on hard tokens and under\-utilizes it on easy ones, contradicting the heterogeneous nature of natural language\. For example, in code, function signatures are typically easier to predict than function bodies, and in mathematical reasoning, routine arithmetic is straightforward whereas multi\-step derivations are more challenging\. Beyond that, a few recent methods introduce simple heuristics for adaptive block sizing and thresholding, such as delimiter \(e\.g\., period\) detection\[[35](https://arxiv.org/html/2609.16450#bib.bib35)\], sliding windows\[[36](https://arxiv.org/html/2609.16450#bib.bib36)\], or entropy\-based boundaries\[[60](https://arxiv.org/html/2609.16450#bib.bib60)\]\. Some methods attempt to learn parallel decoding strategies\[[4](https://arxiv.org/html/2609.16450#bib.bib4)\], but still rely on fixed block sizes\. These limitations call for a principled framework that not only adaptively determines block sizes but also inherently enables early decoding under homogeneous token difficulty within each block\.

![Refer to caption](https://arxiv.org/html/2609.16450v1/ebdecode_cropped.png)Figure 1:Conceptual comparison illustrating the differences between vanilla block\-wise decoding \(Left\)\[[1](https://arxiv.org/html/2609.16450#bib.bib1)\]and the proposed EB\-Decode framework \(Right\)\.In this work, wefor the first timepropose a principled “early\-bird” decoding framework, motivated by our observation that tokens with similar low entropy and close semantic meanings tend to cluster and can be jointly decoded much earlier before reaching the confidence threshold, due to their similar difficulty levels\. To enable such principled EB decoding, two key challenges arise:First, how can we automatically learn adaptive block sizes instead of relying on heuristics? Unlike prior delimiter\-based methods, our observation shows that token entropy exhibits a staircase pattern across token positions, indicating that at certain key steps, adjacent or even non\-contiguous tokens share similar entropy or uncertainty\. This motivates the design of a lightweight network that leverages entropy and positional semantics to dynamically predict block sizes on the fly\.Second, how can we decode the predicted block of tokens earlier before reaching the confidence threshold? Our observation shows that confidence\-based decoding wastes computation on tokens that have already converged: many tokens become correct in early denoising steps but are repeatedly remasked because their confidence has not yet reached the threshold\. Moreover, under fixed block sizes, confident tokens outside the current block are excluded from early finalization\. These observations motivate a position\-aware, learnable parallel sampler that leverages per\-token statistics \(i\.e\., entropy, position, and step\) to determine which positions can be finalized at earlier steps, while naturally supporting variable\-length blocks\. To the best of our knowledge, this work is the first to tackle the above challenges toward a principled EB decoding framework\. Our contributions are summarized as follows:

- •We propose a principled early\-bird decoding framework for efficient dLLM inference acceleration, termedEB\-Decode, that automatically clusters tokens with similar difficulty into blocks and enables their early decoding with fewer denoising steps than confidence\-based methods\.
- •Enabler 1: We adopt alearnable block size \(LBS\)prediction network to dynamically cluster contiguous or non\-contiguous tokens with similar difficulty and semantic coherence on the fly\.
- •Enabler 2: We introduce a position\-awarelearnable parallel sampling \(LPS\)network to identify and finalize converged tokens at early denoising steps under variable\-length blocks\.
- •Extensive experiments across three dLLMs and four representative benchmarks demonstrate the effectiveness of EB\-Decode: it achieves3\.53–18\.76×\\timesthroughput improvement over the vanilla decoder with comparable accuracy and only∼\\sim6\.21%routing overhead, anddeliversan additional1\.20×\\timesspeedup when combined with KV\-cache pipelines\.

## 2Related Works

dLLMs\.Unlike AR models, dLLMs generate text through an iterative denoising process over discrete tokens\[[2](https://arxiv.org/html/2609.16450#bib.bib2),[42](https://arxiv.org/html/2609.16450#bib.bib42),[44](https://arxiv.org/html/2609.16450#bib.bib44),[34](https://arxiv.org/html/2609.16450#bib.bib34),[61](https://arxiv.org/html/2609.16450#bib.bib61)\], achieving likelihood comparable to their AR counterparts\. At the billion\-parameter scale, LLaDA\[[40](https://arxiv.org/html/2609.16450#bib.bib40)\]performs on par with LLaMA3, and subsequent extensions further improve its alignment, sparsity, and scaling\[[63](https://arxiv.org/html/2609.16450#bib.bib63),[64](https://arxiv.org/html/2609.16450#bib.bib64),[5](https://arxiv.org/html/2609.16450#bib.bib5)\]\. Dream\-7B\[[56](https://arxiv.org/html/2609.16450#bib.bib56)\]adds AR\-based initialization and adaptive noise rescheduling\. We provide more literature review of dLLM in Appendix[A](https://arxiv.org/html/2609.16450#A1)\.

Efficient Inference of dLLMs\.Recent work improves dLLM decoding along several directions\. Cache\-based methods reuse key\-value projections across denoising steps to avoid redundant computation\[[33](https://arxiv.org/html/2609.16450#bib.bib33),[53](https://arxiv.org/html/2609.16450#bib.bib53),[38](https://arxiv.org/html/2609.16450#bib.bib38),[21](https://arxiv.org/html/2609.16450#bib.bib21)\]\. Distillation\-based methods train the model to commit more tokens per step\[[9](https://arxiv.org/html/2609.16450#bib.bib9),[30](https://arxiv.org/html/2609.16450#bib.bib30)\]\. Efficient sampling methods redesign the denoising trajectory to better allocate computation across steps\. For example, SlowFast\[[51](https://arxiv.org/html/2609.16450#bib.bib51)\]alternates between exploratory and accelerated phases; DUS\[[37](https://arxiv.org/html/2609.16450#bib.bib37)\]front\-loads computation to early steps via dilated scheduling; Adaptive acceptance methods replace the static confidence threshold with learned or per\-position confidence\[[23](https://arxiv.org/html/2609.16450#bib.bib23),[4](https://arxiv.org/html/2609.16450#bib.bib4),[39](https://arxiv.org/html/2609.16450#bib.bib39),[28](https://arxiv.org/html/2609.16450#bib.bib28)\]\. Dynamic block methods resize the block at runtime by aligning boundaries with confidence or entropy shifts\[[35](https://arxiv.org/html/2609.16450#bib.bib35),[60](https://arxiv.org/html/2609.16450#bib.bib60),[36](https://arxiv.org/html/2609.16450#bib.bib36)\]\. An orthogonal line targets test\-time quality through revocable draft\-and\-verify decoding\[[20](https://arxiv.org/html/2609.16450#bib.bib20)\], temporal\-dynamics voting across denoising steps\[[48](https://arxiv.org/html/2609.16450#bib.bib48)\], and inference\-time remasking\[[47](https://arxiv.org/html/2609.16450#bib.bib47)\]\. In contrast, our proposed principled EB\-Decode explicitly leverages the token\-level statistics, and learns to dynamically cluster tokens with similar difficulty and adapt the sampling schedule to identify and finalize converged tokens at early denoising steps under variable\-length blocks\.

Early\-Bird \(EB\) Phenomenon\.Early prediction is important for efficient training and inference\. For training, the EB ticket hypothesis\[[57](https://arxiv.org/html/2609.16450#bib.bib57)\]shows that small subnetworks \(i\.e\., lottery tickets\[[14](https://arxiv.org/html/2609.16450#bib.bib14)\]\) can be identified early in training via mask\-distance convergence, achieving accuracy comparable to overparameterized networks\. This EB phenomenon has been consistently observed in BERT\[[7](https://arxiv.org/html/2609.16450#bib.bib7)\], GCNs\[[58](https://arxiv.org/html/2609.16450#bib.bib58)\], LLMs\[[18](https://arxiv.org/html/2609.16450#bib.bib18)\], and diffusion models\[[52](https://arxiv.org/html/2609.16450#bib.bib52)\]\. For inference, early\-exit mechanisms enable dynamic computation by terminating inference once predictions become sufficiently confident\. They were first applied to RNNs through a learned halting unit\[[17](https://arxiv.org/html/2609.16450#bib.bib17)\], and later generalized to CNNs\[[46](https://arxiv.org/html/2609.16450#bib.bib46)\]and Transformers\[[54](https://arxiv.org/html/2609.16450#bib.bib54),[62](https://arxiv.org/html/2609.16450#bib.bib62),[43](https://arxiv.org/html/2609.16450#bib.bib43)\], where intermediate layers can halt computation once a confidence or prediction\-agreement criterion is satisfied\. In this work, wefor the first timeobserve the EB phenomenon during the decoding phase in dLLMs and leverage it to enable efficient EB decoding\.

## 3Preliminaries of dLLMs

dLLMs formulate text generation as a diffusion process over discrete token sequences, consisting of a forward masking process and a reverse denoising process\. Let𝐱0=\(x01,…,x0L\)\\mathbf\{x\}\_\{0\}=\(x^\{1\}\_\{0\},\\ldots,x^\{L\}\_\{0\}\)denote a clean token sequence of lengthLLdrawn from a vocabulary𝒱\\mathcal\{V\}, wherex0ix^\{i\}\_\{0\}is the token at positionii\. The forward process samples a noise levelt∈\[0,1\]t\\in\[0,1\]and produces a corrupted sequence𝐱t=\(xt1,…,xtL\)\\mathbf\{x\}\_\{t\}=\(x^\{1\}\_\{t\},\\ldots,x^\{L\}\_\{t\}\)by independently replacing each token with the special\[MASK\]symbol with probabilitytt, so that𝐱t\\mathbf\{x\}\_\{t\}converges to a fully masked sequence ast→1t\\rightarrow 1\. The reverse process is parameterized by a bidirectional Transformer that defines a mask predictorpθ\(⋅∣𝐱t\)p\_\{\\theta\}\(\\cdot\\mid\\mathbf\{x\}\_\{t\}\), which recovers the distribution over the original token at every masked position simultaneously\. The model is trained by minimizing

ℒ\(θ\)=𝔼t∼𝒰⁡\(0,1\),𝐱0,𝐱t\[1t∑i=1L𝟏\{xit=\[MASK\]\}\(−logpθ\(x0i∣𝐱t\)\)\],\\mathcal\{L\}\(\\theta\)=\\mathbb\{E\}\_\{t\\sim\\mathcal\{U\}\(0,1\),\\,\\mathbf\{x\}\_\{0\},\\,\\mathbf\{x\}\_\{t\}\}\\left\[\\frac\{1\}\{t\}\\sum\_\{i=1\}^\{L\}\\mathbf\{1\}\_\{\\\{x^\{i\}\_\{t\}=\\texttt\{\[MASK\]\}\\\}\}\\bigl\(\-\\log p\_\{\\theta\}\(x^\{i\}\_\{0\}\\mid\\mathbf\{x\}\_\{t\}\)\\bigr\)\\right\],\(1\)where𝟏\{⋅\}\\mathbf\{1\}\_\{\\\{\\cdot\\\}\}is the indicator function that restricts the cross\-entropy to currently masked positions, and the factor1/t1/tcompensates for the expected mask ratio at noise leveltt\. This objective upper\-bounds the negative log\-likelihood of the model distribution\.

Block\-wise Decoding\.At inference time, dLLMs commonly adopt a block\-wise decoding strategy\[[1](https://arxiv.org/html/2609.16450#bib.bib1),[53](https://arxiv.org/html/2609.16450#bib.bib53)\]: the response is partitioned into contiguous fixed\-size blocks decoded from left to right, where positions inside the active block are predicted in parallel and committed based on confidence, while the rest are remasked for further refinement\. The full update rule is deferred to Appendix[B](https://arxiv.org/html/2609.16450#A2)\. However, reliance on fixed block sizes and static commit rules ignores variation in token difficulty along the sequence, motivating our empirical analysis in Sec\.[4](https://arxiv.org/html/2609.16450#S4)\.

## 4The Observations of EB Phenomenon in dLLM Decoding

To understand the inefficiency of existing dLLM decoding, we analyze the denoising trajectory and identify two empirical phenomena overlooked by fixed\-block, threshold\-based decoding: astaircase entropy patternacross token positions and anearly convergence phenomenonacross denoising steps\.

Observation 1: Staircase Entropy Pattern\.To examine how token difficulty varies across positions, we measure the Shannon entropyHtiH^\{i\}\_\{t\}at each masked positioniiand denoising steptt\. We visualize this on a GSM8K sample with generation length 256, decoded by the block\-wise diffusion method\[[53](https://arxiv.org/html/2609.16450#bib.bib53)\]\.

![Refer to caption](https://arxiv.org/html/2609.16450v1/ebdecode-page13.png)Figure 2:Comparison of entropy heatmaps between\(a\)fixed block size and\(b\)our non\-contiguous learnable block size method, where each colored region represents a block\. Gray regions denote tokens that have been unmasked\.As shown in Fig\.[2](https://arxiv.org/html/2609.16450#S4.F2)\(a\), the entropy does not decrease smoothly across token positions\. Instead, it exhibits a staircase pattern, consisting of long, flat low\-entropy plateaus that typically correspond to predictable syntax or boilerplate content, such as reasoning connectives like“To solve this problem, we first need to …”in mathematical reasoning tasks, or structural scaffolding around known function signatures likedef function\_name\(args\):in code generation tasks\. This suggests that semantically coherent tokens within each plateau share similar difficulty, and these regions exhibit similarly low entropy across denoising steps\. However, this conflicts with fixed block boundaries, under which such semantically coherent regions are split in a manner agnostic to semantic structure, preventing stabilized regions across block boundaries from being decoded jointly and leaving the available parallelism unexploited\. In addition, these flat plateaus are occasionally interrupted by short high\-entropy spikes corresponding to harder\-to\-predict tokens\. Although such hard tokens sit within an otherwise semantically coherent region, they are not well\-suited to being decoded jointly with the surrounding easy tokens\. Yet fixed\-block methods enforce a strict left\-to\-right order that forces these hard tokens to be resolved inside the current block before any subsequent content can be generated\. In contrast, as shown in Fig\.[2](https://arxiv.org/html/2609.16450#S4.F2)\(b\), a block selection strategy that allows non\-contiguous grouping could defer these hard tokens to a later block, letting the model first decode the surrounding easy context and leverage the additional right\-side information when eventually resolving them\.

\(a\) Per\-token Denoising Trajectory

\(b\) Block\-relative Lead\-time on GSM8K

Figure 3:\(a\)Per\-token denoising trajectory, where each column represents a token position across steps within one segment\. Light green cells indicate tokens whose predictions become correct before they are committed, illustrating the inefficiency of confidence\-based decoding\.\(b\)Mean lead\-time \(in denoising steps\) and fraction \(%\) of tokens with lead≥k\\geq kdenoising steps before commit\.Observation 2: Early Convergence Phenomenon\.Even when block selections are well aligned with token difficulty, decoding still has to decide*when*each token within a block should be committed\. To examine this, we track the per\-step prediction at each masked position for six representative token positions of a GSM8K sample decoded using the vanilla block\-wise method\[[40](https://arxiv.org/html/2609.16450#bib.bib40)\]\. As shown in Fig\.[3](https://arxiv.org/html/2609.16450#S4.F3)\(a\), we observe that many tokens reach the correct prediction several denoising steps before their per\-token confidence crosses the threshold, in principle allowing them to be committed early\. These early\-converged tokens, however, are repeatedly remasked before being finalized, introducing step\-level redundancy\. Fig\.[3](https://arxiv.org/html/2609.16450#S4.F3)\(b\) quantifies this redundancy by showing that under both the vanilla\[[40](https://arxiv.org/html/2609.16450#bib.bib40)\]and confidence\-based method\[[53](https://arxiv.org/html/2609.16450#bib.bib53)\], many tokens remain unaccepted for multiple denoising steps after their predictions become correct\. As reported in Fig\.[3](https://arxiv.org/html/2609.16450#S4.F3)\(b\), the mean lead times of the two baseline methods are 13\.4 and 2\.3 steps, respectively\. The fractions of tokens are 96\.3% and 62\.5% when the lead time is≥1\\geq 1step, and 60\.5% and 3\.5% when the lead time is≥10\\geq 10steps\. This behavior arises because existing methods evaluate each masked position independently and cannot distinguish correct\-but\-uncertain predictions from genuinely incorrect ones, highlighting the need for a principled and learnable strategy to commit correct tokens earlier\.

![Refer to caption](https://arxiv.org/html/2609.16450v1/method4.png)Figure 4:Overview of the proposed EB\-Decode framework\. \(a\) Inference pipeline: LBS first predicts and filters the active block out, and LPS then selects the tokens to commit within this block\. \(b\) LBS router training with a weighted cross\-entropy loss and a regularization term\. \(c\) LPS router training, where features and labels are first recorded by decoding with LBS and then used to train the router\.
## 5The Proposed EB\-Decode Framework

Overview\.The two observations in Sec\.[4](https://arxiv.org/html/2609.16450#S4)reveal two fundamental issues in current dLLM decoding: \(1\) a fixed block size that ignores the staircase pattern of token difficulty and the non\-contiguous nature of tokens that should be grouped, and \(2\) a fixed confidence threshold that overlooks the adaptive nature of how early predictions actually converge\. As illustrated in Fig\.[4](https://arxiv.org/html/2609.16450#S4.F4)\(a\), our proposed EB\-Decode framework replaces the aforementioned two static choices with learnable counterparts while keeping the base dLLM frozen\. Specifically, LBS replaces the fixed block size and decides*where*to decode at each step, as described in Sec\.[5\.1](https://arxiv.org/html/2609.16450#S5.SS1), while LPS replaces the fixed confidence threshold and decides*which*positions within a given variable\-length block to finalize early, as described in Sec\.[5\.2](https://arxiv.org/html/2609.16450#S5.SS2)\. Both modules reuse the base dLLM’s forward pass and introduce negligible training and inference overhead\.

### 5\.1Enabler 1: Learnable Block Size \(LBS\)

The staircase entropy pattern identified in Sec\.[4](https://arxiv.org/html/2609.16450#S4)reveals that token difficulty is spatially clustered: low\-entropy regions alternate with sharp uncertainty spikes that do not align with fixed block boundaries\. A fixed partition either splits an easy plateau across two blocks, wasting a decoding pass, or lumps easy and hard tokens together, causing hard tokens to bottleneck their easier neighbors\. To resolve this, we introduce LBS, a lightweight routerℛϕ\\mathcal\{R\}\_\{\\phi\}that at each step selects a variable\-length, non\-contiguous active blockℬ⊆ℳ\\mathcal\{B\}\\subseteq\\mathcal\{M\}from the currently masked positionsℳ=\{i:xi=\[MASK\]\}\\mathcal\{M\}=\\\{i:x^\{i\}=\\texttt\{\[MASK\]\}\\\}\.

LBS Router\.As shown in Fig\.[4](https://arxiv.org/html/2609.16450#S4.F4)\(b\), for each masked positionii, the router takes two complementary inputs: \(1\)*The normalized entropy*, defined asH~i=H~\(pi\)=\(−∑v∈𝒱pi\(v\)logpi\(v\)\)/log\|𝒱\|\\tilde\{H\}^\{i\}=\\tilde\{H\}\(p^\{i\}\)=\(\-\\sum\_\{v\\in\\mathcal\{V\}\}p^\{i\}\(v\)\\log p^\{i\}\(v\)\)/\\log\|\\mathcal\{V\}\|, which measures local prediction uncertainty\. Entropy alone, however, does not distinguish between qualitatively different sources of uncertainty: a function word may be uncertain among a few interchangeable alternatives, while a rare content word may be uncertain because its identity depends on context not yet resolved—two situations that call for different routing decisions; \(2\)*The top\-1 predicted token IDs \(x^i\\hat\{x\}^\{i\}\)*\. Inspired by[Lu et al\. \[35\]](https://arxiv.org/html/2609.16450#bib.bib35), we embed token IDs using the frozen base\-model token embedding table\. This allows the router to differentiate between semantically trivial tokens and semantically informative ones\. In summary, the router maps these input signals to a per\-masked\-token inclusion probabilitypϕip^\{i\}\_\{\\phi\}of whether a masked token should be included in the final selected block:

pϕi=σ⁡\(Oϕi\)=σ⁡\(ℛϕ​\(H~i,x^i\)\)∈\(0,1\),i∈ℳ,p^\{i\}\_\{\\phi\}=\\sigma\(O^\{i\}\_\{\\phi\}\)=\\sigma\(\\mathcal\{R\}\_\{\\phi\}\(\\tilde\{H\}^\{i\},\\hat\{x\}^\{i\}\)\)\\in\(0,1\),i\\in\\mathcal\{M\},\(2\)whereσ\\sigmais the sigmoid function\. The router is implemented as a lightweight two\-layer Transformer encoder, with full architectural details described in Appendix[C\.1](https://arxiv.org/html/2609.16450#A3.SS1)\.

LBS Router Training\.We train the LBS router in a block\-wise masking style\. A single frozen forward pass through the dLLM yields entropyH~i\\tilde\{H\}^\{i\}and token IDx^i\\hat\{x\}^\{i\}at every masked position, which are then passed into the LBS router to get the per\-token probabilitypip^\{i\}\. We train the router by minimizing a weighted cross\-entropy loss \(ℒwCE\\mathcal\{L\}\_\{\\mathrm\{wCE\}\}\) with a regularization term \(ℒReg\\mathcal\{L\}\_\{\\mathrm\{Reg\}\}\):

ℒLBS=ℒwCE−ℒReg=1\|ℳ\|​∑i∈ℳσ⁡\(Oϕi\)⋅ℓCEi−1\|ℳ\|​∑i∈ℳσ⁡\(Oϕi\)\\mathcal\{L\}\_\{\\mathrm\{LBS\}\}=\\mathcal\{L\}\_\{\\mathrm\{wCE\}\}\-\\mathcal\{L\}\_\{\\mathrm\{Reg\}\}=\\frac\{1\}\{\|\\mathcal\{M\}\|\}\\sum\_\{i\\in\\mathcal\{M\}\}\\sigma\(O^\{i\}\_\{\\phi\}\)\\cdot\\ell^\{i\}\_\{\\mathrm\{CE\}\}\\;\-\\;\\frac\{1\}\{\|\\mathcal\{M\}\|\}\\,\\displaystyle\\sum\_\{i\\in\\mathcal\{M\}\}\\sigma\(O^\{i\}\_\{\\phi\}\)\(3\)whereℓCEi\\ell^\{i\}\_\{\\mathrm\{CE\}\}denotes the per\-masked\-token cross\-entropy loss of the frozen dLLM at positionii, andOϕi=ℛϕ​\(H~i,x^i\)O^\{i\}\_\{\\phi\}=\\mathcal\{R\}\_\{\\phi\}\(\\tilde\{H\}^\{i\},\\hat\{x\}^\{i\}\)is the router output\. The first termℒwCE\\mathcal\{L\}\_\{\\mathrm\{wCE\}\}trains the router to assign high inclusion probability \(i\.e\., likelihood of being included in the current block\) to positions where the dLLM already predicts correctly \(i\.e\., lowℓCEi\\ell^\{i\}\_\{\\mathrm\{CE\}\}\), and assign low inclusion probability to positions where the dLLM is either incorrect or underconfident \(i\.e\., highℓCEi\\ell^\{i\}\_\{\\mathrm\{CE\}\}\)\. The second term acts as a regularization role that prevents the degenerate solution of selecting no tokens—which would trivially minimize the first term—by rewarding larger total inclusion; the subtraction ensures the regularizer and the loss pull in opposite directions, stabilizing the block size\. The training procedure is also summarized in Alg\.[1](https://arxiv.org/html/2609.16450#alg1); the full pseudocode is detailed in Appendix[C\.2](https://arxiv.org/html/2609.16450#A3.SS2)\.

Algorithm 1EB\-Decode Training1:Frozen dLLM

ℱθ\\mathcal\{F\}\_\{\\theta\}, prompt\-response pair

𝒟\\mathcal\{D\}
2:// LBS routerℛϕ\\mathcal\{R\}\_\{\\phi\}

3:for

xx∈𝒟\\in\\mathcal\{D\}do

4:

H~i,x^i←ℱθ​\(Mask​\(x\)\)\\tilde\{H\}^\{i\},\\hat\{x\}^\{i\}\\leftarrow\\mathcal\{F\}\_\{\\theta\}\(\\text\{Mask\}\(x\)\)⊳\\trianglerightLBS Inputs

5:

Oϕi←ℛϕ​\(H~i,x^i\)O^\{i\}\_\{\\phi\}\\leftarrow\\mathcal\{R\}\_\{\\phi\}\(\\tilde\{H\}^\{i\},\\hat\{x\}^\{i\}\)
6:Update

ϕ\\phivia

ℒLBS\\mathcal\{L\}\_\{\\mathrm\{LBS\}\}⊳\\trianglerightEq\.\([3](https://arxiv.org/html/2609.16450#S5.E3)\)

7:endfor

8:// LPS routerℛψ\\mathcal\{R\}\_\{\\psi\}

9:for

x∈𝒟x\\in\\mathcal\{D\}do

10:Decode with LBS; record trace

𝒯\\mathcal\{T\}
11:endfor

12:for

\(fi,posi\)⊂𝒯\(f^\{i\},\\mathrm\{pos\}^\{i\}\)\\subset\\mathcal\{T\}do⊳\\trianglerightLPS Inputs

13:

Oψi←ℛψ​\(fi,posi\)O^\{i\}\_\{\\psi\}\\leftarrow\\mathcal\{R\}\_\{\\psi\}\(f^\{i\},\\mathrm\{pos\}^\{i\}\)
14:Update

ψ\\psivia

ℒLPS\\mathcal\{L\}\_\{\\mathrm\{LPS\}\}⊳\\trianglerightEq\.\([4](https://arxiv.org/html/2609.16450#S5.E4)\)

15:endfor

16:return

ℛϕ,ℛψ\\mathcal\{R\}\_\{\\phi\},\\mathcal\{R\}\_\{\\psi\}

Algorithm 2EB\-Decode Inference1:

ℱθ\\mathcal\{F\}\_\{\\theta\},

ℛϕ\\mathcal\{R\}\_\{\\phi\},

ℛψ\\mathcal\{R\}\_\{\\psi\}, and thresholds

τ\\tau,

τψ\\tau\_\{\\psi\}
2:

x←Prompt&\[MASK\]x\\leftarrow\\text\{Prompt \}\\&\\texttt\{\[MASK\]\}
3:whileany

xi=\[MASK\]x^\{i\}=\\texttt\{\[MASK\]\}do

4:

H~i,x^i,fi←ℱθ​\(x\)\\tilde\{H\}^\{i\},\\hat\{x\}^\{i\},f^\{i\}\\leftarrow\\mathcal\{F\}\_\{\\theta\}\(x\)
5:

fi=\(κi,H~i,Δi,ρ\)f^\{i\}=\(\\kappa^\{i\},\\tilde\{H\}^\{i\},\\Delta^\{i\},\\rho\)⊳\\trianglerightFeature Details

6:// LBS

7:ifno active blockthen

8:

ℬ←Filters⁡\(ℛϕ​\(H~i,x^i\)\)\\mathcal\{B\}\\leftarrow\\mathrm\{Filters\}\\bigl\(\\mathcal\{R\}\_\{\\phi\}\(\\tilde\{H\}^\{i\},\\hat\{x\}^\{i\}\)\\bigr\)
9:endif

10:// LPS

11:

Oψi←ℛψ​\(fi,posi\)O^\{i\}\_\{\\psi\}\\leftarrow\\mathcal\{R\}\_\{\\psi\}\(f^\{i\},\\mathrm\{pos\}^\{i\}\)
12:

𝒜←\{i∈ℬ:κi≥τ∨pψi\>τψ\}\\mathcal\{A\}\\leftarrow\\\{i\\in\\mathcal\{B\}:\\kappa^\{i\}\\geq\\tau\\,\\vee\\,\{\\color\[rgb\]\{0,0,0\}p^\{i\}\_\{\\psi\}\}\>\\tau\_\{\\psi\}\\\}
13:

xi←x^ix^\{i\}\\leftarrow\\hat\{x\}^\{i\}for

i∈𝒜i\\in\\mathcal\{A\}⊳\\trianglerightAccept & Commit

14:If

ℬ\\mathcal\{B\}resolved, move to Line 5

15:endwhile

16:return

xx

LBS Router Inference\.As shown in Alg\.[2](https://arxiv.org/html/2609.16450#alg2), at each step, entropyH~i\\tilde\{H\}^\{i\}and token IDx^i\\hat\{x\}^\{i\}are computed from the dLLM output logits over masked positions and passed toℛϕ\\mathcal\{R\}\_\{\\phi\}, producing a candidate block𝒞=\{i:pϕi\>0\.5\}\\mathcal\{C\}=\\\{i:p^\{i\}\_\{\\phi\}\>0\.5\\\}\. This candidate block is then refined by three filters, each addressing a specific failure mode\. \(1\)*EOS removal*\. End\-of\-sequence tokens have near\-zero entropy, so the router nearly always selects them\. However, unmasking EOS tokens too early in the inference process would truncate generation prematurely and severely damages accuracy on tasks that require step\-by\-step reasoning like math and coding\[[22](https://arxiv.org/html/2609.16450#bib.bib22)\]\. Thus, we remove all EOS tokens from𝒞\\mathcal\{C\}entirely\. We quantify the dominance of EOS predictions at the sequence tail in Appendix[H](https://arxiv.org/html/2609.16450#A8)\. \(2\)*Max\-gap constraint*\(gmaxg\_\{\\max\}\)\. Ideally, we want to identify a homogeneous difficulty plateau as the block, instead of cherry\-picking isolated easy tokens scattered across a hard region\. If consecutive selected positions are separated by more thangmaxg\_\{\\max\}tokens, the candidate block straddles an unresolved hard region, violating the spatial coherence assumption and forcing the model to denoise positions that have not yet been committed\. Thus, starting from the leftmost selected position, any candidate whose gap to its predecessor exceedsgmaxg\_\{\\max\}is discarded along with all subsequent selected positions\. \(3\)*Min\-tokens constraint*\(LminL\_\{\\min\}\)\. After applying the above two filters, if the resulting candidate set contains fewer thanLminL\_\{\\min\}tokens, LBS falls back to selecting the first 32 masked positions in left\-to\-right order\. This constraint is critical for accuracy\. If the number of tokens is too small after filtering, the router can degenerate to selecting scattered easy tokens anywhere in the sequence\. In summary, the router makes per\-token decisions, resulting in an inherently non\-contiguous candidate block that adapts to the entropy pattern; contiguity is instead imposed at inference time via filtering\.

### 5\.2Enabler 2: Learnable Parallel Sampling \(LPS\)

LBS adapts block boundaries to local difficulty but keeps the commit rule fixed, leaving intra\-block parallelism underutilized\. The early convergence observation in Sec\.[4](https://arxiv.org/html/2609.16450#S4)shows that a masked position’s top\-1 prediction often becomes correct several steps before its confidence reaches the threshold, making additional refinement steps unnecessary\. To bridge this gap, we introduce LPS, a lightweight routerℛψ\\mathcal\{R\}\_\{\\psi\}that learns when a masked position within a variable\-length block is ready to be finalized, enabling early commitment before reaching the confidence threshold, as shown in Fig\.[4](https://arxiv.org/html/2609.16450#S4.F4)\(c\)\.

Position\-aware LPS Router\.Unlike many existing decoding methods\[[40](https://arxiv.org/html/2609.16450#bib.bib40),[56](https://arxiv.org/html/2609.16450#bib.bib56),[53](https://arxiv.org/html/2609.16450#bib.bib53),[35](https://arxiv.org/html/2609.16450#bib.bib35)\]that rely on top\-1 confidenceκi\\kappa^\{i\}as a static commit criterion, LPS learns this decision, using confidence as the primary input and augmenting it with two signals\. First, to make the LPS router position\-aware, we incorporate the token’s normalized intra\-block position embedding,posi=\(i−b\)/\|ℬ\|∈\[0,1\]\\mathrm\{pos\}^\{i\}=\\nicefrac\{\{\(i\-b\)\}\}\{\{\|\\mathcal\{B\}\|\}\}\\in\[0,1\], wherebbis the left boundary index and\|ℬ\|\|\\mathcal\{B\}\|is the block size\. This encoding captures the token’s relative location within the block and enables the router to correlate decisions across positions rather than scoring them independently\. To support variable\-length block inputs, our LPS router adopts a two\-layer Transformer architecture to leverage positional information, rather than an MLP\-based design\[[4](https://arxiv.org/html/2609.16450#bib.bib4)\]that cannot handle variable\-length inputs\. Model details are provided in Appendix[D\.1](https://arxiv.org/html/2609.16450#A4.SS1)\. Second, to enable the LPS router to distinguish correct\-but\-underconfident tokens from genuinely uncertain ones, we incorporate three auxiliary uncertainty features: \(1\) the normalized entropyH~ti\\tilde\{H\}^\{i\}\_\{t\}for absolute prediction uncertainty, which alone cannot tell whether the top\-1 candidate dominates; \(2\) the confidence gapΔti\\Delta^\{i\}\_\{t\}between the top\-1 and top\-2 predictions for relative label ambiguity, i\.e\., the margin over the strongest competitor; and \(3\) the block\-wise mask ratioρt\\rho\_\{t\}to capture global decoding progress, as the commit policy should differ across denoising stages\. Overall, for each masked positioni∈ℳi\\in\\mathcal\{M\}in the active blockℬ\\mathcal\{B\}, the input to the LPS router is defined as the concatenation of all featuresfti=\(κti,H~ti,Δti,ρt\)f^\{i\}\_\{t\}=\(\\kappa^\{i\}\_\{t\},\\tilde\{H\}^\{i\}\_\{t\},\\Delta^\{i\}\_\{t\},\\rho\_\{t\}\), and the output isOψi=ℛψ​\(fti,posi\)O^\{i\}\_\{\\psi\}=\\mathcal\{R\}\_\{\\psi\}\(f^\{i\}\_\{t\},\\mathrm\{pos\}^\{i\}\), from which the commit probability is obtained aspψi=σ⁡\(Oψi\)p^\{i\}\_\{\\psi\}=\\sigma\(O^\{i\}\_\{\\psi\}\)\.

Self\-supervised LPS Router Training\.We train LPS using agenerate\-then\-replaypipeline that requires no human label\. We first run the base dLLM under its default decoding schedule and record the resulting completed sequencex0x\_\{0\}\. We treatx0x\_\{0\}as a convergence target rather than absolute ground truth, sotheLPS router learns to predict when a token has stabilized to the model’s own final prediction\. During replay, at each intermediate steptt, we recompute the featuresftif^\{i\}\_\{t\}and assignyti=𝟏\[x^ti=x0i\]y^\{i\}\_\{t\}=\\mathbf\{1\}\[\\hat\{x\}^\{i\}\_\{t\}=x^\{i\}\_\{0\}\], indicating whether the current top\-1 prediction already matchesx0x\_\{0\}\. To keep the replay aligned with the original generation trajectory, we apply an oracle rule that unmasks positions where the current prediction matchesx0x\_\{0\}, and replaces mismatched positions with their corresponding tokens fromx0x\_\{0\}\. Since prematurely committing an incorrect token is irreversible, we use a weighted binary cross\-entropy loss that penalizes false positives more heavily than false negatives during training:

ℒLPS=1\|ℳt\|​∑i∈ℳtwti⋅BCE⁡\(σ⁡\(ℛψ​\(fti,posti\)\),yti\),\\mathcal\{L\}\_\{\\mathrm\{LPS\}\}=\\frac\{1\}\{\|\\mathcal\{M\}\_\{t\}\|\}\\sum\_\{i\\in\\mathcal\{M\}\_\{t\}\}\{\\color\[rgb\]\{0,0,0\}w^\{i\}\_\{t\}\}\\cdot\\mathrm\{BCE\}\\Bigl\(\\sigma\\bigl\(\\mathcal\{R\}\_\{\\psi\}\(f^\{i\}\_\{t\},\\mathrm\{pos\}^\{i\}\_\{t\}\)\\bigr\),y^\{i\}\_\{t\}\\Bigr\),\(4\)whereσ\\sigmaindicates a sigmoid function, negative samples \(incorrect early commitments\) receive weightswti\>1w\_\{t\}^\{i\}\>1, causing false positives to dominate the gradient\. The training procedure is briefly summarized in Alg\.[1](https://arxiv.org/html/2609.16450#alg1), with full trace\-generation and training details provided in Appendix[D\.2](https://arxiv.org/html/2609.16450#A4.SS2)and[D\.3](https://arxiv.org/html/2609.16450#A4.SS3)\.

LPS Router Inference\.As shown in Alg\.[2](https://arxiv.org/html/2609.16450#alg2), at each decoding step, the base dLLM produces top\-1 predictions and confidences for all masked positions\. If no active block exists, LBS selects a newℬ\\mathcal\{B\}; otherwise, the previousℬ\\mathcal\{B\}is reused\. Withinℬ\\mathcal\{B\}, a positioniiis committed if its base confidence exceeds the thresholdτ\\tauor its LPS score exceeds a thresholdτψ\\tau\_\{\\psi\}tuned on a held\-out validation split\. In this way, LPS preserves the standard threshold rule for tokens the model is already confident about, while committing correct\-but\-underconfident tokens that would otherwise be deferred\. If no position inℬ\\mathcal\{B\}satisfies either condition, we fall back to committing the highest\-confidence masked position to prevent decoding from stalling, following standard practice in confidence\-based decoding\. Onceℬ\\mathcal\{B\}is fully resolved, decoding proceeds to the next block\. Full pseudocode is provided in Appendix[E](https://arxiv.org/html/2609.16450#A5)\.

## 6Experiments

### 6\.1Experimental Setup

Models, Benchmarks, and Metrics\.Models\.We evaluate EB\-Decode on three open\-source dLLMs: LLaDA\-8B\-Instruct\[[40](https://arxiv.org/html/2609.16450#bib.bib40)\], Dream\-v0\-Instruct\-7B\[[56](https://arxiv.org/html/2609.16450#bib.bib56)\], and LLaDA\-1\.5\[[63](https://arxiv.org/html/2609.16450#bib.bib63)\]\.Benchmarks\.Our evaluation covers mathematical reasoning on GSM8K\[[11](https://arxiv.org/html/2609.16450#bib.bib11)\]\(0\-shot\) and MATH500\[[31](https://arxiv.org/html/2609.16450#bib.bib31)\]\(0\-shot\), as well as code generation on HumanEval\[[6](https://arxiv.org/html/2609.16450#bib.bib6)\]\(0\-shot\) and MBPP\[[3](https://arxiv.org/html/2609.16450#bib.bib3)\]\(3\-shot\)\. For GSM8K and MATH500, we append a chain\-of\-thought prompt suffix\[[50](https://arxiv.org/html/2609.16450#bib.bib50)\]to each question following\[[9](https://arxiv.org/html/2609.16450#bib.bib9)\]\.Metrics\.We report task accuracy, throughput, and the associated speedups compared to the base models\. Detailed experimental settings are provided in Appendix[F](https://arxiv.org/html/2609.16450#A6)\.

Table 1:Comparison of EB\-Decode and baseline methods across four benchmarks and three base models\. We report tokens per second per GPU \(TPS\), speedup \(Sp\.up\), and accuracy \(Acc\.%\)\. Speedup is measured relative to the vanilla decoding method\. The best throughput and speedup are highlighted inbold\. “—” indicates not applicable, as Learn2PDprovides no open\-source implementation forLLaDA\-1\.5\.Baselines\.We compare EB\-Decode against four inference\-stage baselines: \(1\) the vanilla block\-wise decoding method used by each base dLLM\[[40](https://arxiv.org/html/2609.16450#bib.bib40),[56](https://arxiv.org/html/2609.16450#bib.bib56),[63](https://arxiv.org/html/2609.16450#bib.bib63)\], which serves as the no\-acceleration reference; \(2\) Fast\-dLLM\[[53](https://arxiv.org/html/2609.16450#bib.bib53)\], which commits multiple tokens per step based on a confidence threshold\(for a fair comparison, we use its parallel decoding without KV cache, and report its results with KV cache in Tab\.[6](https://arxiv.org/html/2609.16450#S6.T6)\); \(3\) AdaBlock\-dLLM\[[35](https://arxiv.org/html/2609.16450#bib.bib35)\], which adaptively places block boundaries using delimiter\-token confidence; and \(4\) Learn2PD\[[4](https://arxiv.org/html/2609.16450#bib.bib4)\], which employs a lightweight filter for parallel decoding\.A detailed comparison between EB\-Decode and these and related methods is given in Appendix[G](https://arxiv.org/html/2609.16450#A7)\.

Router Training Details\.Both routers are lightweight, with about0\.60\.6M trainable parameters for LBS and0\.030\.03M for LPS\.The LBS router is trained on dLLM\-generated responses from92,00092\{,\}000prompts randomly sampled from GSM8K\[[11](https://arxiv.org/html/2609.16450#bib.bib11)\], the PRM12K training set\[[31](https://arxiv.org/html/2609.16450#bib.bib31)\], and a subset of the Numina\-Math dataset\[[27](https://arxiv.org/html/2609.16450#bib.bib27)\], following the setup of\[[9](https://arxiv.org/html/2609.16450#bib.bib9)\]\. The router is trained for 6 epochs using AdamW with a learning rate of5​e−55\\mathrm\{e\}\{\-5\}\. The actual training of the LBS router takes only about22hours\. The LPS router is trained on10,00010\{,\}000prompts randomly sampled from AQUA\-RAT\[[32](https://arxiv.org/html/2609.16450#bib.bib32)\]\. Training uses AdamW with a learning rate of1​e−31\\mathrm\{e\}\{\-3\}and is early\-stopped when the validation recall does not improve for100100epochs\. Trace generation takes about22hours, while router training itself requires only around1010minutes\. Further details are provided in Appendix[C](https://arxiv.org/html/2609.16450#A3)and[D](https://arxiv.org/html/2609.16450#A4)\.

### 6\.2Comparison of EB\-Decode with SOTA Baselines

Tab\.[1](https://arxiv.org/html/2609.16450#S6.T1)summarizes the comparison between our proposed EB\-Decode and four baselines in terms of accuracy and throughput across three dLLMs and four benchmarks\. We observe that both EB\-Decode with LBS only and the full EB\-Decode with LBS\+LPS consistently achieve better accuracy\-efficiency trade\-offs\. EB\-Decode with LBS alone already improves throughput across all settings, while even improving accuracy in several cases\. For example,on LLaDA\-8B\-Instruct with HumanEval, LBS increases throughput from 23\.70 to 68\.83 TPS while simultaneously improving accuracy from 43\.90% to 45\.73%\. This result suggests that adaptive block boundaries not only unlock greater parallelism, but also provide difficult tokens with richer right\-context information during decoding\. The full EB\-Decode improves throughput by3\.53–18\.76×\\timesover the vanilla decoder, while maintaining comparable accuracy \(around±1%\\pm 1\\%\)\.EB\-Decode also outperforms the strongest baseline in every setting, delivering up to 1\.58×\\timeshigher throughput than Fast\-dLLM while maintaining comparable or even higher accuracy\.These results demonstratethe effectiveness of our EB\-Decode\.

### 6\.3Ablation Studies of EB\-Decode

Table 2:Ablation of LBS components on LLaDA\-8B\-Instruct and GSM8K\-CoT\. Each row progressively adds one component\.
Table 3:Ablation of LPS components on LLaDA\-8B\-Instruct and GSM8K\-CoT\. Each row progressively adds one component\.

Table 4:Accuracy and TPS on LLaDA\-8B\-Instruct and GSM8K\-CoT with the three filters applied progressively without the LBS router, where blocks are selected randomly\. Even with all three filters, both accuracy and TPS remain below those of LBS\.LBS Design Choices\.Tab\.[3](https://arxiv.org/html/2609.16450#S6.T3)additively builds up the LBS filter described in Sec\.[5\.1](https://arxiv.org/html/2609.16450#S5.SS1)\. Starting from the LBS router’s selected block, we incrementally add EOS removal, the max\-gap constraint, and the min\-tokens constraint to further refine the selected block\. The results in the table empirically demonstrate that all filters are crucial in maximizing the task accuracy while maintaining high TPS\.To check whether these gains stem from the filters alone, Tab\.[4](https://arxiv.org/html/2609.16450#S6.T4)removes the LBS router while keeping the same filters\. Even with all three filters, this variant reaches only76\.80%76\.80\\%accuracy at23\.9623\.96TPS,2\.882\.88points lower and3\.62×3\.62\\timesslower than LBS with the same filters, showing that the learned block selection itself brings large gains\.We also train an LBS router of comparable size on LLaDA\-8B\-Instruct that takes only normalized entropy as input\. Its converged training loss is higher, and accuracy across the four benchmarks drops by 1\.2 points on average at similar TPS\.This suggests that semantic information can indeed help the LBS router pick a more optimal block\.

LPS Design Choices\.Tab\.[3](https://arxiv.org/html/2609.16450#S6.T3)additively builds up the LPS router described in Sec\.[5\.2](https://arxiv.org/html/2609.16450#S5.SS2), on top of LBS selected blocks\. Starting fromanMLP taking only confidence as input, we replace the encoder with a small Transformer, add the intra\-block position embedding, and add the three auxiliary uncertainty features \(entropy, top\-1/top\-2 gap, mask ratio\)\. The results in the table empirically demonstrate the benefit of using a Transformer architecture over an MLP, and show that all components contribute to maximizing the TPS while preserving the task accuracy\.

Table 5:Accuracy and speedup on LLaDA\-8B\-Instruct and GSM8K\-CoT at different generation lengths\. EB\-Decode delivers higherspeedupat both lengths with accuracy comparable tothe vanilla decoder\.Performance on Different Generation Lengths\.Tab\.[5](https://arxiv.org/html/2609.16450#S6.T5)compares EB\-Decode against the baseline at generation lengthsL∈\{256,512\}L\\in\\\{256,512\\\}\. EB\-Decode achieves the highest throughput and comparable accuracy at both lengths\. Also, the speedup grows as the generation length increases, indicating that EB\-Decode may be especially effective for long\-sequence generation\.

Table 6:Accuracy and TPS on LLaDA\-8B\-Instruct and GSM8K\-CoT across different cache strategies, for both EB\-Decode and Fast\-dLLM\.Effect of KV\-Cache Configuration\.Tab\.[6](https://arxiv.org/html/2609.16450#S6.T6)shows the result of using different KV\-cache strategies on top of EB\-Decode on 4×\\timesA100\. Since LBS uses non\-contiguous active blocks, we apply cache at the block boundaries: prefix cache\[[53](https://arxiv.org/html/2609.16450#bib.bib53)\]reuses keys/values for tokens to the left of the leftmost selected position, and dual cache\[[53](https://arxiv.org/html/2609.16450#bib.bib53)\]additionally reuses keys/values for tokens to the right of the rightmost selected position\. Both variants further improve the throughput of EB\-Decode while preserving accuracy, demonstrating that EB\-Decode can integrate with standard KV\-cache pipelines for additional speedup\.With KV caching being considered on top of both methods, EB\-Decode still outperforms Fast\-dLLM by1\.28×1\.28\\times,1\.22×1\.22\\times, and1\.38×1\.38\\timeswithout cache, with the prefix cache, and with the dual cache, respectively, showing that its advantage holds even when the baseline also uses KV caching\.

Overhead of LBS and LPS Routers\.We measure the wall\-clock overhead of the two routers as a fraction of the per\-step backbone forward time on A100\. LBS accounts for roughly3\.80%3\.80\\%and LPS for2\.41%2\.41\\%of the per\-step latency \(6\.21% in total\), confirming that the speedups come from jointly decoding low\-entropy tokens earlier before they reach the confidence threshold and thereby using fewer decoding steps, while the routers themselves introduce only negligible inference overhead\.

### 6\.4Generality and Broader Comparisons

Table 7:Router transfer without retraining on GSM8K\-CoT, \(a\) across models and \(b\) across generation lengths\. “Native” uses routers trained in the target setting, and “Transferred” uses the router\(s\) listed in the second column\.Router Transferability\.The routers of EB\-Decode are largely transferable, so they can be reused without retraining when the deployment setting changes\. We examine this transferability from three aspects: across models, across generation lengths, and across tasks\. \(1\) Across models as shown in Tab\.[7](https://arxiv.org/html/2609.16450#S6.T7)\(a\), we consider two transfer targets\. For LLaDA\-1\.5, which belongs to the same model series as LLaDA\-8B\-Instruct, transferring LBS alone or together with LPS matches the natively trained routers in accuracy and throughput\. For Dream\-v0\-Instruct\-7B, which comes from a different model series, LBS does not transfer, because it embeds predicted token IDs with the base model’s embedding table and vocabularies differ across series\. LPS, whose features are vocabulary\-independent, still transfers\. \(2\) Across generation lengths as shown in Tab\.[7](https://arxiv.org/html/2609.16450#S6.T7)\(b\), an LPS router trained at one length remains on par at the other\. Only LPS is evaluated here, because LBS takes the entire masked sequence as input and uses positional embeddings tied to the generation length \(Appendix[C\.1](https://arxiv.org/html/2609.16450#A3.SS1)\)\. We therefore recommend using LBS at its training length\. \(3\) Across tasks, both routers are trained only on mathematical data, so the HumanEval and MBPP results in Tab\.[1](https://arxiv.org/html/2609.16450#S6.T1)already reflect transfer to unseen code tasks\.

Table 8:Accuracy and throughput of LBS on block\-forward dLLMs, which compute logits only for the current active window\. “Thr\.” denotes the commit threshold; accuracy gains over the baseline are shown in parentheses\.Table 9:Comparison with AR models on GSM8K\-CoT \(L=512L\{=\}512\)\. Acc\. follows the standard answer extraction of the GSM8K harness, while Flex\. Acc\. takes the content of\\boxed\{\}or the last number in the response and is thus robust to the output format\.Compatibility with Block\-Forward dLLMs\.Unlike the bidirectional dLLMs in Tab\.[1](https://arxiv.org/html/2609.16450#S6.T1), which forward the full sequence at every step, block\-forward dLLMs such as LLaDA 2\.0\[[5](https://arxiv.org/html/2609.16450#bib.bib5)\]and SDAR\[[10](https://arxiv.org/html/2609.16450#bib.bib10)\]adopt block\-causal attention and compute logits only for the current active window\. This does not prevent EB\-Decode from applying\. LBS selects the next easy\-to\-decode block, so its selections concentrate near the decoding frontier, which the active window already covers\. Adapting to block\-causal attention then requires a single parameter change, tighteninggmaxg\_\{\\max\}from55to11so that each block is filled contiguously\. Restricted to the window logits, LBS improves accuracy in all eight configurations of Tab\.[8](https://arxiv.org/html/2609.16450#S6.T8), at both commit thresholds tested\. Throughput stays close to the baseline, since LBS reuses logits the engine already computes and adds no extra forward\. LPS needs no adaptation, since its input covers only the tokens inside the selected block\.

Comparison with Autoregressive Models\.To examine whether EB\-Decode narrows the efficiency gap between dLLMs and autoregressive \(AR\) models, Tab\.[9](https://arxiv.org/html/2609.16450#S6.T9)compares it with Qwen3\-8B\[[55](https://arxiv.org/html/2609.16450#bib.bib55)\]and LLaMA3\-8B\-Instruct\[[16](https://arxiv.org/html/2609.16450#bib.bib16)\], with all models run on the sametransformersbackend with batch size11on4×4\\timesA100 GPUs\. The low Acc\. of the AR models reflects a format mismatch with the standard answer\-extraction rule of the GSM8K harness, while Flex\. Acc\. extracts the answer from\\boxed\{\}or the last number in the response and is thus robust to the output format\. While vanilla dLLM decoding is less than half as fast as either AR model, EB\-Decode runs about2\.4×2\.4\\timesfaster than the faster of the two, and the remaining accuracy gap to Qwen3\-8B in Flex\. Acc\. stems mostly from the base model rather than the decoding method\. Beyond this matched setting, AR models benefit from recent serving systems such as vLLM\[[26](https://arxiv.org/html/2609.16450#bib.bib26)\], where Qwen3\-8B and LLaMA3\-8B\-Instruct reach5689\.705689\.70and4965\.194965\.19TPS through paged KV\-cache memory management, continuous batching, and optimized attention kernels\[[12](https://arxiv.org/html/2609.16450#bib.bib12)\], whereas serving frameworks for dLLMs such as dInfer\[[39](https://arxiv.org/html/2609.16450#bib.bib39)\]are still in early development\.

## 7Conclusion

Diffusion large language models promise parallel decoding but still suffer from inefficient inference under fixed block sizes and fixed confidence thresholds\. To understand the source of this inefficiency, we identify two empirical phenomena: a staircase entropy pattern across token positions and an early convergence phenomenon across denoising steps\. Building on these observations, we propose EB\-Decode, a plug\-in framework that replaces the fixed block size and the fixed confidence threshold with two lightweight learnable routers, LBS and LPS, while keeping the base dLLM frozen\. Experiments on three open\-source dLLMs and four benchmarks show that EB\-Decode delivers up to18\.76×18\.76\\timesthroughput improvement over the vanilla decoder with comparable accuracy, offering a practical step toward narrowing the efficiency gap between dLLMs and autoregressive models\.

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## Appendix ADetailed dLLM Backgrounds

This section extends the dLLM literature summary inSec\.[2](https://arxiv.org/html/2609.16450#S2)of the main text with additional context on representative methods\.

D3PM\[[2](https://arxiv.org/html/2609.16450#bib.bib2)\]generalizes discrete diffusion using structured transition matrices over the token vocabulary\. Subsequent works refine the masked diffusion paradigm\. MDLM\[[42](https://arxiv.org/html/2609.16450#bib.bib42)\]derives a Rao\-Blackwellized objective that reduces to a weighted mixture of masked language modeling losses\. MD4\[[44](https://arxiv.org/html/2609.16450#bib.bib44)\]further simplifies and generalizes the continuous\-time objective\. SEDD\[[34](https://arxiv.org/html/2609.16450#bib.bib34)\]proposes a score\-entropy objective\.[Zheng et al\. \[61\]](https://arxiv.org/html/2609.16450#bib.bib61)shows that masked diffusion training and sampling are time\-agnostic\. Together, these advances bring masked dLLMs to likelihood performance comparable to AR models\. LLaDA\[[40](https://arxiv.org/html/2609.16450#bib.bib40)\]performs on par with LLaMA3 at the billion\-parameter scale\. Subsequent extensions add variance\-reduced preference optimization\[[63](https://arxiv.org/html/2609.16450#bib.bib63)\], sparse experts\[[64](https://arxiv.org/html/2609.16450#bib.bib64)\], and AR\-to\-diffusion conversion that scales to 100B parameters\[[5](https://arxiv.org/html/2609.16450#bib.bib5)\]\. Dream\-7B\[[56](https://arxiv.org/html/2609.16450#bib.bib56)\]introduces AR\-based LLM initialization and context\-adaptive, token\-level noise rescheduling\. Semi\-AR methods such as BD3\-LMs\[[1](https://arxiv.org/html/2609.16450#bib.bib1)\]and SDAR\[[10](https://arxiv.org/html/2609.16450#bib.bib10)\]combine the coherence of AR generation with the parallelism of diffusion\. They maintain AR dependencies across blocks while sampling tokens in parallel within each block\.

## Appendix BBlock\-wise Decoding for dLLMs

This section details the block\-wise decoding recipe summarized in Sec\.[3](https://arxiv.org/html/2609.16450#S3), reusing the discrete\-diffusion notation introduced therein, including the vocabulary𝒱\\mathcal\{V\}, the mask symbol\[MASK\], the mask predictorpθp\_\{\\theta\}, the clean response sequence𝐱0\\mathbf\{x\}\_\{0\}with its position\-iitokenx0ix^\{i\}\_\{0\}, and the partially denoised sequence𝐱t\\mathbf\{x\}\_\{t\}at noise levelt∈\[0,1\]t\\in\[0,1\]\. We further writextix^\{i\}\_\{t\}for the token at positioniiin𝐱t\\mathbf\{x\}\_\{t\}, which is either a vocabulary item in𝒱\\mathcal\{V\}or the mask symbol\[MASK\]\.

Inference\-time denoising trajectory\.At inference time, given a promptcc, decoding starts from a fully masked response sequence and iteratively applies the reverse process fromt=1t=1tot=0t=0over a sequence of denoising steps\. The promptccis concatenated with the response and processed jointly by the bidirectional Transformer\[[40](https://arxiv.org/html/2609.16450#bib.bib40)\], so that the predictive distribution at every masked position conditions on both𝐱t\\mathbf\{x\}\_\{t\}andcc\. The final output is the fully unmasked sequence𝐱0\\mathbf\{x\}\_\{0\}\.

Block partition and in\-block prediction\.The response is divided into contiguous blocksℬ1,ℬ2,…\\mathcal\{B\}\_\{1\},\\mathcal\{B\}\_\{2\},\\ldotsof fixed sizeBB, indexed byjj, and processed from left to right\. Within the active blockℬj\\mathcal\{B\}\_\{j\}, the model predicts all masked positions in parallel, commits a subset of tokens, and remasks the remaining uncertain ones for further refinement\. The per\-position top\-1 confidence at masked positioni∈ℬji\\in\\mathcal\{B\}\_\{j\}over candidate vocabulary itemv∈𝒱v\\in\\mathcal\{V\}follows the same definition as in the main text:

κi=maxv∈𝒱⁡pθ​\(x0i=v\|𝐱t,c\)\.\\kappa^\{i\}=\\max\_\{v\\in\\mathcal\{V\}\}p\_\{\\theta\}\\bigl\(x^\{i\}\_\{0\}=v\\,\\bigm\|\\,\\mathbf\{x\}\_\{t\},\\,c\\bigr\)\.\(5\)Given a confidence thresholdτ\\tauor an optional per\-step commit budgetmm, the standard block update rule is

xti←\{arg​maxv∈𝒱⁡pθ​\(x0i=v\|𝐱t,c\),κi≥τ​or​i∈Top​\-​m​\(ℬj\),\[MASK\],otherwise,x^\{i\}\_\{t\}\\leftarrow\\begin\{cases\}\\displaystyle\\operatorname\*\{arg\\,max\}\_\{v\\in\\mathcal\{V\}\}p\_\{\\theta\}\\bigl\(x^\{i\}\_\{0\}=v\\,\\bigm\|\\,\\mathbf\{x\}\_\{t\},\\,c\\bigr\),&\\kappa^\{i\}\\geq\\tau\\;\\;\\text\{or\}\\;\\;i\\in\\mathrm\{Top\}\\text\{\-\}m\(\\mathcal\{B\}\_\{j\}\),\\\\ \\texttt\{\[MASK\]\},&\\text\{otherwise,\}\\end\{cases\}\(6\)whereTop​\-​m​\(ℬj\)\\mathrm\{Top\}\\text\{\-\}m\(\\mathcal\{B\}\_\{j\}\)denotes themmmasked positions inℬj\\mathcal\{B\}\_\{j\}with the highest top\-1 confidence at the current step\. Once all positions inℬj\\mathcal\{B\}\_\{j\}have been committed, the active block advances toℬj\+1\\mathcal\{B\}\_\{j\+1\}and the same procedure repeats\. Throughout this recipe, the block sizeBB, the thresholdτ\\tau, and the budgetmmremain fixed regardless of how token difficulty varies along the sequence, which is precisely the limitation that motivates our EB\-Decode framework\. Building on the same block\-wise procedure and keeping the base dLLM frozen, EB\-Decode replaces the two static design choices with learnable counterparts\. Specifically, LBS replaces the fixed block sizeBBwith a learnable block selector as described in Sec\.[5\.1](https://arxiv.org/html/2609.16450#S5.SS1), while LPS replaces the static commit rule with a position\-aware, learnable acceptance strategy as described in Sec\.[5\.2](https://arxiv.org/html/2609.16450#S5.SS2)\.

## Appendix CDetailed Setup for Learnable Block Size

### C\.1Router Architecture

The LBS routerℛϕ\\mathcal\{R\}\_\{\\phi\}is a small Transformer encoder with 600k parameters \(not including the frozen token embedding tableWeW\_\{e\}\) that maps per\-masked\-position features to scalar inclusion logits\.

Entropy branch\.The normalized entropy sequence for all masked tokensH~=\{H~i:i∈ℳ\}\\tilde\{H\}=\\\{\\tilde\{H\}^\{i\}:i\\in\\mathcal\{M\}\\\}is first zero\-padded to the generation lengthL\{\\color\[rgb\]\{0,0,0\}L\}and then processed by a two\-layer MLPFent\{\\color\[rgb\]\{0,0,0\}F\_\{\\mathrm\{ent\}\}\}:Linear​\(1→64\)→GELU→Linear​\(64→64\)→LayerNorm\\text\{Linear\}\(1\\to 64\)\\to\\text\{GELU\}\\to\\text\{Linear\}\(64\\to 64\)\\to\\text\{LayerNorm\}, producing a 64\-dimensional entropy embedding\.

Token branch\.The top\-1 predicted token IDsx^=\{x^i:i∈ℳ\}\\hat\{x\}=\\\{\\hat\{x\}^\{i\}:i\\in\\mathcal\{M\}\\\}arelooked up in the frozen base\-model token embedding tableWeW\_\{e\}, zero\-padded toL\{\\color\[rgb\]\{0,0,0\}L\}, and projected toℝ64\\mathbb\{R\}^\{64\}via a learned linear layerWtok∈ℝ64×dbaseW\_\{\\text\{tok\}\}\\in\\mathbb\{R\}^\{64\\times\{\\color\[rgb\]\{0,0,0\}d\_\{\\mathrm\{base\}\}\}\}\.

Fusion and positional embedding\.The two 64\-dimensional embeddings are concatenated to form a 128\-dimensional vector, then passed through a fusion layerWcombine∈ℝ128×128W\_\{\\text\{combine\}\}\\in\\mathbb\{R\}^\{128\\times 128\}\. The final input embedding to the Transformer encoder is calculated as:

Zϕ=Wcombine​\[Fent​\(H~\);Wtok​We​\(x^\)\]\+Epos,Zϕ∈ℝL×128\{\\color\[rgb\]\{0,0,0\}Z\_\{\\phi\}\}=W\_\{\\text\{combine\}\}\\\!\\left\[\\,\{\\color\[rgb\]\{0,0,0\}F\_\{\\mathrm\{ent\}\}\}\\\!\\left\(\\tilde\{H\}\\right\)\\;;\\;W\_\{\\text\{tok\}\}\\,W\_\{e\}\\bigl\(\\hat\{x\}\\bigr\)\\,\\right\]\+\{\\color\[rgb\]\{0,0,0\}E\_\{\\mathrm\{pos\}\}\},\{\\color\[rgb\]\{0,0,0\}Z\_\{\\phi\}\}\\in\\mathbb\{R\}^\{\{\\color\[rgb\]\{0,0,0\}L\}\\times 128\}\(7\)whereEpos∈ℝL×128\{\\color\[rgb\]\{0,0,0\}E\_\{\\mathrm\{pos\}\}\}\\in\\mathbb\{R\}^\{\{\\color\[rgb\]\{0,0,0\}L\}\\times 128\}is a learnable positional embedding\.

Transformer encoder\.The sequenceZϕ\{\\color\[rgb\]\{0,0,0\}Z\_\{\\phi\}\}is then fed into a 2\-layer Transformer encoder with hidden dimensiondϕ=128d\_\{\\phi\}\{=\}128, 4 attention heads, FFN width 256, and GELU activations\.

Output head\.A linear headℝ128→ℝ1\\mathbb\{R\}^\{128\}\\to\\mathbb\{R\}^\{1\}maps the encoder output to logitsOϕ∈ℝL×1O\_\{\\phi\}\\in\\mathbb\{R\}^\{\{\\color\[rgb\]\{0,0,0\}L\}\\times 1\}\. The padding is removed before returning, yielding an output of shape\(\|ℳ\|,1\)\(\|\\mathcal\{M\}\|,1\)\. The final inclusion probability at positioniiispϕi=σ⁡\(Oϕi\)p^\{i\}\_\{\\phi\}=\\sigma\(O^\{i\}\_\{\\phi\}\)\.

### C\.2Training Procedure

The full per\-step training procedure for the LBS router, summarized in the main text and the simplifiedAlg\.[1](https://arxiv.org/html/2609.16450#alg1), is givenin Alg\.[3](https://arxiv.org/html/2609.16450#alg3)\.Training is performed on8×8\\timesA100 GPUs\.

Algorithm 3LBS Router Training1:dLLM\-generated response dataset

𝒟\\mathcal\{D\}, frozen dLLM

ℱθ\\mathcal\{F\}\_\{\\theta\}, router

ℛϕ\\mathcal\{R\}\_\{\\phi\}, generation length

LL, block length

LBL\_\{B\}, mask ratio

r∈\(0,1\]r\\in\(0,1\], learning rate

ηlr\\eta\_\{\\mathrm\{lr\}\}
2:

3:

H~\(p\)=\(−∑v∈𝒱p\(v\)logp\(v\)\)/log\|𝒱\|\\tilde\{H\}\(p\)=\(\-\\sum\_\{v\\in\\mathcal\{V\}\}p\(v\)\\log p\(v\)\)/\\log\|\\mathcal\{V\}\|
4:

x^​\(p\)=arg⁡maxx⁡p⁡\(x\)\\hat\{x\}\(p\)=\\arg\\max\_\{x\}p\(x\)
5:

σ⁡\(p\)=1/\(1\+e−p\)\\sigma\(p\)=1/\(1\+e^\{\-p\}\)
6:for

x∈𝒟x\\in\\mathcal\{D\}do

7:Sample

n∼\{0,1,…,L/LB−1\}n\\sim\\\{0,\\,1,\\ldots,\\,L/L\_\{B\}\-1\\\}
8:

xmasked←Blockwise\-Masking​\(x,r,n,L,LB\)x\_\{\\mathrm\{masked\}\}\\leftarrow\\text\{Blockwise\-Masking\}\(x,r,n,L,L\_\{B\}\);

ℳ←\{i:xmaskedi=\[MASK\]\}\\mathcal\{M\}\\leftarrow\\\{i:x\_\{\\mathrm\{masked\}\}^\{i\}=\\texttt\{\[MASK\]\}\\\}
9:foreach

i∈ℳi\\in\\mathcal\{M\}do

10:

pi←softmax⁡\(ℱθ​\(xmasked\)i\)p^\{i\}\\leftarrow\\mathrm\{softmax\}\(\\mathcal\{F\}\_\{\\theta\}\(x\_\{\\mathrm\{masked\}\}\)^\{i\}\)
11:

ℓCEi=CrossEntropy​\(pi,xi\)\\ell^\{i\}\_\{\\mathrm\{CE\}\}=\\text\{CrossEntropy\}\(p^\{i\},x^\{i\}\)⊳\\trianglerightper\-token CE against ground truth

12:

Oϕi←ℛϕ​\(H~​\(pi\),x^​\(pi\)\)O^\{i\}\_\{\\phi\}\\leftarrow\\mathcal\{R\}\_\{\\phi\}\(\\tilde\{H\}\(p^\{i\}\),\\hat\{x\}\(p^\{i\}\)\)
13:endfor

14:

ℒwCE=1\|ℳ\|​∑i∈ℳσ⁡\(Oϕi\)⋅ℓCEi\\mathcal\{L\}\_\{\\mathrm\{wCE\}\}=\\frac\{1\}\{\|\\mathcal\{M\}\|\}\\displaystyle\\sum\_\{i\\in\\mathcal\{M\}\}\\sigma\(O^\{i\}\_\{\\phi\}\)\\cdot\\ell^\{i\}\_\{\\mathrm\{CE\}\}
15:

ℒReg=1\|ℳ\|​∑i∈ℳσ⁡\(Oϕi\)\\mathcal\{L\}\_\{\\mathrm\{Reg\}\}=\\frac\{1\}\{\|\\mathcal\{M\}\|\}\\,\\displaystyle\\sum\_\{i\\in\\mathcal\{M\}\}\\sigma\(O^\{i\}\_\{\\phi\}\)
16:

ℒLBS←ℒwCE−ℒReg\\mathcal\{L\}\_\{\\mathrm\{LBS\}\}\\leftarrow\\mathcal\{L\}\_\{\\mathrm\{wCE\}\}\-\\mathcal\{L\}\_\{\\mathrm\{Reg\}\}
17:

ℛϕ←ℛϕ−ηlr​∇ϕℒLBS\\mathcal\{R\}\_\{\\phi\}\\leftarrow\\mathcal\{R\}\_\{\\phi\}\-\\eta\_\{\\mathrm\{lr\}\}\\,\\nabla\_\{\\phi\}\\,\\mathcal\{L\}\_\{\\mathrm\{LBS\}\}⊳\\trianglerightupdate only router;ℱθ\\mathcal\{F\}\_\{\\theta\}is frozen

18:endfor

19:return

ℛϕ\\mathcal\{R\}\_\{\\phi\}

Training stability\.Tab\.[10](https://arxiv.org/html/2609.16450#A3.T10)reports the training and validation losses of the LBS router\. The training loss is negative because Eq\. \([3](https://arxiv.org/html/2609.16450#S5.E3)\) subtracts the regularization term from the weighted cross\-entropy loss\. Both losses drop sharply in the first epoch, after which the changes are much smaller, indicating stable training\.

Table 10:Training and validation losses of the LBS router across epochs\.

## Appendix DDetailed Setup for Learnable Parallel Sampling

### D\.1Router Architecture

The LPS routerℛψ\\mathcal\{R\}\_\{\\psi\}is a small Transformer encoder that operates over all positions of the active block jointly\. Taking the per\-position input\(fi,posi\)\(f^\{i\},\\mathrm\{pos\}^\{i\}\)defined in Sec\.[5\.2](https://arxiv.org/html/2609.16450#S5.SS2), the input embeddingZψZ\_\{\\psi\}is a linear projection followed by LayerNorm and GELU, mapping each feature vector into a hidden space of dimensiondψ=32d\_\{\\psi\}=32, so thatZψ∈ℝ\|ℬ\|×32Z\_\{\\psi\}\\in\\mathbb\{R\}^\{\|\\mathcal\{B\}\|\\times 32\}\. The encoder then processesZψZ\_\{\\psi\}with22pre\-norm Transformer blocks with44attention heads, a feed\-forward width of6464, GELU activations, and dropout of0\.10\.1\. A final linear head projects each hidden state to a scalar logitOψiO^\{i\}\_\{\\psi\}, from which the commit probability ispψi=σ⁡\(Oψi\)p^\{i\}\_\{\\psi\}=\\sigma\(O^\{i\}\_\{\\psi\}\), giving a total of roughly2727k trainable parameters\. This keeps the router cost negligible compared with the base dLLM forward pass\.

### D\.2Trace Generation

To remove any train/test distribution skew, we collect traces under the exact decoding configuration that the router will face at inference\. For each experiment we run the corresponding frozen base dLLM with generation length512512, base block size3232, and confidence thresholdτ=0\.9\\tau\{=\}0\.9, matching the inference setting in Sec\.[6\.1](https://arxiv.org/html/2609.16450#S6.SS1)\. Since the distribution differs across base models, we train a separate router for each base model\. Every record captures, at each positioniiof the active block, the four feature componentsfti=\(κti,H~ti,Δti,ρt\)f^\{i\}\_\{t\}=\(\\kappa^\{i\}\_\{t\},\\tilde\{H\}^\{i\}\_\{t\},\\Delta^\{i\}\_\{t\},\\rho\_\{t\}\), the predicted tokenx^ti\\hat\{x\}^\{i\}\_\{t\}, the correctness labelyti=𝟏\[x^ti=x0i\]y^\{i\}\_\{t\}=\\mathbf\{1\}\[\\hat\{x\}^\{i\}\_\{t\}=x^\{i\}\_\{0\}\]obtained by comparing against the completed sequencex0x\_\{0\}, and the mask indicatormtim^\{i\}\_\{t\}which shows which position is still masked\. The replay pass follows the oracle progress rule described in Sec\.[5\.2](https://arxiv.org/html/2609.16450#S5.SS2), so each intermediate state stays consistent withx0x\_\{0\}and the labelsytiy^\{i\}\_\{t\}provide a clean supervision signal without external annotation\.

Table 11:Training and validation losses and validation recall of the LPS router across epochs\.Algorithm 4LPS Router Training1:Frozen dLLM

ℱθ\\mathcal\{F\}\_\{\\theta\}; prompt corpus

𝒟\\mathcal\{D\}; LBS Router

ℛϕ\\mathcal\{R\}\_\{\\phi\}; LPS Router

ℛψ\\mathcal\{R\}\_\{\\psi\}; negative weight

w−=2w^\{\-\}=2; target False\-Positive rate

η∈\(0,1\)\\eta\\\!\\in\\\!\(0,1\); learning rate

ηlr\\eta\_\{\\mathrm\{lr\}\}; early\-stop patience

PpatienceP\_\{\\mathrm\{patience\}\}\.

2:Router parameters

ℛψ\\mathcal\{R\}\_\{\\psi\}and acceptance threshold

τψ\\tau\_\{\\psi\}\.

3:Stage 1: build the trace dataset𝒯\\mathcal\{T\}\.

4:

𝒯←∅\\mathcal\{T\}\\leftarrow\\emptyset⊳\\triangleright𝒯\\mathcal\{T\}stores per\-position records\(fti,posi,yti\)\(f^\{i\}\_\{t\},\\,\\mathrm\{pos\}^\{i\},\\,y^\{i\}\_\{t\}\)\.

5:for allprompt

q∈𝒟q\\in\\mathcal\{D\}do⊳\\trianglerightRun LBS decoding onqqto obtainx0x\_\{0\}and the per\-step trajectory\.

6:for allstep

ttand masked position

i∈ℬi\\in\\mathcal\{B\}at step

ttdo

7:

pti←softmax​\(ℱθ​\(𝐱t\)\)ip^\{i\}\_\{t\}\\leftarrow\\mathrm\{softmax\}\\bigl\(\\mathcal\{F\}\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\)\\bigr\)\_\{i\}⊳\\trianglerightpredictive distribution at positioniifrom𝐱t\\mathbf\{x\}\_\{t\}\.

8:

κti←maxv⁡pti​\(v\)\\kappa^\{i\}\_\{t\}\\leftarrow\\max\_\{v\}p^\{i\}\_\{t\}\(v\),

x^ti←arg⁡maxv​pti​\(v\)\\hat\{x\}^\{i\}\_\{t\}\\leftarrow\\arg\\max\_\{v\}p^\{i\}\_\{t\}\(v\)⊳\\trianglerighttop\-1 confidence and predicted token\.

9:

H~ti,Δti,ρt←\\tilde\{H\}^\{i\}\_\{t\},\\,\\Delta^\{i\}\_\{t\},\\,\\rho\_\{t\}\\leftarrowentropy, top\-1/top\-2 gap, block\-wise mask ratio

10:

fti←\(κti,H~ti,Δti,ρt\)f^\{i\}\_\{t\}\\leftarrow\(\\kappa^\{i\}\_\{t\},\\,\\tilde\{H\}^\{i\}\_\{t\},\\,\\Delta^\{i\}\_\{t\},\\,\\rho\_\{t\}\),

posi←\(i−b\)/\|ℬ\|\\mathrm\{pos\}^\{i\}\\leftarrow\{\\color\[rgb\]\{0,0,0\}\\nicefrac\{\{\(i\-b\)\}\}\{\{\|\\mathcal\{B\}\|\}\}\}
11:

yti←𝟏\[x^ti=x0i\]y^\{i\}\_\{t\}\\leftarrow\\mathbf\{1\}\[\\hat\{x\}^\{i\}\_\{t\}=x^\{i\}\_\{0\}\]⊳\\trianglerightcorrectness label vs\.x0x\_\{0\}\.

12:Append

\(fti,posi,yti\)\(f^\{i\}\_\{t\},\\,\\mathrm\{pos\}^\{i\},\\,y^\{i\}\_\{t\}\)to

𝒯\\mathcal\{T\}
13:endfor

14:endfor

15:Stage 2: LPS routerℛψ\\mathcal\{R\}\_\{\\psi\}training\.

16:repeat

17:for allmini\-batch

𝒮⊂𝒯train\\mathcal\{S\}\\subset\\mathcal\{T\}\_\{\\mathrm\{train\}\}do

18:

Oψi←ℛψ​\(fti,posti\)O^\{i\}\_\{\\psi\}\\leftarrow\\mathcal\{R\}\_\{\\psi\}\(f^\{i\}\_\{t\},\\,\\mathrm\{pos\}^\{i\}\_\{t\}\)for each record

\(fti,posi,yti\)∈𝒮\(f^\{i\}\_\{t\},\\,\\mathrm\{pos\}^\{i\},\\,y^\{i\}\_\{t\}\)\\in\\mathcal\{S\}
19:

ℒLPS←1\|𝒮\|​∑\(i,t\)∈𝒮wti​BCE​\(σ⁡\(Oψi\),yti\)\{\\color\[rgb\]\{0,0,0\}\\mathcal\{L\}\_\{\\mathrm\{LPS\}\}\}\\leftarrow\\dfrac\{1\}\{\|\\mathcal\{S\}\|\}\\displaystyle\\sum\_\{\{\\color\[rgb\]\{0,0,0\}\(i,t\)\}\\in\\mathcal\{S\}\}\{\\color\[rgb\]\{0,0,0\}w^\{i\}\_\{t\}\}\\,\\mathrm\{BCE\}\(\\sigma\(O^\{i\}\_\{\\psi\}\),\\,y^\{i\}\_\{t\}\), where

wti=\{1,yti=1w−,yti=0\{\\color\[rgb\]\{0,0,0\}w^\{i\}\_\{t\}\}=\\begin\{cases\}1,&y^\{i\}\_\{t\}=1\\\\ w^\{\-\},&y^\{i\}\_\{t\}=0\\end\{cases\}⊳\\trianglerightmini\-batch form ofEq\.\([4](https://arxiv.org/html/2609.16450#S5.E4)\)

20:

ℛψ←ℛψ−ηlr∇ℒLPS\\mathcal\{R\}\_\{\\psi\}\\leftarrow\\mathcal\{R\}\_\{\\psi\}\-\\eta\_\{\\mathrm\{lr\}\}\\,\\nabla\{\\color\[rgb\]\{0,0,0\}\\mathcal\{L\}\_\{\\mathrm\{LPS\}\}\}
21:endfor

22:Evaluate validation recall on

𝒯val\\mathcal\{T\}\_\{\\mathrm\{val\}\}under

FP≤η\\mathrm\{FP\}\\leq\\eta
23:untilvalidation recall has not improved for

PpatienceP\_\{\\mathrm\{patience\}\}consecutive epochs

24:get

𝒯vals\.t\.\|FP𝒯val​\(τψ\)−η\|≤δ\\mathcal\{T\}\_\{\\mathrm\{val\}\}\\quad\\text\{s\.t\.\}\\quad\|\\mathrm\{FP\}\_\{\\mathcal\{T\}\_\{\\mathrm\{val\}\}\}\(\\tau\_\{\\psi\}\)\-\\eta\|\\leq\\delta⊳\\trianglerightwithinη±0\.5\\eta\\pm 0\.5pp false\-positive rate on validation set\.

25:return

\(ℛψ,τψ\)\(\\mathcal\{R\}\_\{\\psi\},\\,\\tau\_\{\\psi\}\)

### D\.3Training Procedure

Valid trajectories are split at the trace level into80%80\\%training,10%10\\%validation, and10%10\\%test sets, with the split deterministic under seed4242so that every router variant sees identical data\. The positions outside the active block are masked out of both the loss and the attention computation to prevent boundary artifacts from contaminating the supervision\.

Optimization uses AdamW with an initial learning rate of1​e−31\\mathrm\{e\}\{\-3\}, weight decay0\.010\.01, gradient clipping at norm1\.01\.0, and automatic mixed precision\. The schedule consists of a linear warmup over0\.5%0\.5\\%of the total steps followed by a per\-epoch geometric decay with ratio0\.970\.97\. We train with a batch size of512512step records for up to20002000epochs, with early stopping triggered when the validation recall at the false\-positive budget fails to improve for100100consecutiveepochs\.

The training objective is the masked weighted binary cross\-entropy defined in Eq\.\([4](https://arxiv.org/html/2609.16450#S5.E4)\), where negative samples carry a weight ofw−=2w^\{\-\}=2\. We use asymmetric weights because a false positive commits a wrong token irreversibly, while a false negative only delays acceptance to a later step\. Premature acceptance therefore has a higher cost\.

We tune the LPS thresholdτψ\\tau\_\{\\psi\}on the validation set at the end of each improving epoch by searching over a250250\-point grid in\[0\.8,0\.9999\]\[0\.8,\\,0\.9999\]and keeping the largest recall under a false\-positive\-rate budget ofη=10−2\\eta=10^\{\-2\}\. The full procedure is summarized in Alg\.[4](https://arxiv.org/html/2609.16450#alg4)\.

Training stability\.As shown in Tab\.[11](https://arxiv.org/html/2609.16450#A4.T11), the training and validation losses converge smoothly and the validation recall rises steadily\. The validation loss stays close to the training loss throughout, indicating no overfitting\.

## Appendix EEB\-Decode Inference Procedure

The complete inference procedure that combines LBS withLPS is given in Alg\.[5](https://arxiv.org/html/2609.16450#alg5)\. The simplified high\-level view is in Alg\.[2](https://arxiv.org/html/2609.16450#alg2)\.

Algorithm 5Inference with LBS Block Selection and LPS Parallel Commit1:Prompt

pp, frozen dLLM

ℱθ\\mathcal\{F\}\_\{\\theta\}; LBS router

ℛϕ\\mathcal\{R\}\_\{\\phi\}; LPS router

ℛψ\\mathcal\{R\}\_\{\\psi\}; generation length

LL; min block size

LminL\_\{\\min\}; fallback block size

LBL\_\{B\}; max gap

gmaxg\_\{\\max\}; confidence threshold

τ\\tau; LPS acceptance threshold

τψ\\tau\_\{\\psi\}
2:

x←\[p\|\[MASK\],…,\[MASK\]⏟L\]x\\leftarrow\[p\\;\|\\;\\underbrace\{\\texttt\{\[MASK\]\},\\ldots,\\texttt\{\[MASK\]\}\}\_\{L\}\];

ℬ←∅\\mathcal\{B\}\\leftarrow\\emptyset
3:while

∃i\\exists\\,isuch that

xi=\[MASK\]x^\{i\}=\\texttt\{\[MASK\]\}do

4:

pt←softmax⁡\(ℱθ​\(x\)\)p\_\{t\}\\leftarrow\\mathrm\{softmax\}\\bigl\(\\mathcal\{F\}\_\{\\theta\}\(x\)\\bigr\);

ℳ←\{i:xi=\[MASK\]\}\\mathcal\{M\}\\leftarrow\\\{i:x^\{i\}=\\texttt\{\[MASK\]\}\\\}⊳\\trianglerightget distribution and masked places

5:foreach

i∈ℳi\\in\\mathcal\{M\}do

6:

κi←max⁡pti\\kappa^\{i\}\\leftarrow\\max p^\{i\}\_\{t\};

x^i←arg⁡max⁡pti\\hat\{x\}^\{i\}\\leftarrow\\arg\\max p^\{i\}\_\{t\}
7:Compute

H~i\\tilde\{H\}^\{i\},

Δi\\Delta^\{i\},

ρ\\rho;⊳\\trianglerightcompute entropy, top\-1/top\-2 gap, block\-wise mask ratio

8:

fi←\(κi,H~i,Δi,ρ\)f^\{i\}\\leftarrow\(\\kappa^\{i\},\\,\\tilde\{H\}^\{i\},\\,\\Delta^\{i\},\\,\\rho\)⊳\\trianglerightLPS feature tuple

9:endfor

10:if

ℬ=∅\\mathcal\{B\}=\\emptysetthen⊳\\trianglerightLBS: select a new active block

11:if

\|ℳ\|≤Lmin\|\\mathcal\{M\}\|\\leq L\_\{\\min\}then

12:

ℬ←\\mathcal\{B\}\\leftarrowℳ\\mathcal\{M\}⊳\\trianglerightsize fallback

13:else

14:

𝒞←\{i∈ℳ:σ⁡\(ℛϕ​\(H~i,x^i\)\)\>0\.5∧x^i≠EOS\}\\mathcal\{C\}\\leftarrow\\\{i\\in\\mathcal\{M\}:\\sigma\(\\mathcal\{R\}\_\{\\phi\}\(\\tilde\{H\}^\{i\},\\,\\hat\{x\}^\{i\}\)\)\>0\.5\\ \\wedge\\ \\hat\{x\}^\{i\}\\neq\\text\{EOS\}\\\}⊳\\trianglerightrouter \+ EOS removal

15:

ℬ←MaxGapFilter⁡\(𝒞,gmax\)\\mathcal\{B\}\\leftarrow\\mathrm\{MaxGapFilter\}\(\\mathcal\{C\},\\,g\_\{\\max\}\)⊳\\trianglerightspatial coherence

16:if

\|ℬ\|<Lmin\|\\mathcal\{B\}\|<L\_\{\\min\}then

17:

ℬ←\\mathcal\{B\}\\leftarrowfirst

LBL\_\{B\}positions of

ℳ\\mathcal\{M\}⊳\\trianglerightmin\-tokens fallback

18:endif

19:endif

20:endif

21:

Oψi←ℛψ​\(fi,\(i−b\)/\|ℬ\|\)O^\{i\}\_\{\\psi\}\\leftarrow\\mathcal\{R\}\_\{\\psi\}\(f^\{i\},\\,\{\\color\[rgb\]\{0,0,0\}\\nicefrac\{\{\(i\-b\)\}\}\{\{\|\\mathcal\{B\}\|\}\}\}\)for

i∈ℬi\\in\\mathcal\{B\}⊳\\trianglerightLPS router,bbis the left boundary index ofℬ\\mathcal\{B\}

22:

𝒜←\{i∈ℬ:xi=\[MASK\]∧\(κi≥τ∨pψi\>τψ\)\}\\mathcal\{A\}\\leftarrow\\bigl\\\{i\\in\\mathcal\{B\}:x^\{i\}=\\texttt\{\[MASK\]\}\\,\\wedge\\,\\bigl\(\\kappa^\{i\}\\geq\\tau\\,\\vee\\,\{\\color\[rgb\]\{0,0,0\}p^\{i\}\_\{\\psi\}\}\>\\tau\_\{\\psi\}\\bigr\)\\bigr\\\}⊳\\trianglerightunion accept rule

23:if

𝒜=∅\\mathcal\{A\}=\\emptysetthen

24:

𝒜←\{arg⁡maxi∈ℬ,xi=\[MASK\]⁡κi\}\\mathcal\{A\}\\leftarrow\\\{\\arg\\max\_\{i\\in\\mathcal\{B\},\\,x^\{i\}=\\texttt\{\[MASK\]\}\}\\kappa^\{i\}\\\}⊳\\trianglerightanti\-stall fallback

25:endif

26:

xi←x^ix^\{i\}\\leftarrow\\hat\{x\}^\{i\}for

i∈𝒜i\\in\\mathcal\{A\}⊳\\trianglerightparallel commit

27:if

xi≠\[MASK\]x^\{i\}\\neq\\texttt\{\[MASK\]\}for all

i∈ℬi\\in\\mathcal\{B\}then

28:

ℬ←∅\\mathcal\{B\}\\leftarrow\\emptyset⊳\\trianglerightIfℬ\\mathcal\{B\}resolved, move to Line 9

29:endif

30:endwhile

31:return

xx

## Appendix FExperimental Setup

Decoding configuration\.All methods share the same inference configuration of generation length512512, base block size3232, and base confidence thresholdτ=0\.9\\tau\{=\}0\.9\.No method in Tab\.[1](https://arxiv.org/html/2609.16450#S6.T1)uses a KV cache\. In the official implementation of Fast\-dLLM\[[53](https://arxiv.org/html/2609.16450#bib.bib53)\], the KV cache is recomputed with one full\-sequence forward pass at the start of each block and reused within that block, whereas the vanilla LLaDA decoding loop maintains no cache\.

EB\-Decode hyperparameters\.For LBS in our EB\-Decode, we setgmax=5g\_\{\\max\}=5for allthreedLLMs, andLmin=8,8,16L\_\{\\min\}=8,8,16for LLaDA\-8B\-Instruct, Dream\-v0\-Instruct\-7B, LLaDA\-1\.5 respectively\. When not paired with LPS, LBS adopts confidence\-based decoding following Fast\-dLLM\[[53](https://arxiv.org/html/2609.16450#bib.bib53)\]\. For LPS, we choose the acceptance thresholdτψ\\tau\_\{\\psi\}under a false\-positive\-rate budget of1%±0\.51\\%\\pm 0\.5% points on the validation set, givingτψ=0\.89\\tau\_\{\\psi\}=0\.89,0\.900\.90, and0\.870\.87for LLaDA\-8B\-Instruct, Dream\-v0\-Instruct\-7B, LLaDA\-1\.5 respectively\.

Throughput measurement\.All methods, including every baseline, are measured with the HuggingFacetransformersbackend at batch size11, following prior dLLM acceleration works\[[35](https://arxiv.org/html/2609.16450#bib.bib35),[4](https://arxiv.org/html/2609.16450#bib.bib4),[9](https://arxiv.org/html/2609.16450#bib.bib9)\], and throughput \(TPS\) is measured per GPU as the number of generated tokens divided by the end\-to\-end latency\.

Hardware\.All runs are performed on4×4\\timesNVIDIA H100 GPUs unless specified otherwise\.The experiments in Tabs\.[6](https://arxiv.org/html/2609.16450#S6.T6)and[7](https://arxiv.org/html/2609.16450#S6.T7)are run on4×4\\timesNVIDIA A100 GPUs, and those in Tab\.[8](https://arxiv.org/html/2609.16450#S6.T8)on NVIDIA A100 GPUs with a maximum generation length of512512\.

## Appendix GPositioning Against Prior Adaptive dLLM Decoding Methods

To make explicit how EB\-Decode differs from prior adaptive dLLM decoding methods, Tab\.[12](https://arxiv.org/html/2609.16450#A7.T12)compares them along two axes:*where to decode*, i\.e\., how the block is formed, and*when to commit*, i\.e\., how tokens are accepted\. Prior methods either fix the block size or form contiguous blocks with hand\-designed heuristics, and most of them commit tokens with a static confidence threshold\. EB\-Decode is the only one that learns both axes and supports non\-contiguous, variable\-length blocks, so hard tokens can be deferred and resolved with richer right\-side context once the surrounding easy tokens are decoded \(Fig\.[1](https://arxiv.org/html/2609.16450#S1.F1)\)\.

Table 12:Positioning against prior dLLM decoding methods along the two decoding axes\.
## Appendix HEntropy at the Sequence Tail

Fig\.[2](https://arxiv.org/html/2609.16450#S4.F2)\(a\) shows a persistent band of low entropy at high token positions across all decoding steps, which may appear at odds with the overall rising\-entropy trend\. This region corresponds to the EOS zone at the tail of the sequence\. We report in Tab\.[13](https://arxiv.org/html/2609.16450#A8.T13)the three most frequent predictions at the last five positions across all decoding steps, where<\|endoftext\|\>is overwhelmingly dominant, appearing in the top\-5 predictions at96\.30%96\.30\\%of the positions sampled\. These positions therefore remain highly certain at almost every step, forming the persistent low\-entropy band\.

Table 13:Top\-3 predicted tokens at the last55positions of the sequence, aggregated over all decoding steps on LLaDA\-8B\-Instruct\. “Top\-5 occurrence” is the fraction of sampled positions at which the token appears among the top\-5 predictions\.
## Appendix ILimitations

We outline the main limitations of EB\-Decode and the directions that we leave for future work\.

Generation Length\.Our main results are reported at generation lengths of 256 and 512 tokens, which match the standard configurations used in prior dLLM acceleration works\[[53](https://arxiv.org/html/2609.16450#bib.bib53),[35](https://arxiv.org/html/2609.16450#bib.bib35),[4](https://arxiv.org/html/2609.16450#bib.bib4)\]\. As shown in Tab\.[5](https://arxiv.org/html/2609.16450#S6.T5), EB\-Decode already exhibits a scaling trend in which the speedup grows fromL=256L\{=\}256toL=512L\{=\}512, suggesting that the framework remains effective as the sequence becomes longer\. A more comprehensive study at substantially longer generation lengths is a natural extension we plan to pursue in future work\.

Serving Conditions\.Our throughput is measured at batch size11, following prior dLLM acceleration works\[[53](https://arxiv.org/html/2609.16450#bib.bib53),[35](https://arxiv.org/html/2609.16450#bib.bib35),[4](https://arxiv.org/html/2609.16450#bib.bib4)\]\. Evaluation under realistic serving conditions with continuous batching and system\-level optimizations is left to future work, as it depends as much on dLLM serving infrastructure as on the decoding algorithm\.

Method Complexity\.EB\-Decode introduces two routers and several hyperparameters, namelygmaxg\_\{\\max\},LminL\_\{\\min\},τ\\tau, andτψ\\tau\_\{\\psi\}, which makes it more involved to deploy and reuse as a baseline than a single\-threshold heuristic\. Each component contributes to the final performance \(Tabs\.[3](https://arxiv.org/html/2609.16450#S6.T3)–[4](https://arxiv.org/html/2609.16450#S6.T4)\), and the trained routers are largely reusable without retraining \(Sec\.[6\.4](https://arxiv.org/html/2609.16450#S6.SS4)\), but reducing the number of moving parts remains a worthwhile direction\.