Ripple-Pivot Search: Active Parallel Decoding for Diffusion Large Language Models

# Active Parallel Decoding for Diffusion Large Language Models
Source: [https://arxiv.org/html/2608.11742](https://arxiv.org/html/2608.11742)
Yushi Ye Xu Chen Haoyun Jiang Jinsong Lan Haihong TangAffiliation:Cooperative Medianet Innovation Center, Shanghai Jiao Tong UniversityAffiliation:Cooperative Medianet Innovation Center, Shanghai Jiao Tong UniversityBo HanIvor TsangYanfeng WangBo ZhengJiangchao YaoAffiliation:Cooperative Medianet Innovation Center, Shanghai Jiao Tong UniversityAffiliation:Alibaba Group TMLR Group, Department of Computer Science, Hong Kong Baptist UniversityAffiliation:A\*STAR CFAR and Nanyang Technological UniversityAffiliation:School of Artificial Intelligence, Shanghai Jiao Tong University\{stephen\-ye, Sunarker\}@sjtu\.edu\.cn

###### Abstract

Diffusion Large Language Models \(dLLMs\) have emerged as a competitive alternative to autoregressive language models, offering the potential for substantially faster inference through parallel decoding\. Existing parallel decoding schedulers typically commit positions only after they meet a per\-position criterion, overlooking how early commitments may benefit subsequent decoding\. We identify a ripple effect in dLLM decoding: proactively committing a*mid\-entropy*pivot position can induce a pronounced reduction in uncertainty across the remaining masked positions\. This uncertainty reduction allows subsequent steps to unmask more tokens in parallel, thereby accelerating the overall decoding process\. To exploit the ripple effect, we propose Ripple\-Pivot Search \(RPS\), a novel training\-free decoding method that seeks mid\-entropy positions as promising candidate pivots \(*where to decode*\), and determines their token assignment that yields the greatest downstream benefit via lookahead evaluation \(*what to decode*\)\. Across 3 dLLMs and 4 reasoning and code\-generation benchmarks, RPS achieves 4–10×\\timeswall\-clock speedup over the standard decoder while preserving generation quality, and improves accuracy over the previous lookahead baseline by up to 5\.49% while delivering higher throughput in most settings\. When integrated with KV caching, RPS further achieves up to 18×\\timeswall\-clock speedup over the standard decoder\.

## 1Introduction

Diffusion large language models \(dLLMs\)\([15](https://arxiv.org/html/2608.11742#bib.bib6);[24](https://arxiv.org/html/2608.11742#bib.bib5);[21](https://arxiv.org/html/2608.11742#bib.bib2);[7](https://arxiv.org/html/2608.11742#bib.bib12);[4](https://arxiv.org/html/2608.11742#bib.bib13)\)have gained prominence as a viable alternative to autoregressive language models\([5](https://arxiv.org/html/2608.11742#bib.bib9);[20](https://arxiv.org/html/2608.11742#bib.bib10);[23](https://arxiv.org/html/2608.11742#bib.bib11)\), offering the potential for faster inference by decoding multiple tokens in parallel\. In each denoising step, the model produces predictions at all masked positions simultaneously, and a decoding scheduler selects which positions to unmask and assigns tokens at those positions\. In practice, however, committing more positions per step increases the risk of error accumulation\([12](https://arxiv.org/html/2608.11742#bib.bib14)\), making the balance between decoding speed and generation quality a central challenge in dLLM inference\.

To navigate this speed\-quality trade\-off, a growing body of work on parallel decoding\([10](https://arxiv.org/html/2608.11742#bib.bib8);[18](https://arxiv.org/html/2608.11742#bib.bib4);[14](https://arxiv.org/html/2608.11742#bib.bib15);[22](https://arxiv.org/html/2608.11742#bib.bib1);[25](https://arxiv.org/html/2608.11742#bib.bib25)\)has focused on designing schedulers that commit multiple positions per step under reliability constraints, thereby amortizing the cost of each forward pass\. The scheduler is thus responsible for two decisions:*where*to unmask and*what*token to assign\. Most schedulers couple the two, committing each position to its greedy prediction once a per\-position criterion is satisfied, such as sufficient confidence\([17](https://arxiv.org/html/2608.11742#bib.bib3);[15](https://arxiv.org/html/2608.11742#bib.bib6)\), low predictive entropy\([21](https://arxiv.org/html/2608.11742#bib.bib2)\), cross\-step stability\([11](https://arxiv.org/html/2608.11742#bib.bib23)\), or bounded cumulative entropy\([3](https://arxiv.org/html/2608.11742#bib.bib22)\)\. More recent lookahead\-based methods\([19](https://arxiv.org/html/2608.11742#bib.bib7);[9](https://arxiv.org/html/2608.11742#bib.bib20)\)further test whether committing an additional position can benefit subsequent decoding\. However, these methods employ lookahead mainly to determine whether and where to commit\. Once a position is selected, its token is typically fixed to the model’s current top\-1 prediction\. As a result, the search explores different commitment positions but not alternative token assignments, potentially overlooking effective non\-greedy decoding trajectories\.

Method familyWhere to decodeWhat to decodeConfidence\-based\([17](https://arxiv.org/html/2608.11742#bib.bib3);[15](https://arxiv.org/html/2608.11742#bib.bib6)\)Confidence\-qualified
positionsTop\-1 \(greedy\)Entropy\-based\([21](https://arxiv.org/html/2608.11742#bib.bib2);[3](https://arxiv.org/html/2608.11742#bib.bib22)\)Entropy\-qualified
positionsTop\-1 \(greedy\)Stability\-based\([11](https://arxiv.org/html/2608.11742#bib.bib23);[16](https://arxiv.org/html/2608.11742#bib.bib24)\)Cross\-step stable
positionsTop\-1 \(greedy\)Lookahead\-based\([19](https://arxiv.org/html/2608.11742#bib.bib7);[9](https://arxiv.org/html/2608.11742#bib.bib20)\)Lookahead\-selected
positionsTop\-1 \(greedy\)RPS \(ours\)Mid\-entropy
pivotLookahead selected
\(non\-greedy\)![Refer to caption](https://arxiv.org/html/2608.11742v1/figure1a.png)Figure 1:Left:Comparison of parallel decoding schedulers\.Right:Starting from a partially decoded block, we commit each remaining masked position to its oracle token and measure the resulting entropy reductionΔ​Hi\\Delta H\_\{i\}at other masked positions \(y\-axis\) against the pre\-commit entropyHbeforeH\_\{\\mathrm\{before\}\}of the resolved position \(x\-axis\)\. Colors indicate the oracle token’s rank under the model distribution, and the shaded band marks the mid\-entropy regime\.The benefit of early commitment depends critically on which unresolved position is selected\. To characterize this, we conduct an oracle analysis over candidate commitments in Fig\.[1](https://arxiv.org/html/2608.11742#S1.F1)\(right\)\. This analysis reveals a distinct pattern that we refer to as the*ripple effect*: proactively committing a pivot position in themid\-entropy regimeinduces the strongest downstream uncertainty reduction\. Intuitively, such positions are not fully determined, but are already sufficiently tied to the current partial decoding state; resolving them can therefore influence other masked positions more strongly than positions that are either already certain or still weakly constrained\. Moreover, the correct token is not the model’s top\-1 prediction in 85% of mid\-entropy cases, revealing a mismatch with existing lookahead\-based schedulers\. As shown in Fig\.[1](https://arxiv.org/html/2608.11742#S1.F1)\(left\), these methods use lookahead to decide*where*to commit while fixing*what*to the current top\-1 prediction\. They therefore search over commitment positions but not token assignments, potentially missing beneficial non\-greedy trajectories\.

Motivated by these findings, we proposeRipple\-Pivot Search \(RPS\), a training\-free parallel decoding method that exploits beneficial early commitments in the mid\-entropy regime\. RPS first applies a two\-stage pivot filter to exclude positions that are either overly certain or insufficiently supported by the current predictive distribution, thereby focusing in the mid\-entropy regime\. Given this pivot, RPS then constructs an adaptive set of plausible token assignments, evaluates their downstream benefits with a single lookahead forward pass, and selects the most promising token for early commitment\. This allows RPS to choose the assignment expected to best amplify the ripple effect\.

Furthermore, to prevent premature commitments from accumulating errors, RPS commits a token only when it improves over leaving the pivot unmasked\. In summary, our contributions are:

- •We identify a*ripple effect*in dLLM decoding: proactively committing a position in the mid\-entropy regime induces the strongest downstream uncertainty reduction, enabling more parallel commits in subsequent steps\. Curicially, the correct token in this regime is frequently not the current top\-1 prediction, motivating token assignment beyond the greedy decoding\.
- •We propose Ripple\-Pivot Search \(RPS\), a novel training\-free decoding method that seeks mid\-entropy pivots with high potential for downstream uncertainty reduction, and determines its token assignment through lookahead\-based evaluation on an adaptive candidate set, committing when the best candidate outperforms leaving the pivot unmasked\.
- •We conduct extensive experiments on three dLLMs across four reasoning and code\-generation benchmarks\. RPS achieves a4–10×\\timesinference speedup over the standard one\-token\-per\-step baseline while preserving generation quality, and improves accuracy over the previous lookahead baseline by up to5\.49%while delivering higher throughput in most settings\. When integrated with KV caching, RPS further achieves up to18×\\timeswall\-clock speedup over the standard decoder\.

## 2Related Work

#### Criterion\-based Parallel Decoding\.

The most direct approach to accelerating dLLM inference commits multiple positions per decoding step based on per\-position prediction statistics\. Fast\-dLLM\([17](https://arxiv.org/html/2608.11742#bib.bib3)\)unmasks every position whose predictive confidence exceeds a fixed threshold and introduces block\-wise KV caching to reduce per\-forward cost\. KLASS\([11](https://arxiv.org/html/2608.11742#bib.bib23)\)augments confidence with a KL divergence criterion that tracks prediction stability across consecutive steps, unmasking only positions that are both confident and temporally consistent\. EB\-Sampler\([3](https://arxiv.org/html/2608.11742#bib.bib22)\)controls the number of tokens unmasked per step by bounding the cumulative entropy of the newly committed positions\. Learn2PD\([2](https://arxiv.org/html/2608.11742#bib.bib21)\)further replaces fixed heuristics with a lightweight learned filter that predicts whether each current token prediction matches the final output, yielding an adaptive criterion for parallel unmasking\.

#### Lookahead\-based Parallel Decoding\.

Lookahead\-based dLLM decoding accelerates inference by proposing token assignments and evaluating their downstream effects\. WINO\([10](https://arxiv.org/html/2608.11742#bib.bib8)\)drafts all positions admitted by a relaxed confidence threshold and evaluates the committed tokens under the enriched context, remasking those whose verification confidence falls below a stricter threshold\. LoPA\([19](https://arxiv.org/html/2608.11742#bib.bib7)\)forms lookahead branches from the highest\-confidence positions that remain masked after the standard decoding update and selects the branch with the greatest future confidence\. From an information\-theoretic perspective, ETE\([9](https://arxiv.org/html/2608.11742#bib.bib20)\)argues that prioritizing high\-confidence positions limits the information revealed per decoding round\. It therefore explores high\-information positions near a prescribed confidence level and selects the one that unlocks the most downstream high\-confidence tokens\. Yet these methods primarily use lookahead to decide*where*to commit with keeping greedy token assignments\. By contrast, RPS focuses on mid\-entropy regime and uses lookahead to jointly decide*where*to commit and*what*token to assign\.

## 3Preliminary

#### Notation\.

Consider a masked discrete diffusion language model with vocabulary𝒱\\mathcal\{V\}, which contains a special mask token\[MASK\]\[\\texttt\{MASK\}\]\. Given a promptyy, a response of lengthLLis represented asx∈𝒱Lx\\in\\mathcal\{V\}^\{L\}, where each position is either decoded or masked\. Letℳ⊆\{1,…,L\}\\mathcal\{M\}\\subseteq\\\{1,\\ldots,L\\\}denote the set of currently masked response positions\. Given the input\[y∥x\]\[y\\\|x\], the model returns predictive distributions\{pi=pθ\(⋅∣y,x\)\}i∈ℳ\\\{p\_\{i\}=p\_\{\\theta\}\(\\cdot\\mid y,x\)\\\}\_\{i\\in\\mathcal\{M\}\}over𝒱\\mathcal\{V\}in a single forward pass\. WriteH⁡\(pi\)H\(p\_\{i\}\)for the predictive entropy at positioniiandPimax≜maxv∈𝒱⁡pi​\(v\)P\_\{i\}^\{\\max\}\\triangleq\\max\_\{v\\in\\mathcal\{V\}\}p\_\{i\}\(v\)for its*confidence*, i\.e\., its top\-11probability\.

#### Decoding\.

Standard practice\([15](https://arxiv.org/html/2608.11742#bib.bib6);[4](https://arxiv.org/html/2608.11742#bib.bib13);[17](https://arxiv.org/html/2608.11742#bib.bib3)\)adopts a semi\-autoregressive schedule: the length\-LLresponse is partitioned into contiguous blocks of sizeBBand decoded left\-to\-right, fully unmasking each block before the next\. Within each block, every decoding step shares the same form\. Given the current\{pi\}i∈ℳ\\\{p\_\{i\}\\\}\_\{i\\in\\mathcal\{M\}\}, a commit set𝒮⊆ℳ\\mathcal\{S\}\\subseteq\\mathcal\{M\}is selected and each position in𝒮\\mathcal\{S\}is assigned its greedy \(top\-1\) prediction,

xi←arg⁡maxv∈𝒱​pi​\(v\)for each​i∈𝒮\.x\_\{i\}\\;\\leftarrow\\;\\arg\\max\_\{v\\in\\mathcal\{V\}\}p\_\{i\}\(v\)\\quad\\text\{for each \}i\\in\\mathcal\{S\}\.\(1\)Different schedulers are characterized by how they specify𝒮\\mathcal\{S\}\. The three most common rules are:

𝒮=\{top​\-​ki∈ℳ⁡\(\{P1max,…,PLmax\}\),highest\-confidence decoding\([15](https://arxiv.org/html/2608.11742#bib.bib6)\);top​\-​ki∈ℳ⁡\(\{−H⁡\(p1\),…,−H⁡\(pL\)\}\),lowest\-entropy decoding\([21](https://arxiv.org/html/2608.11742#bib.bib2)\);\{i∈ℳ:Pimax≥τ\},confidence\-aware decoding\([17](https://arxiv.org/html/2608.11742#bib.bib3)\)\.\\displaystyle\\mathcal\{S\}=\\begin\{cases\}\\operatorname\{top\\text\{\-\}k\}\_\{i\\in\\mathcal\{M\}\}\\big\(\\left\\\{P\_\{1\}^\{\\max\},\\dots,P\_\{L\}^\{\\max\}\\right\\\}\\big\),&\\text\{highest\-confidence decoding~\\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{DBLP:journals/corr/abs\-2502\-09992\}\{\\@@citephrase\{, \}\}\{\}\)\}\};\\\\ \\\\ \\operatorname\{top\\text\{\-\}k\}\_\{i\\in\\mathcal\{M\}\}\\big\(\\left\\\{\\\!\-H\(p\_\{1\}\),\\dots,\\\!\-H\(p\_\{L\}\)\\right\\\}\\big\),&\\text\{lowest\-entropy decoding~\\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{DBLP:journals/corr/abs\-2508\-15487\}\{\\@@citephrase\{, \}\}\{\}\)\}\};\\\\ \\\\ \\big\\\{i\\in\\mathcal\{M\}:P\_\{i\}^\{\\max\}\\geq\\tau\\big\\\},&\\text\{confidence\-aware decoding~\\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{DBLP:journals/corr/abs\-2505\-22618\}\{\\@@citephrase\{, \}\}\{\}\)\}\}\.\\\\ \\end\{cases\}\(2\)All three rules differ in how𝒮\\mathcal\{S\}is constructed but share the same token\-assignment mechanism: each committed position receives its greedy prediction \(Eq\.[1](https://arxiv.org/html/2608.11742#S3.E1)\)\. Positions not selected into𝒮\\mathcal\{S\}remain masked until they satisfy the criterion as decoding progresses\.

## 4Method

### 4\.1Ripple\-Pivot Search

Our RPS introduces a per\-step pivot search on top of the standard decoding process as illustrated in Fig\.[2](https://arxiv.org/html/2608.11742#S4.F2), consisting of*pivot selection*\(where to commit\) and*lookahead scoring*\(what to commit\)\.

#### Pivot selection\.

Pivot selection must balance benefit and cost: pivots should lie in the mid\-entropy regime, where early resolution is most beneficial, while keeping lookahead token search compact\. Although such positions are uncertain, the correct token typically remains among the top\-ranked candidates\. Motivated by this observation, RPS truncates each position’s support to the top\-kmaxk\_\{\\max\}tokens, discarding most vocabulary items irrelevant to the decision, which is formulated as follows

i⋆=arg⁡maxi∈ℳ:μi≥τpivot\{−∑v∈𝒯ipi\(v\)logpi\(v\)\},μi=∑v∈𝒯ipi\(v\),i^\{\\star\}=\\underset\{i\\in\\mathcal\{M\}:\\,\\mu\_\{i\}\\geq\\tau\_\{\\text\{pivot\}\}\}\{\\arg\\max\}\\left\\\{\-\\sum\_\{v\\in\\mathcal\{T\}\_\{i\}\}p\_\{i\}\(v\)\\log p\_\{i\}\(v\)\\right\\\},\\qquad\\mu\_\{i\}=\\sum\_\{v\\in\\mathcal\{T\}\_\{i\}\}p\_\{i\}\(v\),\(3\)where the truncated support𝒯i⊆𝒱\\mathcal\{T\}\_\{i\}\\subseteq\\mathcal\{V\}contains thekmaxk\_\{\\max\}tokens with the highest probabilities under predictive distributionpip\_\{i\}, andμi\\mu\_\{i\}denotes the corresponding retained probability mass\. With Eq\. \([3](https://arxiv.org/html/2608.11742#S4.E3)\), the probability\-mass constraint\(μi≥τpivot\)\(\\mu\_\{i\}\\geq\\tau\_\{\\text\{pivot\}\}\)excludes positions in the high\-entropy regime, and maximizing truncated entropy over the retained positions avoids those that are already nearly determined\. The resulting pivot therefore naturally falls in the mid\-entropy regime to propagate useful information\. If no position satisfies the probability\-mass constraint, all remaining masked positions are still in a highly uncertain state where reliable intervention is infeasible\. In this case, RPS skips pivot search at the current step and proceeds with standard decoding alone\.

Figure 2:Overview of RPS\.Left: One\-step example of RPS decoding\. The upper region illustrates pivot selection, while the lower region shows lookahead scoring\.Right: Customized attention mask for lookahead forward pass\. The packed sequence contains the shared context with candidate branches\. Light\-blue cells indicate allowed attention and blank cells indicate blocked attention\.
#### Lookahead scoring\.

With the pivoti⋆i^\{\\star\}identified, RPS then constructs an adaptive candidate set to determine the token assignment, where the set is defined as𝒞=\{v\|pi⋆\(v\)≥r∗Pi⋆max,∀v∈𝒯i⋆\}∪\{\[MASK\]\}\\mathcal\{C\}=\\left\\\{v\|p\_\{i^\{\\star\}\}\(v\)\\geq r\*P\_\{i^\{\\star\}\}^\{\\text\{max\}\},~\\forall v\\in\\mathcal\{T\}\_\{i^\{\\star\}\}\\right\\\}\\cup\\\{\[\\texttt\{MASK\}\]\\\}by retaining tokens that satisfy the reachability ratiorrrelative to the top\-1 probability\. The reason that we include\[MASK\]\[\\texttt\{MASK\}\]here is to allow the pivot to remain masked\. As shown in Fig\.[2](https://arxiv.org/html/2608.11742#S4.F2), we build a corresponding lookahead branchBcB\_\{c\}for each candidate tokenc∈𝒞c\\in\\mathcal\{C\}by assigningcctoi⋆i^\{\\star\}\(the branch induced by assigning\[MASK\]\[\\texttt\{MASK\}\]is referred to as the anchor branchBanchorB\_\{\\text\{anchor\}\}\), and all branches are evaluated jointly in a single lookahead forward pass with an isolated attention mask\. Finally, RPS selects the token assignment for the pivot based on the following equation:

c⋆=arg⁡maxc∈𝒞\{−1\|ℳ\|−1∑i∈ℳ∖\{i⋆\}H\(pic\)\+λlogpanchor\(c\)\},c^\{\\star\}=\\underset\{c\\in\\mathcal\{C\}\}\{\\arg\\max\}\\left\\\{\-\\frac\{1\}\{\|\\mathcal\{M\}\|\-1\}\\sum\_\{i\\in\\mathcal\{M\}\\setminus\\\{i^\{\\star\}\\\}\}H\\\!\\left\(p\_\{i\}^\{c\}\\right\)\+\\lambda\\log p\_\{\\text\{anchor\}\}\(c\)\\right\\\},\(4\)wherepicp\_\{i\}^\{c\}denotes the predictive distribution at positioniiunder the branch assigningccat the pivot,panchor​\(c\)p\_\{\\text\{anchor\}\}\(c\)denotes the anchor branch’s probability forccat the pivot, and is set toPanchormaxP\_\{\\text\{anchor\}\}^\{\\max\}whenc=\[MASK\]c=\[\\texttt\{MASK\}\],λ≥0\\lambda\\geq 0is the plausibility weight balancing current plausibility against future benefit\.

In Eq\. \([4](https://arxiv.org/html/2608.11742#S4.E4)\), the first addition term inside the braces is the mean entropy over the remaining masked positions of each branch, reflecting how much entropy the pivot assignment reduces\. Specifically, lower is better, signaling a stronger ripple effect for more parallel commits in subsequent steps\. The second addition term inside the braces acts as plausibility regularization to avoid trivial tokens—for instance, a premature end\-of\-sequence prediction may suppress downstream entropy simply by collapsing future uncertainty\. Moreover, as the anchor branch keeps the pivot masked,panchor​\(c\)p\_\{\\text\{anchor\}\}\(c\)is evaluated without conditioning on the assignment, providing an useful independent quality signal\.

#### Integration with standard decoding\.

At each decoding step, the standard commit rule first produces a commit set𝒮\\mathcal\{S\}and assigns greedy predictions to these positions\. RPS then performs its pivot search on the remaining masked positionsℳ\\mathcal\{M\}: it selects the pivot, constructs the anchor and candidate branches, and evaluates them jointly in a single forward pass using the branch\-isolating attention mask in Fig[2](https://arxiv.org/html/2608.11742#S4.F2)\. Each branch is isolated from all others so that it faithfully simulates what the model would predict if only that particular token were assigned at the pivot\. To avoid redundant computation, the predictive distributions from the selected branch are carried over to the next decoding step\. The full per\-step procedure is summarized in Appendix[A](https://arxiv.org/html/2608.11742#A1)\.

### 4\.2Analysis of the Lookahead Objective

We next analyze the lookahead objective in Eq\. \([4](https://arxiv.org/html/2608.11742#S4.E4)\)\. This analysis does not attempt to derive the empirically motivated pivot rule; instead, it characterizes how the two scoring terms relate to the speed–quality trade\-off after a pivot has been selected\. For brevity, write the downstream\-position countn=\|ℳ\|−1n=\|\\mathcal\{M\}\|\-1and the mean downstream entropyH¯c=1n​∑i∈ℳ∖\{i⋆\}H⁡\(pic\)\\bar\{H\}\_\{c\}=\\frac\{1\}\{n\}\\sum\_\{i\\in\\mathcal\{M\}\\setminus\\\{i^\{\\star\}\\\}\}H\(p\_\{i\}^\{c\}\)for branchcc\.

###### Proposition 1\(Entropy\-certified parallelism\)\.

Assumen\>0n\>0\. For a confidence thresholdτ∈\[1/2,1\)\\tau\\in\[1/2,1\), define the eligible\-commit countNτ\(c\)=∑i∈ℳ∖\{i⋆\}𝕀\[maxvpic\(v\)≥τ\]N\_\{\\tau\}\(c\)=\\sum\_\{i\\in\\mathcal\{M\}\\setminus\\\{i^\{\\star\}\\\}\}\\mathbb\{I\}\\\!\\left\[\\max\_\{v\}p\_\{i\}^\{c\}\(v\)\\geq\\tau\\right\]\. With binary entropyh⁡\(τ\)=−τ​log⁡τ−\(1−τ\)​log⁡\(1−τ\)h\(\\tau\)=\-\\tau\\log\\tau\-\(1\-\\tau\)\\log\(1\-\\tau\),

Nτ​\(c\)≥max⁡\{0,n−⌊n​H¯ch⁡\(τ\)⌋\}\.N\_\{\\tau\}\(c\)\\geq\\max\\\!\\left\\\{0,\\;n\-\\left\\lfloor\\frac\{n\\bar\{H\}\_\{c\}\}\{h\(\\tau\)\}\\right\\rfloor\\right\\\}\.\(5\)

Nτ​\(c\)N\_\{\\tau\}\(c\)counts the positions eligible for commitment by a confidence\-aware decoder in the next step\. Proposition[1](https://arxiv.org/html/2608.11742#Thmproposition1)establishes that reducing the mean downstream entropyH¯c\\bar\{H\}\_\{c\}monotonically tightens a certified lower bound on this number\. The future\-benefit term in Eq\. \([4](https://arxiv.org/html/2608.11742#S4.E4)\) therefore has a direct speed interpretation rather than serving only as a generic uncertainty heuristic\. The guarantee is deliberately conservative: it lower\-bounds one\-step commit opportunities but neither predicts the exact number of commits nor directly bounds end\-to\-end NFE\.

###### Proposition 2\(Plausibility\-adjusted selection margin\)\.

Letaca\_\{c\}denote the effective anchor plausibility used by the scoring rule, whereac=panchor​\(c\)a\_\{c\}=p\_\{\\text\{anchor\}\}\(c\)ifc≠\[MASK\]c\\neq\[\\texttt\{MASK\}\], otherwiseac=maxv⁡panchor​\(v\)a\_\{c\}=\\max\_\{v\}p\_\{\\text\{anchor\}\}\(v\)whenc=\[MASK\]c=\[\\texttt\{MASK\}\]\. Maximizing Eq\. \([4](https://arxiv.org/html/2608.11742#S4.E4)\) is equivalent to

c⋆=arg⁡minc∈𝒞​\{H¯c−λ​log⁡ac\}\.c^\{\\star\}=\\arg\\min\_\{c\\in\\mathcal\{C\}\}\\left\\\{\\bar\{H\}\_\{c\}\-\\lambda\\log a\_\{c\}\\right\\\}\.\(6\)Moreover, for anyc,d∈𝒞c,d\\in\\mathcal\{C\}, candidateccscores at least as high asddif and only if

H¯d−H¯c≥λ​log⁡adac\.\\bar\{H\}\_\{d\}\-\\bar\{H\}\_\{c\}\\geq\\lambda\\log\\frac\{a\_\{d\}\}\{a\_\{c\}\}\.\(7\)

Eq\. \([6](https://arxiv.org/html/2608.11742#S4.E6)\) is a Lagrangian relaxation of minimizing downstream entropy under a candidate\-surprisal budget\. Proposition[2](https://arxiv.org/html/2608.11742#Thmproposition2)makes the resulting safeguard explicit: relative to a more plausible candidate, a less plausible candidate must compensate for its plausibility deficit with a proportionally larger entropy reduction\. Whend=\[MASK\]d=\[\\texttt\{MASK\}\], the same margin governs whether RPS commits or abstains\. Together, Propositions[1](https://arxiv.org/html/2608.11742#Thmproposition1)and[2](https://arxiv.org/html/2608.11742#Thmproposition2)separate the two roles of the scoring function: the entropy term promotes certifiable next\-step parallelism, while the anchor term controls the evidence required to take that acceleration opportunity\. Appendix[B](https://arxiv.org/html/2608.11742#A2)provides both proofs\.

### 4\.3Discussion

As compared in Fig\.[1](https://arxiv.org/html/2608.11742#S1.F1)\(left\), although RPS shares the high\-level goal of accelerating dLLM decoding with lookahead evaluation, recent lookahead methods\([19](https://arxiv.org/html/2608.11742#bib.bib7);[9](https://arxiv.org/html/2608.11742#bib.bib20)\)mainly refine*where*to commit: LoPA searches high\-confidence residual positions, whereas ETE targets positions near a prescribed confidence level\. RPS instead selects mid\-entropy pivots via truncated entropy\. More importantly, unlike LoPA and ETE, which retain greedy top\-1 assignment, RPS also refines*what*to commit by lookahead evaluation over plausible token candidates, motivated by our finding that the correct token is often non\-top\-1 where the ripple effect is strongest\.

## 5Experiments

#### Setup\.

We evaluate two representative dLLM families: LLaDA family, which is trained from scratch, and Dream family, which is adapted from an autoregressive language model\. Our evaluation covers mathematical reasoning \(GSM8K\([8](https://arxiv.org/html/2608.11742#bib.bib16)\)and MATH500\([13](https://arxiv.org/html/2608.11742#bib.bib17)\)\) and code generation \(HumanEval\([6](https://arxiv.org/html/2608.11742#bib.bib19)\)and MBPP\([1](https://arxiv.org/html/2608.11742#bib.bib18)\)\)\. We compare RPS with the standard one\-token\-per\-stepDefaultdecoder and five parallel decoding or sampling baselines:Confidence\([17](https://arxiv.org/html/2608.11742#bib.bib3)\),KLASS\([11](https://arxiv.org/html/2608.11742#bib.bib23)\),EB\-Sampler\([3](https://arxiv.org/html/2608.11742#bib.bib22)\),WINO\([10](https://arxiv.org/html/2608.11742#bib.bib8)\), andLoPA\([19](https://arxiv.org/html/2608.11742#bib.bib7)\)\. For evaluation, we use lm\-evaluation\-harness111[https://github\.com/EleutherAI/lm\-evaluation\-harness](https://github.com/EleutherAI/lm-evaluation-harness)with its standard task implementations and prompting configurations: 5\-shot for GSM8K, 4\-shot for MATH500, 0\-shot for HumanEval, and 3\-shot for MBPP\. Unless stated otherwise, the generation length is 256 and the block length is 32\. We measure generation quality by accuracy and efficiency by both the number of function evaluations \(NFE\), which captures sequential model evaluations, and tokens per second \(TPS\), which captures end\-to\-end throughput\. Speedups are computed relative to Default under the same model, task, and generation length\.

#### Implementation details\.

We build on the Fast\-dLLM inference stack\([17](https://arxiv.org/html/2608.11742#bib.bib3)\)and evaluate LLaDA\-8B\-Instruct, Dream\-v0\-Instruct\-7B, and LLaDA\-1\.5, for 12 model–benchmark configurations in total\. For brevity, we refer to LLaDA\-8B\-Instruct and Dream\-v0\-Instruct\-7B as LLaDA and Dream, respectively, throughout this section\. We report the main\-text results on LLaDA and Dream, with additional results on LLaDA\-1\.5 provided in Appendix[D](https://arxiv.org/html/2608.11742#A4)\. Unless otherwise noted, RPS useskmax=10k\_\{\\max\}=10, reachability ratior=0\.1r=0\.1, and probability\-mass thresholdτpivot=0\.9\\tau\_\{\\text\{pivot\}\}=0\.9for LLaDA and 0\.95 for Dream\. We select the plausibility weightλ\\lambdafrom the interval\[0\.1,0\.5\]\[0\.1,0\.5\]identified in §[5\.2](https://arxiv.org/html/2608.11742#S5.SS2.SSS0.Px1)\. Appendix[C](https://arxiv.org/html/2608.11742#A3)reports baseline configurations and hardware details\.

### 5\.1Main Results

#### Comparison with parallel decoding baselines\.

We compare RPS with the standard one\-token\-per\-step decoder and several state\-of\-the\-art parallel decoding methods across two model families and four benchmarks, with results reported in Table[1](https://arxiv.org/html/2608.11742#S5.T1)\. RPS remains in the highest\-throughput regime among parallel decoders, delivering 4\.24–9\.80×\\timesTPS speedup over Default while largely preserving generation quality\. In several settings, acceleration even comes with improved accuracy: on LLaDA, RPS exceeds Default by 2\.2% on MBPP, and on HumanEval it improves accuracy by 1\.22 % for both LLaDA and Dream\. These results demonstrate that jointly selecting a mid\-entropy pivot and evaluating plausible token assignments with an explicit plausibility safeguard enables aggressive parallel commitment while preserving or improving generation quality in most settings\. Among the parallel baselines, LoPA is closest to RPS in decoding speed, as it likewise uses lookahead to guide early commitment\. However, LoPA consistently loses accuracy relative to Default on the code\-generation benchmarks\. The gap is especially clear on HumanEval, where RPS outperforms LoPA by 4\.27 % on LLaDA and 5\.49 % on Dream at comparable throughput\. Section[5\.3](https://arxiv.org/html/2608.11742#S5.SS3.SSS0.Px2)provides a dedicated failure\-mode analysis of this behavior\.

Table 1:Results on LLaDA and Dream\.Boldindicates the best result\.Table 2:Generation\-length robustness on LLaDA and Dream\.
#### Robustness to generation length\.

The preceding experiments establish a strong quality–efficiency trade\-off for RPS at the default generation length of 256\. We next examine whether this advantage persists across different generation lengths\. As shown in Table[2](https://arxiv.org/html/2608.11742#S5.T2), RPS maintains the strongest quality–efficiency trade\-off among the compared baselines across all evaluated lengths\. A model\-specific exception is HumanEval atL=128L=128, where all LLaDA decoders exhibit lower accuracy than at longer lengths, while Dream does not show the same degradation\. Inspection of the generated programs indicates that the LLaDA outputs are generally syntactically complete but often implement overly simplified or incomplete logic, suggesting that the restricted budget affects solution formation rather than merely truncating the code\. Under this constraint, LoPA’s aggressive early commitments further amplify the quality loss, reducing accuracy from 30\.49 for Default to 22\.56\. In contrast, RPS retains 28\.66 accuracy while achieving 6\.49×\\timesNFE and 5\.26×\\timesTPS speedups, limiting the additional degradation caused by parallel decoding\.

### 5\.2Hyperparameter Ablations

#### Plausibility\-weight sensitivity\.

We isolate the contribution of the plausibility safeguard by sweepingλ∈\{0\.0,0\.1,0\.2,0\.3,0\.4,0\.5\}\\lambda\\in\\\{0\.0,0\.1,0\.2,0\.3,0\.4,0\.5\\\}on LLaDA GSM8K, as shown in Fig\.[3](https://arxiv.org/html/2608.11742#S5.F3)\(left\)\. Enabling the safeguard with any positiveλ\\lambdasubstantially improves accuracy over entropy\-only scoring \(λ=0\\lambda=0\); in particular,λ=0\.1\\lambda=0\.1yields a 2\.1% gain\. This consistent improvement demonstrates that branch\-external plausibility is necessary to prevent lookahead from favoring decisive but incorrect token assignments\. Meanwhile, accuracy varies by only 1\.1% acrossλ∈\[0\.1,0\.5\]\\lambda\\in\[0\.1,0\.5\], with little change in TPS speedup\. The scoring function is therefore robust once the safeguard is enabled and does not require extensive scenario\-specific hyperparameter tuning\.

Figure 3:Left:Accuracy and TPS speedup of RPS on GSM8K with LLaDA under different plausibility weightsλ\\lambda\.Middle:Accuracy and TPS trade\-off of different pivot\-selection and scoring strategies on GSM8K with LLaDARight:Failure\-length comparison on HumanEval for Dream and LLaDA\. For cases where exactly one of LoPA and RPS succeeds, we report the fraction for which the failing completion contains fewer non\-empty source lines than the successful completion\.
#### Search\-space hyperparameters\.

Table 3:Hyperparameter sensitivity on LLaDA GSM8K\. One parameter is varied at a time;boldmarks the setting used in the main experiments\.We further analyze the three hyperparameters governing the pivot\-search space \(Table[3](https://arxiv.org/html/2608.11742#S5.T3)\)\. The candidate budgetkmaxk\_\{\\max\}caps the support: overly large values admit weak candidates and hurt accuracy, while overly small values limit useful alternatives and slow decoding\. The reachability ratiorrfurther prunes tokens relative to the top\-1 probability and is robust over a broad range, though aggressive pruning may remove plausible assignments\. The thresholdτpivot\\tau\_\{\\text\{pivot\}\}controls pivot reliability: low values admit overly diffuse positions, whereas high values make selection too conservative and reduce speed\. Since these parameters impose structural search constraints rather than task\-specific preferences, we fixkmaxk\_\{\\max\}andrrglobally and use a singleτpivot\\tau\_\{\\text\{pivot\}\}across all tasks within each model family, without any tuning\.

### 5\.3Further Analysis

#### Pivot strategy and scoring ablation\.

Fig\.[3](https://arxiv.org/html/2608.11742#S5.F3)\(middle\) organizes the compared configurations into four accuracy–speed regimes\. Only the complete RPS design lies in the desirable high\-accuracy and fast quadrant\. Max Confidence retains accuracy but remains slow because it selects positions that are already nearly resolved and thus induces weaker ripple effects\. Conversely, unconstrained Max Entropy is fast but inaccurate because it admits positions whose distributions are too diffuse for reliable candidate search\. The proposed reachability constraint targets the tractable mid\-entropy region between these extremes, while plausibility\-aware scoring further lifts accuracy with negligible throughput change\. Together, the pivot\-selection strategy and lookahead scoring enable the best quality–speed trade\-off among the compared designs\.

#### Failure\-mode analysis on HumanEval\.

To better understand why RPS attains higher accuracy than LoPA on zero\-shot HumanEval, we examine the programs generated by the two methods\. We focus on cases where exactly one method succeeds and compare the number of non\-empty source lines in the failing and successful completions\. Fig\.[3](https://arxiv.org/html/2608.11742#S5.F3)\(right\) reveals a clear asymmetry: a shorter failing program is substantially more common when LoPA fails than when RPS fails, suggesting that LoPA is more prone to terminating a plausible\-looking solution before completing the required logic\. Fig\.[4](https://arxiv.org/html/2608.11742#S5.F4)\(left\) illustrates this behavior on HumanEval/57 \(monotonic\)\. LoPA commitsreturnearly in decoding, closing the program before it accounts for monotonically decreasing inputs\. RPS makes more conservative early commitments and later completes the complementary state logic, thereby avoiding this premature termination\. This example reflects a broader failure mode of confidence\-driven lookahead: LoPA can favor a wrong\-but\-decisive token because it immediately collapses downstream uncertainty\. RPS instead combines cautious pivot selection with a plausibility safeguard that screens the quality of an early commitment\. This distinction is especially consequential in code generation, where control\-flow tokens such asreturnandbreakdirectly alter the execution path\. An incorrect commitment can therefore produce a program that is syntactically valid and executable, yet globally incomplete or logically wrong—an error that subsequent decoding cannot easily repair\.

Figure 4:Left:Decoding snapshots of LoPA and RPS on HumanEval/57 \(monotonic\)\. Coloured tokens denote newly committed tokens, underlined tokens denote lookahead commits, and “■\\blacksquare” denotes still\-masked positions\.Right:TPS speedup relative to Default and Confidence with and without Fast\-dLLM prefix caching, evaluated on MBPP with Dream and GSM8K with LLaDA\.
#### Speedup decomposition\.

Table 4:Average forward\-pass breakdown per sample for RPS on LLaDA GSM8K\.As shown in Fig\.[2](https://arxiv.org/html/2608.11742#S4.F2), our packed lookahead evaluates all candidate branches jointly with the normal sequence in a single forward pass\. Although this increases per\-step computation, Table[4](https://arxiv.org/html/2608.11742#S5.T4)shows that a lookahead pass is only 15% slower than a normal pass, while reducing the average number of forward passes from 77\.80 under confidence decoding to 35\.73 with RPS\. Thus, a modest per\-step overhead yields a substantial reduction in decoding iterations\.

#### Compatibility with KV caching\.

We combine RPS with Fast\-dLLM prefix caching to test whether iteration\-level and per\-forward optimizations are complementary\. As in Fig\.[4](https://arxiv.org/html/2608.11742#S5.F4)\(right\), the combination achieves the highest throughput in both settings, with up to 17\.82×\\timesTPS speedup and less than 0\.5% accuracy change\. This complementarity is natural: RPS reduces decoding iterations via proactive commitment, while KV caching lowers the attention cost per iteration\.

## 6Conclusion

We presented Ripple\-Pivot Search \(RPS\), a training\-free parallel decoding method for diffusion language models motivated by the ripple effect\. Proactively committing a pivot position in the mid\-entropy regime can substantially reduce uncertainty across the remaining masked positions, creating more opportunities for parallel commitment in subsequent decoding steps\. RPS exploits this effect by addressing both*where to commit*and*what to commit*\. It seeks mid\-entropy pivots with high potential for downstream uncertainty reduction and determines their token assignments through lookahead evaluation over plausible candidates\. Across three dLLMs and four reasoning and code\-generation benchmarks, RPS achieves 4–10×\\timeswall\-clock speedup over the standard decoder while largely preserving generation quality, and improves accuracy over the previous lookahead baseline by up to 5\.49% while delivering higher throughput in most settings\. When combined with KV caching, RPS further achieves up to 18×\\timeswall\-clock speedup over the standard decoder\. Together, these results demonstrate the effectiveness of jointly searching over commitment positions and token assignments for efficient dLLM decoding\.

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## Appendix ARPS Algorithm Pseudocode

Algorithm 1RPS decoding1:prompt

yy; generation length

LL; block size

BB; hyperparameters

τpivot,kmax,r,λ,τ\\tau\_\{\\text\{pivot\}\},k\_\{\\max\},r,\\lambda,\\tau
2:

x←\[MASK\]Lx\\leftarrow\\texttt\{\[MASK\]\}^\{L\}⊳\\trianglerightInitialize response with all masks

3:for

b=1,…,L/Bb=1,\\dots,L/Bdo⊳\\trianglerightSemi\-autoregressive block loop

4:

\{pi\}←\\\{p\_\{i\}\\\}\\leftarrowforward pass over

\[y∥x\]\[y\\\|x\]⊳\\trianglerightBlock\-opening forward

5:

ℳ←\\mathcal\{M\}\\leftarrowmasked positions in block

bb
6:while

ℳ≠∅\\mathcal\{M\}\\neq\\emptysetdo

7:

𝒮←\{i∈ℳ:Pimax≥τ\}\\mathcal\{S\}\\leftarrow\\\{i\\in\\mathcal\{M\}:P\_\{i\}^\{\\max\}\\geq\\tau\\\}; if

𝒮=∅\\mathcal\{S\}=\\emptyset, set

𝒮←\{arg⁡maxi∈ℳ⁡Pimax\}\\mathcal\{S\}\\leftarrow\\\{\\arg\\max\_\{i\\in\\mathcal\{M\}\}P\_\{i\}^\{\\max\}\\\}⊳\\trianglerightStandard commit

8:Commit

xi←arg⁡maxv​pi​\(v\)x\_\{i\}\\leftarrow\\arg\\max\_\{v\}p\_\{i\}\(v\)for

i∈𝒮i\\in\\mathcal\{S\}; update

ℳ←ℳ∖𝒮\\mathcal\{M\}\\leftarrow\\mathcal\{M\}\\setminus\\mathcal\{S\}
9:if

ℳ=∅\\mathcal\{M\}=\\emptysetthenbreak

10:endif

11:

𝒯i←topkmax​\(pi\)\\mathcal\{T\}\_\{i\}\\leftarrow\\mathrm\{top\}\_\{k\_\{\\max\}\}\(p\_\{i\}\)and

μi←∑v∈𝒯ipi​\(v\)\\mu\_\{i\}\\leftarrow\\sum\_\{v\\in\\mathcal\{T\}\_\{i\}\}p\_\{i\}\(v\)for each

i∈ℳi\\in\\mathcal\{M\}
12:

ℳfeas←\{i∈ℳ:μi≥τpivot\}\\mathcal\{M\}\_\{\\text\{feas\}\}\\leftarrow\\\{i\\in\\mathcal\{M\}:\\mu\_\{i\}\\geq\\tau\_\{\\text\{pivot\}\}\\\}⊳\\trianglerightProbability\-mass constraint

13:if

ℳfeas=∅\\mathcal\{M\}\_\{\\text\{feas\}\}=\\emptysetthen

14:

\{pi\}←\\\{p\_\{i\}\\\}\\leftarrowforward pass over

\[y∥x\]\[y\\\|x\];continue

15:endif

16:

i⋆←argmaxi∈ℳfeas\[−∑v∈𝒯ipi\(v\)logpi\(v\)\]i^\{\\star\}\\leftarrow\\arg\\max\_\{i\\in\\mathcal\{M\}\_\{\\text\{feas\}\}\}\\big\[\-\\sum\_\{v\\in\\mathcal\{T\}\_\{i\}\}p\_\{i\}\(v\)\\log p\_\{i\}\(v\)\\big\]⊳\\trianglerightTruncated\-entropy maximization

17:

𝒞←\{c:pi⋆​\(c\)/Pi⋆max≥r\}\\mathcal\{C\}\\leftarrow\\\{c:p\_\{i^\{\\star\}\}\(c\)/P\_\{i^\{\\star\}\}^\{\\max\}\\geq r\\\}, capped at

kmaxk\_\{\\max\}tokens; add

\[MASK\]\[\\texttt\{MASK\}\]
18:if

\|𝒞∖\{\[MASK\]\}\|≤1\|\\mathcal\{C\}\\setminus\\\{\[\\texttt\{MASK\}\]\\\}\|\\leq 1then

19:

\{pi\}←\\\{p\_\{i\}\\\}\\leftarrowforward pass over

\[y∥x\]\[y\\\|x\];continue

20:endif

21:

xahead←\[x∥xb\(c1\)∥⋯∥xb\(c\|𝒞\|\)\]x\_\{\\text\{ahead\}\}\\leftarrow\[x\\;\\\|\\;x\_\{b\}^\{\(c\_\{1\}\)\}\\\|\\;\\cdots\\;\\\|\\;x\_\{b\}^\{\(c\_\{\|\\mathcal\{C\}\|\}\)\}\];

\{pic\}←\\\{p\_\{i\}^\{c\}\\\}\\leftarrowforward pass over

\[y∥xahead\]\[y\\\|x\_\{\\text\{ahead\}\}\]⊳\\trianglerightLookahead forward

22:

c⋆←arg⁡maxc∈𝒞⁡score⁡\(c\)c^\{\\star\}\\leftarrow\\arg\\max\_\{c\\in\\mathcal\{C\}\}\\mathrm\{score\}\(c\)via Eq\. \([4](https://arxiv.org/html/2608.11742#S4.E4)\)⊳\\trianglerightScoring and commit

23:if

c⋆≠\[MASK\]c^\{\\star\}\\neq\[\\texttt\{MASK\}\]then

24:Commit

c⋆c^\{\\star\}at

i⋆i^\{\\star\};

ℳ←ℳ∖\{i⋆\}\\mathcal\{M\}\\leftarrow\\mathcal\{M\}\\setminus\\\{i^\{\\star\}\\\};

\{pi\}←\{pi\(c⋆\)\}\\\{p\_\{i\}\\\}\\leftarrow\\\{p\_\{i\}^\{\(c^\{\\star\}\)\}\\\}⊳\\trianglerightReuse winner’s logits

25:else

26:

\{pi\}←\{pi\(\[MASK\]\)\}\\\{p\_\{i\}\\\}\\leftarrow\\\{p\_\{i\}^\{\(\[\\texttt\{MASK\}\]\)\}\\\}⊳\\trianglerightReuse anchor’s logits

27:endif

28:endwhile

29:endfor

30:return

xx

## Appendix BProofs for the Lookahead Objective

###### Proof of Proposition[1](https://arxiv.org/html/2608.11742#Thmproposition1)\.

Fori∈ℳ∖\{i⋆\}i\\in\\mathcal\{M\}\\setminus\\\{i^\{\\star\}\\\}, define

qi=maxv⁡pic​\(v\),𝒰τ​\(c\)=\{i∈ℳ∖\{i⋆\}:qi<τ\},Uτ​\(c\)=\|𝒰τ​\(c\)\|\.q\_\{i\}=\\max\_\{v\}p\_\{i\}^\{c\}\(v\),\\qquad\\mathcal\{U\}\_\{\\tau\}\(c\)=\\\{i\\in\\mathcal\{M\}\\setminus\\\{i^\{\\star\}\\\}:q\_\{i\}<\\tau\\\},\\qquad U\_\{\\tau\}\(c\)=\|\\mathcal\{U\}\_\{\\tau\}\(c\)\|\.Fixi∈𝒰τ​\(c\)i\\in\\mathcal\{U\}\_\{\\tau\}\(c\), soqi<τ<1q\_\{i\}<\\tau<1\. Ifqi≥1/2q\_\{i\}\\geq 1/2, letvi⋆∈arg⁡maxv​pic​\(v\)v\_\{i\}^\{\\star\}\\in\\arg\\max\_\{v\}p\_\{i\}^\{c\}\(v\)and definep~ic​\(v\)=pic​\(v\)/\(1−qi\)\\widetilde\{p\}\_\{i\}^\{c\}\(v\)=p\_\{i\}^\{c\}\(v\)/\(1\-q\_\{i\}\)forv≠vi⋆v\\neq v\_\{i\}^\{\\star\}\. The entropy decomposition gives

H⁡\(pic\)\\displaystyle H\(p\_\{i\}^\{c\}\)=h⁡\(qi\)\+\(1−qi\)​H​\(p~ic\)\\displaystyle=h\(q\_\{i\}\)\+\(1\-q\_\{i\}\)H\(\\widetilde\{p\}\_\{i\}^\{c\}\)≥h⁡\(qi\)\.\\displaystyle\\geq h\(q\_\{i\}\)\.Ifqi<1/2q\_\{i\}<1/2, thenpic​\(v\)≤qip\_\{i\}^\{c\}\(v\)\\leq q\_\{i\}for everyvv, and hence

H⁡\(pic\)\\displaystyle H\(p\_\{i\}^\{c\}\)=∑vpic​\(v\)​log⁡1pic​\(v\)\\displaystyle=\\sum\_\{v\}p\_\{i\}^\{c\}\(v\)\\log\\frac\{1\}\{p\_\{i\}^\{c\}\(v\)\}≥∑vpic​\(v\)​log⁡1qi=−log⁡qi\.\\displaystyle\\geq\\sum\_\{v\}p\_\{i\}^\{c\}\(v\)\\log\\frac\{1\}\{q\_\{i\}\}=\-\\log q\_\{i\}\.Sinceh⁡\(x\)h\(x\)is non\-increasing on\[1/2,1\]\[1/2,1\]andh⁡\(τ\)≤log⁡2h\(\\tau\)\\leq\\log 2, for everyi∈𝒰τ​\(c\)i\\in\\mathcal\{U\}\_\{\\tau\}\(c\),

H⁡\(pic\)≥\{h⁡\(qi\)≥h⁡\(τ\),qi∈\[1/2,τ\),−log⁡qi\>log⁡2≥h⁡\(τ\),qi<1/2\.H\(p\_\{i\}^\{c\}\)\\geq\\begin\{cases\}h\(q\_\{i\}\)\\geq h\(\\tau\),&q\_\{i\}\\in\[1/2,\\tau\),\\\\ \-\\log q\_\{i\}\>\\log 2\\geq h\(\\tau\),&q\_\{i\}<1/2\.\\end\{cases\}Summing this bound over𝒰τ​\(c\)\\mathcal\{U\}\_\{\\tau\}\(c\)yields

n​H¯c=∑i∈ℳ∖\{i⋆\}H⁡\(pic\)\\displaystyle n\\bar\{H\}\_\{c\}=\\sum\_\{i\\in\\mathcal\{M\}\\setminus\\\{i^\{\\star\}\\\}\}H\(p\_\{i\}^\{c\}\)≥∑i∈𝒰τ​\(c\)H⁡\(pic\)\\displaystyle\\geq\\sum\_\{i\\in\\mathcal\{U\}\_\{\\tau\}\(c\)\}H\(p\_\{i\}^\{c\}\)≥Uτ​\(c\)​h​\(τ\),\\displaystyle\\geq U\_\{\\tau\}\(c\)h\(\\tau\),and therefore

Uτ​\(c\)≤min⁡\{n,⌊n​H¯ch⁡\(τ\)⌋\}\.U\_\{\\tau\}\(c\)\\leq\\min\\\!\\left\\\{n,\\left\\lfloor\\frac\{n\\bar\{H\}\_\{c\}\}\{h\(\\tau\)\}\\right\\rfloor\\right\\\}\.SinceNτ​\(c\)=n−Uτ​\(c\)N\_\{\\tau\}\(c\)=n\-U\_\{\\tau\}\(c\),

Nτ​\(c\)\\displaystyle N\_\{\\tau\}\(c\)=n−Uτ​\(c\)\\displaystyle=n\-U\_\{\\tau\}\(c\)≥max⁡\{0,n−⌊n​H¯ch⁡\(τ\)⌋\},\\displaystyle\\geq\\max\\\!\\left\\\{0,n\-\\left\\lfloor\\frac\{n\\bar\{H\}\_\{c\}\}\{h\(\\tau\)\}\\right\\rfloor\\right\\\},which is Eq\. \([5](https://arxiv.org/html/2608.11742#S4.E5)\)\. ∎

###### Proof of Proposition[2](https://arxiv.org/html/2608.11742#Thmproposition2)\.

Let

S⁡\(c\)=−H¯c\+λ​log⁡ac,S\(c\)=\-\\bar\{H\}\_\{c\}\+\\lambda\\log a\_\{c\},Then

arg⁡maxc∈𝒞​S​\(c\)\\displaystyle\\arg\\max\_\{c\\in\\mathcal\{C\}\}S\(c\)=arg⁡maxc∈𝒞​\{−H¯c\+λ​log⁡ac\}\\displaystyle=\\arg\\max\_\{c\\in\\mathcal\{C\}\}\\\{\-\\bar\{H\}\_\{c\}\+\\lambda\\log a\_\{c\}\\\}=arg⁡minc∈𝒞​\{H¯c−λ​log⁡ac\},\\displaystyle=\\arg\\min\_\{c\\in\\mathcal\{C\}\}\\\{\\bar\{H\}\_\{c\}\-\\lambda\\log a\_\{c\}\\\},which proves Eq\. \([6](https://arxiv.org/html/2608.11742#S4.E6)\)\. For anyc,d∈𝒞c,d\\in\\mathcal\{C\},

S⁡\(c\)≥S⁡\(d\)\\displaystyle S\(c\)\\geq S\(d\)⇔−H¯c\+λ​log⁡ac≥−H¯d\+λ​log⁡ad\\displaystyle\\iff\-\\bar\{H\}\_\{c\}\+\\lambda\\log a\_\{c\}\\geq\-\\bar\{H\}\_\{d\}\+\\lambda\\log a\_\{d\}⇔H¯d−H¯c≥λ⁡\(log⁡ad−log⁡ac\)\\displaystyle\\iff\\bar\{H\}\_\{d\}\-\\bar\{H\}\_\{c\}\\geq\\lambda\(\\log a\_\{d\}\-\\log a\_\{c\}\)⇔H¯d−H¯c≥λ​log⁡adac,\\displaystyle\\iff\\bar\{H\}\_\{d\}\-\\bar\{H\}\_\{c\}\\geq\\lambda\\log\\frac\{a\_\{d\}\}\{a\_\{c\}\},which proves Eq\. \([7](https://arxiv.org/html/2608.11742#S4.E7)\) in both directions\. ∎

## Appendix CImplementation Details

#### Baselines\.

Default uses highest\-confidence unmasking with one token per step\. Confidence follows Fast\-dLLM\([17](https://arxiv.org/html/2608.11742#bib.bib3)\)with thresholdτ=0\.9\\tau=0\.9\. KLASS\([11](https://arxiv.org/html/2608.11742#bib.bib23)\), EB\-Sampler\([3](https://arxiv.org/html/2608.11742#bib.bib22)\), and WINO\([10](https://arxiv.org/html/2608.11742#bib.bib8)\)follow the settings in their original papers and report the best result from the prescribed sweep ranges: for KLASS, we use confidence thresholdτ=0\.9\\tau=0\.9and select the KL thresholdϵKL\\epsilon\_\{\\mathrm\{KL\}\}from\{0\.015,0\.01,0\.005,0\.001\}\\\{0\.015,0\.01,0\.005,0\.001\\\}; for EB\-Sampler, we sweepγ∈\{0\.1,0\.01,0\.001\}\\gamma\\in\\\{0\.1,0\.01,0\.001\\\}with the confidence\-based error proxy; for WINO, we sweep the drafting thresholdτ1∈\{0\.5,0\.6,0\.7,0\.8\}\\tau\_\{1\}\\in\\\{0\.5,0\.6,0\.7,0\.8\\\}while fixing the verification thresholdτ2=0\.9\\tau\_\{2\}=0\.9\. For LoPA, we reimplement its core decoding algorithm without the LoPA\-Dist distributed inference system or its system\-level optimizations, so that all methods run on the same single\-device inference stack\.

#### Hardware\.

All experiments are conducted on a node equipped with 4×\\timesNVIDIA A100\-SXM4\-80GB GPUs\. Speed metrics \(TPS\) are measured on an equivalent single\-GPU basis\.

## Appendix DLLaDA\-1\.5 Results

Table 5:LLaDA\-1\.5\.Performance and inference speedup comparison across 4 benchmarks\.
## Appendix ELimitations

#### Limitations\.

While RPS demonstrates strong acceleration gains while largely preserving generation quality, it still has several limitations\. First, although the plausibility weightλ\\lambdais not highly sensitive within a broad validated plateau \(§[5\.2](https://arxiv.org/html/2608.11742#S5.SS2.SSS0.Px1)\), it still requires per\-task selection within that range and therefore does not constitute zero\-tuning in the strictest sense\. Second, our characterization of the ripple effect is empirical and qualitative rather than theoretically derived\. In particular, the motivating analysis in §[1](https://arxiv.org/html/2608.11742#S1)is conducted on a subset of GSM8K and is intended to illustrate the phenomenon, not to establish that the precise location of the cascade peak quantitatively generalizes across models or tasks\. Third, when combined with prefix cache acceleration \(§[5\.3](https://arxiv.org/html/2608.11742#S5.SS3.SSS0.Px4)\), accuracy degradation may arise from the cache approximation itself\. This effect is not specific to RPS and is also observed for other parallel decoding methods, but mitigating it remains outside the scope of this work\.