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structure (Gandhi et al., 2025; Li et al., 2025a; Ye et al., 2025). Both Wu et al. and Ballon et al. highlighted the overthinking phenomenon, where overly long reasoning chains can degrade rather than improve final answer quality. However, our analysis (Figure 1) shows that response length alone remains an inadequate predictor of answer correctness, as responses with similar lengths vary greatly in correctness. These findings suggest that these heuristics, such as token count, step count, or PRM-based semantic metrics fall short in effec- tively dictating reasoning success. Thus, we propose Long Chain-of-Thought to Tree (LCoT2Tree), the first automated frame- work for structural analysis of reasoning in LLMs. LCoT2Tree transforms sequential LCoTs into hier- archical tree representations (Section 3.2), which enable structural patterns in reasoning chains, in- cluding exploration, backtracking, and verification, to be made explicit and analyzable. By modeling these trees with graph neural networks (GNNs), we not only extract these structural patterns as fea- tures, but also demonstrate that they serve as strong predictors of reasoning success (Section 3.3). Beyond establishing their predictive power, we further investigate which structural patterns specifi- cally contribute to reasoning success or failure, how these patterns vary across tasks and models, and how they can be applied to further enhance LLM reasoning in practice. Concretely, by leveraging a GNN-based explainability method, we unveil key thought patterns ( i.e., critical substructures within the tree) that that explain answer correctness across diverse tasks and models (Section 4). These anal- yses reveal how reasoning behaviors differ by (1) answer correctness, (2) task type, and (3) model variant. Furthermore, we demonstrate that these patterns can be leveraged to improve Best-of-N decoding: incorporating our tree-based predictive classifier into its selection strategy consistently en- hances accuracy across diverse models and tasks (Section 5). We summarize the main contributions of the proposed LCoT2Tree in three aspects: •(Predictability ) We are the first to explicitly con- struct structural representations of LCoT; our pro- posed LCoT2Tree offers stronger signals for rea- soning success and improves binary classification of answer correctness by an average of 5.63% , compared to using length alone. •(Interpretability ) We leverage LCoT2Tree to pin- point the reasoning patterns that oftentimes lead to errors, e.g., over-branching, and to account for disparate behaviors across tasks and models. •(Practicality ) We demonstrate that LCoT2Tree offers a principled path for selecting well- structured reasoning chains, greatly enhancingBest-of-N decoding and also remaining extensi- ble for future decoding strategies. 2 Related Works Reasoning LLMs Advancing reasoning capabil- ities of LLMs has shown benefits in tackling com- plex tasks (Kojima et al., 2022; Wei et al., 2022; Li et al., 2025b). Researchers first demonstrated that CoT prompting can significantly improve per- formance on complex tasks like arithmetic (Wei et al., 2022). To refine the reasoning processes, hierarchical cognitive phases have been introduced, such as multi-path exploration (Wang et al., 2023b; Zhou et al., 2023; Yao et al., 2023), step verifi- cation (Miao et al., 2024; Gou et al., 2024), and iterative refinement (Madaan et al., 2023; Besta et al., 2024). These approaches expand solution spaces and deepen | https://arxiv.org/abs/2505.22148v1 |
reasoning, driving more reliable answers. More recently, models such as Deepseek- R1 (Guo et al., 2025), Kimi-1.5 (Team et al., 2025) and QwQ-32B (Team, 2024) have leveraged rule- based reinforcement learning to embed reason- ing capabilities directly into model parameters, achieving remarkable progress in handling com- plex tasks (Chen et al., 2025; Gandhi et al., 2025). Chain-of-Thought Analysis Numerous studies have explored when CoT prompting is effective. Empirical research has revealed that factors, such as step length (Jin et al., 2024), relevance, the order of reasoning fragments (Wang et al., 2023a), and prompt structure (Li et al., 2025a), heavily influ- ence performance. Expanding on these findings, Feng et al. (2023) and Chen et al. (2024a) proposed that there is an inherent reasoning limit in LLMs when tackling tasks exceeding a complexity thresh- old. In the context of long CoT, research has in- creasingly emphasized the importance of response structures in enhancing reasoning success (Li et al., 2025a; Gandhi et al., 2025; Muennighoff et al., 2025; Ye et al., 2025). Additionally, challenges like the overthinking phenomenon, where overly long responses inadvertently hurt the model perfor- mance, seemingly establish the correlation between length and reasoning success (Chen et al., 2024b; Wu et al., 2025; Cuadron et al., 2025). Beyond these prior works, we are motivated to develop an automated tool to empirically identify the structural patterns that dictate reasoning success in long CoT. Besides the above analyses towards reasoning success, another line of works primarily analyzes towards the semantic rationality of reasoning. Thus, early methods directly compare generated steps to human-annotated explanations (Welleck et al., 2022). However, such methods often fail to cap- ture logical coherence beyond surface-level sim- ilarity. Recent work has introduced LLM-driven PRMs (Ling et al., 2023; Yuan et al., 2024; She et al., 2025; Zhang et al., 2025) to provide holis- tic and step-wise assessment, but they struggle with scaling to long and complex CoT reasoning chains (He et al., 2025). Rather than relying on surface similarity or token-level reward signals, we analyze reasoning success through internal struc- tural patterns derived from hierarchical tree repre- sentations. The structural patterns offer a princi- pled alternative for dictating “good” chains, and are fully complementary to this body of semantic- based research. 3 LCoT2Tree: Automated Long Chain-of-Thoughts to Tree In this section, we empirically study overthinking, highlighting issues with assessing reasoning quality via CoT length. Then, we propose Long Chain-of- Thought to Tree (LCoT2Tree), an automated tool that converts LCoTs into tree structures to reveal cognitive frameworks and enable deeper analysis of LLMs’ reasoning processes. 3.1 Overthinking Phenomenon The “overthinking” phenomenon in reasoning mod- els refers to situations where a model expends excessive computational resources ( e.g., generat- ing overly long sequences or repeating reasoning steps), yet makes little contributions to the cor- rectness of final answer. In some cases, this can even lead to a decline in performance (Chen et al., 2024b; Wu et al., 2025). Figure 2 illustrates this phenomenon by showing the relationship between the output token length and the answer accuracy of DeepSeek-32B ( i.e., DeepSeek-R1-Distill-Qwen- 32B (Guo et | https://arxiv.org/abs/2505.22148v1 |
al., 2025)) on the MATH (Hendrycks et al., 2021) dataset. It demonstrates that as the rea- soning chain becomes unnecessarily long, model performance deteriorates, highlighting how over- thinking can harm the reasoning ability of LLMs. To tackle this issue, researchers have proposed using a length penalty during the training period to constrain the length of generated LCoTs (Team et al., 2025; Yu et al., 2025). However, this strat- egy relies on the oversimplified assumption that shorter or moderately long reasoning chains inher- 0 5 10 15 Length (K)12Data Count (K) 0.500.751.00 Accuracy Token Length vs AccuracyFigure 2: Data count and accuracy of the MATH dataset for DeepSeek-32B across varying response lengths. Ac- curacy notably declines as response length increases. ently lead to better reasoning quality. In this work, we conduct a classification experiment to empiri- cally quantify the actual relationship between these two factors and uncover the limitations of relying on length as an indicator of reasoning quality. Experimental Setup. For our study, we use DeepSeek-32B, DeepSeek-R1 (Guo et al., 2025), QwQ-32B (Team, 2024), Seed-1.5-Thinking- pro (Seed et al., 2025), and Grok-3-mini-beta (xAI, 2025) as the primary models. We evaluate these models on four benchmark datasets: MATH (Level5 question in high school math competi- tions; Hendrycks et al., 2021), GPQA (“main” sub- set in grade-level google-proof question answer- ing; Rein et al., 2024), LiveCodeBench (version 5 in live code benchmark; Jain et al., 2025), and MMLU-Pro (proficient-level multi-discipline lan- guage understanding; Wang et al., 2024). For each dataset, we collect 2,000 model responses, consist- ing of 1,000 correctly answered cases (Positive) and 1,000 incorrectly answered cases (Negative). These samples are divided into training and testing sets at a ratio of 4:1. In our experiments, we train a logistic regression model using LCoT response length as the input fea- ture and answer correctness as the target label. The test set accuracy quantifies the degree of correlation between LCoT length and reasoning quality-higher accuracy suggests a stronger association between these two factors. Results and Analysis. Figure 1 shows the token length distributions of positive and negative sam- ples. It reveals a significant overlap between the two classes, indicating that responses with similar lengths can vary greatly in reasoning quality. More- over, Table 1 presents the classification results, where the accuracy on the MMLU-Pro dataset is only 60.0% for DeepSeek-32B and 58.0% for QwQ-32B. These relatively low accuracies under- score the limitations of using response length alone in predicting reasoning success. Reasoning SketchStep 0 (Fixed): Task decompositionStep 1: Determine divisibility by 4 requirement:…Step 2: Analyze valid B value: …Step 3: Compute probability:… Thought List𝑻𝟎: Okay, so … First, let’s …𝑻𝟏 :Wait, actually, the last …𝑻𝟐 :Alternative,… So, for …𝑻𝟑 :Wait, wait: 10 B should be ……𝑻𝟗 :Let me verify: k … Yes, …Thought Function𝑻𝟏: 1 (Continuous Logic)𝑻𝟐: 2 (Exploration)𝑻𝟑: 4 (Verification)𝑻𝟒: 3 (Backtrack)...𝑻𝟗: 4 (Verification)Thought Step𝑻𝟎 : [Step0] 𝑻𝟏 : [Step1]𝑻𝟐 : [Step1, Step 2, Step 3]𝑻𝟑 : [Step3] 𝑻𝟒 : [Step2]𝑻𝟓 : [Step2] 𝑻𝟔 : [Step2] 𝑻𝟕 : [Step3] 𝑻𝟗 : [Step3]𝑻𝟖 : [Step1, Step 2, Step 3]⓵⓶Extract SketchSplitThought Assign StepIdentify Function⓷⓸⓹The positive three-digit | https://arxiv.org/abs/2505.22148v1 |
integer N as a ones digit of 0. What is the probability that N is divisible by 4?Long Chain-of-thoughtOkay, so I need to figure …First, let’s understand …Wait, actually, the last number should …Alternative,since three-digit number …So, for each A, there are 5 …Wait, wait: 10 B should be ……Let me verify: k is from 10 to 99 …Yes, that make sense …AnswerTo determine the probability that a three-digit integer ……Therefore, the final answer is: (\boxed {\frac{1}{2}}). I hope it is correct. Reasoning TreeBuild Tree ExplorationBacktrackVerification𝑁!"𝑁""𝑁#"𝑁##𝑁#$𝑁$"𝑁%"𝑁&"𝑁'"𝑁("𝑁)$𝑁*"𝑁)"𝑁)#Step 1Step 0Step 2Step 3Continuous logicflowQuestionFigure 3: The workflow for LCoT2Tree. It transforms sequential long chain-of-thought into reasoning tree, which involves five steps: (1) Extract Sketch, (2) Split Thought, (3) Assign Step, (4) Identify Function, and (5) Build Tree. 3.2 LCoT2Tree Tool We present LCoT2Tree, a novel tool that extracts structural insights from LCoTs, addressing the lim- itations of length-based prediction. LCoT2Tree converts the sequential chain of reasoning into a tree structure, enabling a deeper analysis of cog- nitive behaviors such as exploration, backtracking, and reasoning depth. These components increas- ingly recognized as crucial for developing reason- ing LLMs (Chen et al., 2025; Gandhi et al., 2025; Ye et al., 2025). To our knowledge, our work is the first to explicitly extract this structural infor- mation and conduct a quantitative analysis of its correlation with reasoning quality. The LCoT2Tree tool involves five automated stages that transform a LCoT into an organized tree structure using an LLM (DeepSeek-v3; Liu et al., 2024a), as shown in Figure 3: Stage 1: Extract Sketch. Leveraging the LLM with prompting (Figure 6), we condense the LCoT into a concise Reasoning Sketch that outlines its main reasoning steps. The sketch serves as an ab- stract summary, highlighting the essential compo- nents and logical flow of the reasoning process. Stage 2: Split Thought. We first define a “Thought ” as a consecutive segment in the reason- ing chain that involves no logical transition ( e.g., exploration and verification). We utilize common linguistic cues ( e.g., “Wait”, “Alternatively”, and “Let me verify”) indicative of transitions between reasoning steps to segment the full reasoning chain into distinct fragments, yielding a Thoughts List . Stage 3: Assign Step. Each thought in the Thoughts List is matched to one or more steps in theReasoning Sketch , depending on its role in the overall reasoning process. This alignment is carriedout using an LLM with prompting (Figure 7), gen- erating a Thought Step dictionary that maps each thought to its corresponding reasoning depths. Stage 4: Identify Function. By prompting the LLM (Figure 8), we analyze consecutive thought pairs to determine the later thought’s role relative to the former, with possible roles: (1) Continuous Logic; (2) Exploration; (3) Backtracking; and (4) Verification. This assigns a Thoughts Function label to each thought for clearer reasoning-flow purpose understanding. Stage 5: Build Tree. Finally, we organize the segmented thoughts into a hierarchical tree struc- ture. Each node Nj iin the tree corresponds to thei-th thought Ti, where jindicates how many times Tihas appeared. The placement of a node is determined by the Thought Step | https://arxiv.org/abs/2505.22148v1 |
, and each edge rep- resents a transition to a deeper level of reasoning, with the edge type defined by the Thought Function of its child node. When inserting a new thought Ti, we first identify the ordered list of reasoning steps it maps to, denoted as [S1 i, ..., Sn i]. Here, n indicates that the current thought encompasses n reasoning steps. Consequently, we create nnodes N1 i, ..., Nn i, where each node Nj irepresents the portion of the thought aligned with the Sj i-th step. The insertion process follows two rules: (1) If S1 i is greater than the step of the latest node Nj i−1in the tree, the new node N1 iis added as a child of Nj i−1. (2) Otherwise, we backtrack to the most recent node at step S1 i−1. Then we create a new branch from that node and link it to new node N1 i. Once N1 iis determined, the remaining nodes N2 i, ...,Nn iare added sequentially and connected to the previous one. For example, in Figure 3, when inserting T8to the tree, its associated reasoning MATH GPQA LiveCodeBench MMLU-Pro 4 Datasets Average DeepSeek-32BLength-based 74.13% 67.08% 81.59% 59.95% 66.27% 69.80% Tree-based 80.81% 70.37% 82.21% 72.41% 71.14% 75.39% Gain +6.68% +3.29% +0.62% +12.46% +4.87% +5.58% QwQ-32BLength-based 75.82% 62.09% 78.30% 58.00% 66.97% 68.24% Tree-based 77.63% 68.55% 80.05% 72.58% 70.98% 73.96% Gain +1.81% +6.46% +1.75% +14.58% +4.01% +5.72% DeepSeek-R1Length-based 76.94% 69.57% 81.75% 63.00% 70.54% 72.36% Tree-based 80.30% 73.56% 81.80% 75.85% 73.72% 77.05% Gain +3.36% +3.99% +0.05% +12.85% +3.18% +4.69% Seed-1.5- Thinking-proLength-based 67.48% 64.84% 76.39% 63.59% 64.34% 67.33% Tree-based 70.07% 69.81% 77.72% 70.82% 67.68% 71.22% Gain +2.59% +4.97% +1.33% +7.23% +3.34% +3.89% Grok-3- mini-betaLength-based 71.31% 61.48% 84.77% 55.47% 63.87% 67.38% Tree-based 83.18% 66.79% 86.35% 70.68% 71.26% 75.65% Gain +11.87% +5.31% +1.58% +15.21% +7.40% +8.27% Table 1: Comparison of performance across various reasoning LLMs and datasets using the length-based method and our proposed tree-based approach for classifying response correctness based on LCoT information. Classification results are reported as the average over five runs. steps[S1 8, S2 8, S3 8] = [1 ,2,3], as determined by the Thought Step . At that point, the latest node in tree isN1 7, which is at step 3 (greater than 1). Therefore, we backtrack to the latest node at step 0, N1 0, and attach N1 8as its child. After that, N2 8andN3 8are linked sequentially to N1 8andN2 8, respectively. In the end, we extract the tree structure, showing how thoughts are connected and branch through- out the reasoning process. This structural repre- sentation offers three key benefits: (1) highlights key cognitive patterns ( e.g., exploration, backtrack- ing and verification); (2) supports more accurate assessment of reasoning quality; and (3) enables structure-aware analysis of reasoning behaviors. Implementation details, including prompts and case visualizations, are available in Appendix A. 3.3 Effectiveness of LCoT2Tree To assess the effectiveness of the LCoT2Tree tool, we conduct a quantitative evaluation by using graph neural networks (GNNs) to predict answer correct- ness based on the tree structures extracted from LCoTs. This evaluation demonstrates the practical value of | https://arxiv.org/abs/2505.22148v1 |
tree-based representations for understand- ing complex reasoning processes. Experimental Setup. We use the same dataset described in Section 3.1, which contains responses from five reasoning models across four public benchmarks. The key difference is that we extract the tree structure from each LCoT response and use it as input to the GNNs. Our objective is to assess how effectively these tree structures can dis- tinguish between correct and flawed reasoning. Tothis end, we utilize GATv2 (Brody et al., 2022), a GNN architecture suited for modeling hierarchical structures and their relationships. The model takes the nodes, edges, and associated features of each LCoT tree as input and learns a structural embed- ding that represents the overall reasoning pattern. Implementation details and graph construction are provided in Appendix B. We use classification ac- curacy as the evaluation metric. A high accuracy score indicates that the model successfully captures the correlation between reasoning structure and an- swer correctness. Effectiveness across Tasks. Table 1 shows the classification results using tree-based input, com- pared to baseline methods that rely on the length- based feature. We assess how well the tree- based method generalizes across diverse types of reasoning tasks, including MATH, GPQA, Live- CodeBench, MMLU-Pro, and a combined dataset of these benchmarks. Across all tasks, the tree- based method consistently outperforms the length- based baseline. The improvement is particularly notable on MMLU-Pro, a dataset where reasoning correctness is difficult to predict from token length alone. For example, our method achieves substan- tial accuracy gains of +12.46% and +14.58% on DeepSeek-32B and QwQ-32B, respectively. Even on datasets like LiveCodeBench, where the length- based approach already performs strongly, the tree- based method still yields improvements, demon- strating its robustness. Effectiveness across Models. For the generaliz- ability of our method, the tree-based classifier con- sistently achieves higher accuracy than the length- based baseline across all models. Average accuracy gains range from +3.89% (Seed-1.5-Thinking-pro) to +8.27% (Grok-3-mini-beta), indicating that the LCoT2Tree provides more informative and reliable structural representations of reasoning processes. These quantitative evaluation validates the ef- fectiveness of the LCoT2Tree tool across diverse tasks and models. By capturing deeper structural and cognitive patterns in reasoning, it enables more accurate prediction of reasoning success. Overall, LCoT2Tree shows strong potential as an automated tool for analyzing, evaluating, and improving the behavior of reasoning systems. 4 Understand Behaviors of Reasoning Large Language Models In this section, we leverage LCoT2Tree to analyze and understand reasoning behaviors. First, we iden- tify key thought patterns in the reasoning tree that predict errors. Then, we compare behaviors across tasks and models. Our findings show that reasoning varies by (1) output correctness, (2) task type, and (3) model variant, underscoring the importance of structural information in reasoning analysis. Explainability Method. To interpret the model’s predictions on reasoning quality and uncover the influential reasoning patterns, we adapt a graph explainability method called GNNExplainer (Ying et al., 2019). This method uncovers important sub- graphs by maximizing the mutual information be- tween the GNN’s output and the distribution of possible subgraph structures. These extracted sub- graphs also correspond to critical thought | https://arxiv.org/abs/2505.22148v1 |
patterns within the reasoning chain. For example, in models trained to predict incorrect answers, the highlighted subgraphs often reflect flawed reasoning behaviors that lead to poor performance. Similarly, in models trained on MATH tasks, the important subgraphs typically capture common reasoning patterns ob- served in mathematical problem-solving. 4.1 Error Patterns in LCoT The experiments in Section 3.3 suggest that reason- ing trees of model responses exhibit separable struc- tures for correct and incorrect outcomes. To further explore the behaviors that contribute to failures, we employ GNNExplainer to identify the most influ- ential edges in each reasoning tree. This allows us to extract critical subgraphs from incorrect re- sponses and summarize common patterns across …(C) Direct Reasoning(D) Skipped Thinking(A) OverBranching(B) Step Redundancy……Step 𝑖Step 𝑗(𝑗≫𝑖)Figure 4: Visualization and frequency of four structural error patterns across three datasets. (A) Over Branch- ing: abundance of explorations or verifications within a single node, (B) Step Redundancy : over-generation of thoughts within a single reasoning step, (C) Direct Reasoning : following a straight, minimal-branch path from one step to a much deeper step, and (D) Skipped Thinking : jumping multiple steps ahead without inter- mediate logical analysis. Representative example of each class is available in Figure 9. diverse examples (See details in Appendix D.1). The identified error patterns are visualized in the top portion of Figure 4, with detailed examples shown in Figure 9. Additionally, we analyze 100 error responses from three tasks and report the fre- quency of each pattern in the bottom portion of Figure 4. A key observation is that excessive and insufficient branching are both strongly associated with incorrect reasoning. 4.2 Task-Specific Patterns in LCoT In the left part of Table 2, we present the results of a task separability experiment conducted on the DeepSeek-32B model. We classify reasoning trees across task pairs ( e.g., MATH/GPQA, MATH/LCB, MATH/MMLU-Pro, GPQA/LCB), with additional results for QwQ-32B provided in Appendix C.1. The dataset follows the same construction as in Section 3.1, but labels tasks instead of correctness. Results show that our tree-based method effec- tively distinguishes task-specific reasoning patterns, achieving an average accuracy of 84.19%. Notably, in cases where length-based features fall short, such as MATH/GPQA and GPQA/LCB, tree-based rep- resentations yield substantial gains of +33.06% and +24.90%, respectively, highlighting their strength in capturing deeper reasoning patterns. Discovering Task-Specific Reasoning Patterns. Beyond quantitative separation, we further lever- Task-specific Analysis Model-specific Analysis MATH/GPQA MATH/LCB MATH/MMLU GPQA/LCB DS-32/DS-R1 DS-32/Grok Length-based 50.45% 63.72% 69.43% 60.65% 55.17% 61.06% Tree-based 83.51% 89.22% 78.46% 85.55% 67.88% 93.22% Gain +33.06% +25.50% +9.03% +24.90% +12.71% +32.16% Table 2: Comparison of task-specific and model-specific classification accuracy using the length-based method and the proposed tree-based approach. Task-specific analysis is conducted on the DeepSeek-32B model across different datasets, while model-specific analysis is performed on the MATH dataset across multiple model variants. age LCoT2Tree to reveal task-specific reasoning patterns through qualitative analysis. Main con- clusions, based on DeepSeek-32B, are as follows: For MATH (Figure 10), the reasoning trees ex- hibit a diagonally descending structure, reaching deeper steps through repeated backtracking. This reflects a layered, step-by-step problem-solving ap- proach. In | https://arxiv.org/abs/2505.22148v1 |
contrast, the behaviors in code comple- tion (Figure 11) show wide, parallel branches with minimal exploration or verification, reflecting a more straightforward pattern of generation. GPQA (Figure 12) samples reveal high out-degree nodes, where the model repeatedly revisits complex con- cepts, indicating the model’s uncertainty in deal- ing an expert-level question. Meanwhile, the trees of MMLU-Pro (Figure 13) are relatively shallow with minimal branching, reflecting a straightfor- ward deductive reasoning style that aligns with the nature of knowledge-based questions. These obser- vations highlight LCoT2Tree’s ability to provide interpretable insights into the distinct reasoning strategies employed across different task types. De- tailed case studies are provided in Appendix D.2 with visualizations shown in Figure 10 - Figure 13. 4.3 Model-Specific Patterns in LCoT We explore whether different models exhibit distin- guishable reasoning behaviors on the same dataset. The results, shown in the right part of Table 2, demonstrate that LCoT2Tree effectively captures model-specific patterns. In particular, tree-based representations significantly outperform simple length-based features, with gains of +12.71% for DS-32 (DeepSeek-32B) vs. DS-R1 (DeepSeek- R1), and +32.16% for DS-32 vs. Grok (Grok-3- mini-beta). Notably, the relatively lower separa- bility score between DS-32 and DS-R1 (67.88%) can be attributed to the fact that DS-32 is a dis- tilled version of DS-R1. In contrast, DS-32 and Grok show a high separability of 93.22%, suggest- ing fundamentally different reasoning styles driven by architectural and training differences. Addi-tional results (Appendix C.2) show that QwQ-32B aligns more closely with the DeepSeek family than with Grok or Seed (Seed-1.5-Thinking-pro). These findings again highlight the strength of structural representations in revealing fine-grained behavioral distinctions. Discovering Model-Specific Reasoning Patterns. To complement the quantitative analysis, we also conduct a qualitative comparison of reasoning trees across different models on the MATH dataset (Ap- pendix D.3). Our analysis reveals that both DS-R1 (Figure 14) and QwQ-32B (Figure 15) produce reasoning structures similar to DS-32 (Figure 10), consistent with quantitative results. However, DS- R1 tends to prune its reasoning paths earlier, sug- gesting a more aggressive backtracking strategy. In contrast, QwQ-32B shows more extensive ex- ploration in the later stages of reasoning. On the other hand, Seed (Figure 16) and Grok (Figure 17) follow simpler, more linear reasoning paths with fewer thought transitions and minimal branching, reflecting a straightforward reasoning strategy. 4.4 Shortcomings in Understanding LCoT from the Structural Perspective Correct Structure but Wrong Output. Despite structurally valid reasoning paths, models can still produce incorrect answers due to semantic errors like misinterpreting the problem, making calcula- tion mistakes, or failing conditional logic. This indicates that reasoning LLMs do not consistently exhibit behaviors like backtracking or verification when facing ambiguity or errors. These cases ex- pose the limitation of structural analysis alone and suggests that combining structural insights with se- mantic verification is necessary for comprehensive reasoning understanding. Flawed Structure but Correct Output. Using our classifier, we identify a set of responses with correct final answer but weak or flawed reason- ing. These cases often involve reasoning paths that deviate from systematic problem-solving, includ- ing guessing, brute-force enumeration, or overly late-stage corrections. Such cases underscore the limitations | https://arxiv.org/abs/2505.22148v1 |
of using answer correctness alone to assess reasoning quality, as it tolerates shallow or unsound reasoning paths. Addressing these flawed- but-correct patterns can guide LLMs toward pro- ducing reasoning that is not only accurate but also logically sound. 5 Application of LCoT2Tree: Tree-based Best-of-N Decoding Beyond evaluating the quality of model reason- ing (Section 3.3), we put forward practical appli- cation to support the decoding process in LLMs. Specifically, we propose an approach to improve the reasoning quality during the decoding stage by selecting the best model response from multiple candidates with the tree-based classifier. Method. Best-of-N decoding is a widely used strategy for improving the quality of responses gen- erated by LLMs (Wu et al., 2024; Snell et al., 2024; Brown et al., 2024). In this strategy, the model produces Ncandidate outputs, and a final response is selected based on a scoring function. However, conventional scoring methods, based on surface- level heuristics or reward models, often ignore the impact of output structures. This limitation can lead to suboptimal choices, especially in tasks that require deep or structured reasoning. To this end, we incorporate LCoT2Tree into the Best-of-N decoding framework to guide the selec- tion of high-quality reasoning outputs. Our method involves three main steps: (1) For each candidate response, we use LCoT2Tree to build its corre- sponding reasoning tree; (2) A graph-based classi- fier, trained to distinguish between successful and flawed reasoning structures, assigns a score to each candidate based on its structural features; (3) The candidate with the highest score is chosen as the final output. Experiments. We choose LiveCodeBench (LCB) as our primary benchmark. We train the GNN models following the setup in Section 3.3 using LCB-v5 dataset and then evaluate on a challenging subset (filtered by correctness ratio) of the LCB- v6 dataset. We compare our tree-based Best-of-N decoding method with three baselines: (1) ORM- Best (Brown et al., 2024), which selects the re- sponse with the highest score from an outcome reward model (we use Skywork-Reward-Gemma- 2-27B-v0.2 (Liu et al., 2024b)); (2) PRM-Best , DeepSeek-32B QwQ-32B30%40%50%60%70%Accuracy (%)50.7755.3856.9261.54 42.1150.88 47.3752.63ORM PRMLength OursFigure 5: Accuracy comparison of different Best-of-N decoding strategies on the LCB-v6 benchmark. which scores responses based on the product of step-level scores from a process reward model ( i.e., Qwen2.5-Math-PRM-72B (Zhang et al., 2025)); and (3) Length-Best (Wang et al., 2025), which selects the response with the fewest tokens. All experiments use N= 10 candidate responses. Ad- ditional results on MATH with more baselines are presented in Appendix C.3. Results. As shown in Figure 5, our tree-based Best- of-N method outperforms both Length-Best, ORM- Best and PRM-Best on the LCB-v6 benchmark. For DeepSeek-32B, it achieves 61.54% accuracy, exceeding Length-Best by +4.62%, ORM-Best by +10.77% and PRM-Best by +6.16%. QwQ-32B shows similar gains, with our method reaching 52.63%, outperforming the baselines by +5.26%, +10.52% and 1.75%, respectively. These results highlight the advantage of using structural reason- ing signals via LCoT2Tree to improve candidate selection in complex tasks. 6 Conclusion In this work, we introduce a novel framework, named LCoT2Tree, for converting LCoT responses into hierarchical tree structures. | https://arxiv.org/abs/2505.22148v1 |
LCoT2Tree en- ables more interpretable and structural analysis of complex reasoning processes, with significantly im- proving the prediction of reasoning success across a wide range of tasks and models. Beyond evaluation, we apply LCoT2Tree for behavioral analysis, re- vealing error patterns and accounting for disparate behaviors across tasks and models. Furthermore, we extend LCoT2Tree to a practical application by integrating it into the Best-of-N decoding paradigm, leading to more accurate outputs than ORM, PRM and length-based baselines. Collectively, these findings underscore the significance of structural reasoning analysis and establish LCoT2Tree as a promising tool for understanding and improving LLMs reasoning capabilities. Limitations While LCoT2Tree is a powerful framework for analyzing reasoning structures, several limitations remain. First, as discussed in Section 4.4, struc- tural analysis alone cannot capture semantic errors or recognize correct reasoning that deviates from common structural patterns. To address this limi- tation, future work should integrate semantic rea- soning signals with structural analysis to achieve a more holistic understanding of LLM reasoning behaviors. Second, the effectiveness of structural analysis relies in part on the fact that current LLMs often generate reasoning that is incomplete or loosely organized. As models improve and begin to pro- duce more coherent and well-structured reasoning by default, the value of structural diagnostics may decrease. Nevertheless, until such consistency is achieved, structural cues remain a valuable tool for identifying and improving reasoning quality. Finally, the construction of reasoning trees in LCoT2Tree currently depend on off-the-shelf large language models (e.g., DeepSeek-V3 (Liu et al., 2024a)), which makes the pipeline computationally expensive. References Marthe Ballon, Andres Algaba, and Vincent Ginis. 2025. The relationship between reasoning and performance in large language models–o3 (mini) thinks harder, not longer. arXiv preprint arXiv:2502.15631 . Maciej Besta, Nils Blach, Ales Kubicek, Robert Ger- stenberger, Michal Podstawski, Lukas Gianinazzi, Joanna Gajda, Tomasz Lehmann, Hubert Niewiadom- ski, Piotr Nyczyk, and 1 others. 2024. Graph of thoughts: Solving elaborate problems with large lan- guage models. In Proceedings of the AAAI Confer- ence on Artificial Intelligence (AAAI) . Shaked Brody, Uri Alon, and Eran Yahav. 2022. How attentive are graph attention networks? In Inter- national Conference on Learning Representations (ICLR) . Bradley Brown, Jordan Juravsky, Ryan Ehrlich, Ronald Clark, Quoc V Le, Christopher Ré, and Azalia Mirho- seini. 2024. Large language monkeys: Scaling infer- ence compute with repeated sampling. arXiv preprint arXiv:2407.21787 . Qiguang Chen, Libo Qin, Jinhao Liu, Dengyun Peng, Jiannan Guan, Peng Wang, Mengkang Hu, Yuhang Zhou, Te Gao, and Wangxiang Che. 2025. Towards reasoning era: A survey of long chain-of-thought for reasoning large language models. arXiv preprint arXiv:2503.09567 .Qiguang Chen, Libo Qin, Jiaqi Wang, Jingxuan Zhou, and Wanxiang Che. 2024a. Unlocking the capabili- ties of thought: A reasoning boundary framework to quantify and optimize chain-of-thought. Advances in Neural Information Processing Systems (NeurIPS) . Xingyu Chen, Jiahao Xu, Tian Liang, Zhiwei He, Jianhui Pang, Dian Yu, Linfeng Song, Qiuzhi Liu, Mengfei Zhou, Zhuosheng Zhang, and 1 others. 2024b. Do not think that much for 2+ 3=? on the overthinking of o1-like llms. arXiv preprint arXiv:2412.21187 . Alejandro Cuadron, Dacheng Li, Wenjie Ma, Xingyao Wang, Yichuan | https://arxiv.org/abs/2505.22148v1 |
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continue segment in a reasoning chain that involves no logical transition, such as exploration or verification. We then analyze the collected LCoTs to identify common linguistic pat- terns ( i.e., separators) that signal shifts between distinct reasoning steps. The separators set to [“Al- ternatively”, “Hmm”, “Let me verify”, “let’s ver- ify”, “To verify”, “Wait”, “Verify”] for Deepseek- 32B, QwQ-32B, and Deepseek-R1. And extend with [“Let’s confirm”, “Let’s check”, “Another ex- ample”, “But let’s”, “wait”, “No:”, “no:”, “Now”] for Seed-1.5-thinking-pro and Grok-3-mini-beta. According to these markers, the long reasoning chain is divided into individual thoughts, forming a Thoughts List where each item represents a single reasoning fragment. Stage 3: Assign Step. Each thought in the Thoughts List is then aligned with one or more corresponding steps in the Reasoning Sketch , based on its role in the overall reasoning process. This mapping is performed using an LLM with prompt 7, generating a Thought Step dictionary that captures the contextual meaning and reasoning stage ( i.e., depth) associated with each thought. To improve token efficiency, we group and merge adjacent thoughts before feeding them into the LLM. How- ever, due to the large number of thoughts in the Thought List , processing them all at once is infea- sible. Therefore, we segment the list into smaller batches, each containing consecutive thoughts with a combined word count of no more than 600. These batches are then input to the LLM, which returns the corresponding reasoning step for each thought in a single response. Stage 4: Identify Function. We further analyze each pair of consecutive thoughts using LLM withprompt 8 to determine the role of the latter thought in relation to the former ( e.g., continuation, ex- ploration, or verification). This step provides a more precise understanding of the relationships between individual thoughts within the reasoning process. Specifically, the roles are categorized as follows: (1) Continuous Logic – A direct continua- tion or extension of the reasoning in the previous thought. (2) Exploration – Introduces alternative reasoning paths, unrelated concepts, or new topics. (3) Backtracking – Revises, corrects, or adjusts the reasoning from the previous step. (4) Validation – Provides supporting evidence, justification, or ex- amples for the previous thought. If the Thought Listcontains Nthoughts, we perform N−1LLM calls to analyze each adjacent pair. Stage 5: Build Tree. Finally, we organize the segmented thoughts into a hierarchical tree struc- ture. Each node Nj iin the tree corresponds to thei-th thought Ti, where jindicates how many times Tihas appeared. The placement of a node is determined by the Thought Step , and each edge rep- resents a transition to a deeper level of reasoning, with the edge type defined by the Thought Function of its child node. When inserting a new thought Ti, we first identify the ordered list of reasoning steps it maps to, denoted as [S1 i, ..., Sn i]. Here, n indicates that the current thought encompasses n reasoning steps. Consequently, we create nnodes N1 i, ..., Nn i, where each node Nj irepresents the portion of the thought aligned with the Sj i-th | https://arxiv.org/abs/2505.22148v1 |
step. The insertion process follows two rules: (1) If S1 i is greater than the step of the latest node Nj i−1in the tree, the new node N1 iis added as a child of Nj i−1. (2) Otherwise, we backtrack to the most recent node at step S1 i−1. Then we create a new branch from that node and link it to new node N1 i. Once N1 iis placed, the remaining nodes N2 i, ...,Nn iare added sequentially and connected to the previous one. For example, in Figure 3, when inserting T8to the tree, its associated reasoning steps[S1 8, S2 8, S3 8] = [1 ,2,3], as determined by the Thought Step . At that point, the latest node in tree isN1 7, which is at step 3—greater than 1. There- fore, we backtrack to the latest node at step 0, N1 0, and attach N1 8as its child. After that, N2 8andN3 8 are linked sequentially to N1 8andN2 8, respectively. In the end, we successfully extract the whole tree structure using LCoT2Tree. To support inter- pretation, we provide a visualization tool for the generated reasoning tree, which allows users to in- teractively explore the thought process behind each node while viewing the overall tree structure. Ex- ample screenshots of the visualization results are shown in Figures 10–17. In each figure, the right column displays the key reasoning steps identified in Step 1, and each node represents an individual thought. The solid line denotes the edge from fa- ther to child and the dash line denotes the edge from child to father. Edges are colored according to its function in reasoning process. B Classification Implementation Detail B.1 Dataset Construction We use the same dataset as described in Section 3.1, consisting of response samples generated by five reasoning LLMs (LLMs) across four public bench- marks: MATH, GPQA, LiveCodeBench (LCB), and MMLU-Pro. Each sample is labeled as pos- itive or negative with answer correctness, which serves as the ground truth for our binary classi- fication task. To ensure sufficient data volume, we apply repeated sampling for each benchmark, generating up to 2,000 samples per dataset. For instance, the LCB benchmark contains 167 unique problems. By generating 16 responses per problem, we obtain approximately 1,000 correctly answered samples and 1,000 incorrect ones. B.2 Graph Construction For each response, we begin by applying the LCoT2Tree framework to convert it into a struc- tured reasoning tree. Each node Nj iin the tree cor- responds to the i-th thought Ti, where jindicates how many times Tihas appeared. The placement of a node is determined by the Thought Step , and each edge represents a transition to a deeper level of rea- soning, with the edge type defined by the Thought Function of its child node as introduced in 3.2. We then transform the tree into a graph representation. Notably, we construct bidirectional edges, allowing information to flow both from parent to child and from child to parent. This design enables the model to simulate behaviors like backtracking, which are often essential in complex reasoning. In the | https://arxiv.org/abs/2505.22148v1 |
end, each sample produces a single graph instance for classification. B.3 Node and Edge Features We design informative features for both nodes and edges to enhance the performance of our tree-based classification model. For each node in the reason-ing tree, we extract the following features: (1) the index of the current thought, (2) the reason- ing depth of the current node, (3) the cumulative number of tokens used up to current node, (4) the number of child nodes, and (5) the cumulative num- ber of nodes at the same reasoning depth. For edge features, we assign each parent-to-child edge a feature based on its logical role as identi- fied by LCoT2Tree: “1” for continuation, “2” for exploration, “3” for backtracking, and “4” for vali- dation. To distinguish child-to-parent edges (used to capture reverse information flow, such as back- tracking), we assign the same value but multiply it by -1. This setup helps the model differentiate directional semantics during message passing. B.4 Hyperparameters We adopt the GATv2 architecture (Brody et al., 2022) to model reasoning trees, leveraging its dy- namic attention mechanism and improved capabil- ity for capturing hierarchical dependencies. The model comprises two GATv2 layers, each with a hidden size of 64. After message passing, graph- level embeddings are obtained via global mean pooling. These embeddings are then fed into a two-layer MLP with ReLU activation, serving as the classification head to predict whether a given reasoning structure leads to a correct or incorrect answer. To train the model, we use binary cross-entropy loss and the Adam optimizer with a learning rate of 1e-3. The model is trained for up to 100 epochs with a batch size of 32. We split the training dataset into 90% for training and 10% for validation. All experiments are conducted using the PyTorch Geo- metric framework. C Additional Experimental Results C.1 Additional Results on Task-specific Analysis In Table 3, we provide additional results from the task separability experiments using the DeepSeek- 32B and QwQ-32B models. We classify reasoning trees across all task pairs, including MATH/GPQA, MATH/LCB, MATH/MMLU-Pro, GPQA/LCB, GPQA/MMLU-Pro, and LCB/MMLU-Pro. The findings are consistent with the conclusions pre- sented in Section 4.2. MATH/GPQA MATH/LCB MATH/MMLU GPQA/LCB GPQA/MMLU MMLU/LCB DS-32Length-based 50.45% 63.72% 69.43% 60.65% 77.71% 82.34% Tree-based 83.51% 89.22% 78.46% 85.55% 82.89% 92.12% Gain +33.06% +25.50% +9.03% +24.90% +5.18% +9.78% QwQ-32Length-based 52.51% 56.64% 67.38% 61.22% 68.25% 73.03% Tree-based 77.82% 67.88% 80.85% 85.69% 76.70% 87.20% Gain +25.31% +11.24% +12.17% +24.47% +8.45% +14.17% Table 3: Comparison of task-specific classification accuracy using the baseline length-based method and the proposed tree-based representation. MATH GPQA LiveCodeBench MMLU-Pro 4 Datasets DeepSeek-32B ±0.0048 ±0.0037 ±0.0024 ±0.0089 ±0.0043 QwQ-32B ±0.0023 ±0.0025 ±0.0030 ±0.0079 ±0.0039 DeepSeek-R1 ±0.0076 ±0.0029 ±0.0010 ±0.0051 ±0.0018 Seed-1.5-Thinking-pro ±0.0037 ±0.0061 ±0.0041 ±0.0066 ±0.0026 Grok-3-mini-beta ±0.0025 ±0.0020 ±0.0037 ±0.0153 ±0.0041 Table 4: Standard deviation of our proposed tree-based approach on classifying response correctness based on LCoT information corresponding to Table 1. DS-32/DS-R1 DS-32/QwQ-32 DS-32/Seed DS-32/Grok DS-R1/Seed MATHLength-based 55.17% 61.49% 55.58% 61.06% 56.90% Tree-based 67.88% 70.93% 82.15% 93.22% 80.10% Gain +12.71% +9.44% +26.57% +32.16% +23.20% GPQALength-based 50.87% 51.12% 67.96% 49.43% 65.34% Tree-based 75.34% | https://arxiv.org/abs/2505.22148v1 |
61.60% 95.20% 99.42% 84.68% Gain +24.47% +10.48% +27.24% +49.99% +19.34% LCBLength-based 54.49% 54.17% 52.37% 54.49% 53.39% Tree-based 86.32% 71.73% 96.12% 86.32% 82.51% Gain +31.83% +17.56% +43.75% +31.83% +29.12% MMLULength-based 55.36% 60.10% 54.17% 53.23% 59.55% Tree-based 62.86% 64.99% 73.65% 85.62% 71.89% Gain +7.50% +4.89% +19.48% +32.39% +12.34% Table 5: Comparison of model-specific classification accuracy using the baseline length-based method and the proposed tree-based representation. DeepSeek-32B QwQ-32B LiveCodeBench MATH LiveCodeBench MATH V ote - 80.41% - 71.19% Length-Best 56.92% 56.70% 47.37% 55.93% Length-V ote - 67.01% - 57.63% ORM-Best 50.77% 60.82% 42.11% 57.63% ORM-V ote - 68.04% - 67.80% PRM-Best 62.89% 63.92% 50.88% 57.63% PRM-V ote - 62.89% - 55.93% Ours-Best 61.54% 65.98% 52.63% 67.80% Ours-V ote - 82.47% - 71.19% Table 6: Accuracy comparison of different Best-of-N decoding strategies on the two benchmark. C.2 Additional Results on Model-specific Analysis Table 5 presents the detailed analysis of whether different models display distinguishable reasoning behaviors when applied to the same dataset. The re- sults confirm that LCoT2Tree effectively captures model-specific reasoning patterns that generalize across tasks. Specifically, QwQ-32B exhibits rea- soning behaviors more closely aligned with the DeepSeek family, compared to Grok-3-mini-beta and Seed-1.5-Thinking-pro. These findings further underscore the effectiveness of structural represen- tations in revealing subtle differences in model be- havior. C.3 Additional Results on Best-of-N Decoding Table 6 provides a detailed comparison of different Best-of-N decoding strategies on the MATH and LiveCodeBench (LCB) datasets using responses from two LLMs: DeepSeek-32B (DS-32) and Qwen-32B (QwQ-32). For the MATH benchmark, we evaluate on samples from the MATH500 and Level5 subsets that are not included in the training set. For LCB, we use LCB-v6 as the test set. In both cases, we ensure that the selected test sam- ples are challenging—each sample is incorrectly answered at least twice across 10 runs. We set N= 10 and compare our proposed tree-based methods (Ours-Best and Ours-V ote) against several baselines: •V ote (Wang et al., 2023b): Standard majority voting among Noutputs. •Length-Best (Wang et al., 2025): Select the response with the fewest tokens. •Length-V ote (Wu et al., 2025): Majority vot- ing after selecting the kresponses with reli- able CoT length. •ORM-Best (Brown et al., 2024): Select the response with the highest outcome reward model score using Skywork-Reward-Gemma- 2-27B-v0.2 (Liu et al., 2024b). •ORM-V ote (Brown et al., 2024): Weighted Majority voting (Lightman et al., 2023) with the outcome reward model score. •PRM-Best (Zhang et al., 2025), which scores responses based on the product of step-level scores from a process reward model (i.e., Qwen2.5-Math-PRM-72B)•PRM-V ote (Zhang et al., 2025), Weighted Ma- jority voting (Lightman et al., 2023) with the processing reward model score. • Ours-Best: Select the response with the high- est score assigned by our tree-based reasoning quality classifier mentioned in Section 3.3. •Ours-V ote: Weighted Majority voting with the score of our classifier. Our method consistently outperforms traditional heuristics and reward model-based baselines, par- ticularly in the MATH dataset, where precise multi- step reasoning is crucial. Notably, for DeepSeek- 32B on MATH, our tree-based voting method achieves the highest accuracy at 82.47%, signif- icantly surpassing both | https://arxiv.org/abs/2505.22148v1 |
Length-Best (56.70%) and ORM-Best (60.82%). Similar trends are observed for QwQ-32B, with our model showing competi- tive or superior performance. These results confirm that incorporating structural reasoning patterns via LCoT2Tree leads to a reliable output selection in complex reasoning tasks. D Diagnostic Insight into Reasoning Behaviors & Visualization Results D.1 Insight into Error Behaviors In this section, we present a detailed analysis of common error patterns found within reasoning trees. We use GNNExplainer (Ying et al., 2019), a graph-based interpretability method, to identify which edges in a reasoning tree contribute most significantly to the model’s predictions. For each reasoning tree, GNNExplainer assigns an impor- tance weight to every edge, reflecting its influence on the model’s output. These weights are normal- ized to the [0,1]range, and we visualize the tree by adjusting the edge thickness and color intensity ac- cording to these scores. The darker and thicker the edge, the more critical it is to the model’s decision. Illustrative examples are shown in Figure 9. Based on this analysis, we extract and categorize the most usual subgraphs associated with incorrect predictions into four primary error patterns. (A) Over Branching: excessive exploration or verifi- cation from a single node; (B) Step Redundancy: repetitive or unnecessary reasoning within the same step; (C) Direct Reasoning: abrupt transitions from one reasoning step to much deeper steps with min- imal branching; (D) Skipped Thinking: leaping across multiple reasoning steps without proper in- termediate logic. These patterns are visualized in the left part of Figure 9, with real examples provided on the right. Notably, these findings reveal that both overly com- plex and overly simplistic reasoning paths can lead to incorrect outcome, underscoring the need for balanced, coherent, and well-structured reasoning in high-quality LLMs. D.2 Task-specific Reasoning Behaviors We have quantitatively demonstrated that LCoT2Tree effectively facilitates the separation of task-specific reasoning contents, as detailed in Section 4.2. In this section, we leverage LCoT2Tree to pinpiont the disparate behaviors exhibited by the DeepSeek-32B model across various tasks. The key findings are summarized below: For MATH (Figure 10), the reasoning trees typ- ically display a diagonally descending structure, with progressively deeper steps achieved through repeated backtracking. This pattern reflects a struc- tured, hierarchical problem-solving strategy. In the visualization, dashed lines—representing back- tracking—are identified as key structural features that distinguish MATH from other tasks. For LiveCodeBench (Figure 11), the trees of- ten exhibit broad, parallel branching, where many sibling nodes continue with independent linear thoughts that are rarely explored or verified further. This suggests a shallow, scattered reasoning style. Our visualization also reveals that these parallel branches contribute most significantly to classify- ing this task. For GPQA (Figure 12), the reasoning trees con- tain numerous high out-degree nodes, indicating that the model frequently revisits and expands on specific concepts. This behavior suggests intensive cognitive effort and repeated clarification, reflect- ing the model’s attempt to thoroughly understand difficult points—while also hinting at a lack of con- fidence in its reasoning. Finally, for MMLU-Pro(Figure 13), the reason- ing trees are relatively shallow, with fewer nodes and minimal branching. This suggests a more di- rect, | https://arxiv.org/abs/2505.22148v1 |
deductive approach with limited exploration, which is consistent with the knowledge-intensive nature of MMLU-Pro questions rather than deeply compositional reasoning. These observations highlight how LCoT2Tree provides fine-grained insights into the cognitive strategies employed by the model in diverse reason- ing scenarios.D.3 Model-specific Reasoning Behaviors We provide a detailed comparison of how differ- ent LLMs approach the same task by visualizing and analyzing their reasoning trees on the MATH dataset. Focusing on DeepSeek-32B as a reference point, we summarize several key observations: DeepSeek-32B (DS-32; Figure 10) typically pro- duces reasoning trees with a diagonally descend- ing structure, with depth increasing progressively through backtracking. This reflects a structured, step-by-step problem-solving reasoning process. DeepSeek-R1 (Figure 14) exhibits similar struc- tural characteristics to DS-32, but with a notable difference: it tends to terminate detailed explo- ration earlier and backtrack more quickly to begin- ning steps. This indicates a more aggressive prun- ing strategy to streamline the reasoning path. In vi- sualizations, connections between Step 0 and Step 1 serve as critical features distinguishing DeepSeek- R1’s behavior. QwQ-32B (Figure 15) also mirrors the behavior of DS-32 to some extent but differs in the latter stages. Unlike DS-32, which often rushes toward the final answer, QwQ-32B continues to invest cog- nitive effort into deeper exploration. In the visu- alization, expanded right subtrees often emerge as defining characteristics of QwQ-32B’s reasoning tree. In contrast, Seed-1.5-Thinking-pro (Figure 16) and Grok-3-mini-beta (Figure 17) follow a markedly different reasoning strategy. They ex- hibit fewer thought transitions during reasoning. As a result, their trees contain fewer nodes and branches, forming simpler structures. This sug- gests a straightforward problem-solving style with limited iterative refinement. These insights reinforce that LCoT2Tree not only captures reasoning structure at the task level, but also reveals distinctive behavioral patterns across model families. Step1 Prompt in LCoT2Tree tool to extract reasoning sketch from LCoT Analyze the following reasoning text and extract a strictly ordered, atomic sequence of key reasoning steps. Focus on extracting the validated, logically essential progression of thoughts while excluding backtracking, rechecks, or redundant details. Reasoning text: <reasoning_text> {{text}} </reasoning_text> Please read the entire text carefully and generate by following these rules: 1. Find the key steps and the logical flow of reasoning. 2. Each step must represent a single, indivisible logical action that directly advances the reasoning. 3. Determine the correct version of the step, ignoring redundant information. A correct step should be able to push the reasoning logic forward and have no errors in itself. 4. Do not skip steps. Do not merge steps. Use the original phrasing where possible. 5. Do not include verification steps unless it introduces new constraints. 6. Organize the steps into a coherent sequence of key reasoning steps and number it sequentially (1., 2., 3., ...). 7. Maintain strict output format. Output format: <reasoning_process> Step 1. concise statement: Detail step Step 2. concise statement: Detail step Step 3. concise statement: Detail step </reasoning_process> Please list the key reasoning steps of the provided text. Figure 6: The content of Step1 Prompt in LCoT2Tree tool to extract reasoning sketch from LCoT. Step3 Prompt | https://arxiv.org/abs/2505.22148v1 |
in LCoT2Tree tool to assign reasoning step to each thought. Your task is to match each reasoning thought from List B to corresponding step number(s) in the List A. Follow the following process: 1. First understand List B: - For each thought in List B, identify if it describes some specific calculation processes (mathemati- cal operation, logical transformation, or data manipulation) - Ignore the describation that only state conclusions, concepts without showing the actual processing detail 2. Then math to List A: - For each thought from List B, find all steps in List A that: * Show the same underlying calculation (even with different numbers/words) * Represent the partial or same reasoning process - Ignore superficial wording differences - focus on logical equivalence 3. Output requirements: - Return ALL plausible matches where computational processes align - Never return empty arrays (except for thought B0 if needed) - Multiple matches are encouraged when justified - Maintain strict JSON format Input: - List A (Detailed Steps): <list_a> {{reasoning_step}} </list_a> - List B (Reasoning Thoughts): <list_b> {{thoughts}} </list_b> Output Format (strict JSON): “‘json { "B0": ["A1"], "B1": ["A3"], "B2": ["A1", "A4"], ... }”’ Please match the reasoning thoughts in List B to step in the List A. Figure 7: The content of Step3 Prompt in LCoT2Tree tool to assign reasoning step to each thought. Step4 Prompt in LCoT2Tree tool to assign function to each thought. Your task is to classify Text2’s purpose relative to Text1 using these categories: Categories: 1. Continuous Logic - Direct continuation/extension of Text1’s reasoning flow 2. Exploration - Introduces parallel/unrelated concepts from Text1, alternative reasoning paths, or new topics 3. Backtracking - Revises, corrects, or adjusts previous step 4. Validation - Provides supporting evidence, logical justification, or examples for Text1’s claims Input: {{ "Text1": "TEXT1", "Text2": "TEXT2" }} Output Format: Return only JSON format “‘json{"Category": "Name of Category"}”’ Figure 8: The content of Step4 Prompt in LCoT2Tree tool to assign function to each thought. (A) OverBranching (B) Step Redundancy…… …(C) Direct ReasoningStep 𝑖Step 𝑗(𝑗≫𝑖) (D) Skipped ThinkingStep 𝑖Step 𝑗(𝑗≫𝑖) Figure 9: Visualization results of tree structure corresponding to different error patterns. The edge is labeled with the importance generated by GNNExplainer. The darker the color and the thicker the edge, the more important it is. Figure 10: Visualization results of tree structure of a response from DeepSeek-32B onMATH dataset extracted using LCoT2Tree. The reasoning trees exhibit a downward-sloping hierarchical structure, with progressively deeper steps achieved through repeated backtracking. Figure 11: Visualization results of tree structure of a response from DeepSeek-32B onLiveCodeBench dataset extracted using LCoT2Tree. The reasoning patterns tend to show broad, parallel branching, where many sibling nodes initiate independent linear thought without subsequent exploration or verification. Figure 12: Visualization results of tree structure of a response from DeepSeek-32B onGPQA dataset extracted using LCoT2Tree. The reasoning trees contain many high out-degree nodes, indicating that the model often revisits and elaborates on complex concepts. Figure 13: Visualization results of tree structure of a response from DeepSeek-32B onMMLU-Pro dataset extracted using LCoT2Tree. The reasoning trees contain fewer nodes and minimal branching, indicating | https://arxiv.org/abs/2505.22148v1 |
a more direct and deductive reasoning style with less exploration. Figure 14: Visualization results of tree structure of a response from DeepSeek-R1 onMATH dataset extracted using LCoT2Tree. It exhibits similar behavior to DS-32, but with an important distinction: it tends to truncate detailed exploration earlier and backtrack to beginning steps more quickly to optimize its reasoning path. Figure 15: Visualization results of tree structure of a response from QwQ-32B onMATH dataset extracted using LCoT2Tree. QwQ-32B mirrors the behavior of DeepSeek-32B to some extent, but differs in how it allocates attention in the latter stages of reasoning. Figure 16: Visualization results of tree structure of a response from Seed-1.5-Thinking-pro onMATH dataset extracted using LCoT2Tree. The reasoning trees contain fewer nodes and branches, forming simpler structures. Figure 17: Visualization results of tree structure of a response from Grok-3-mini-beta onMATH dataset extracted using LCoT2Tree. The reasoning trees contain fewer nodes and branches, forming simpler structures. | https://arxiv.org/abs/2505.22148v1 |
arXiv:2505.22165v1 [cs.CL] 28 May 2025Unifying Continuous and Discrete Text Diffusion with Non-simultaneous Diffusion Processes Bocheng Li*1,2, Zhujin Gao*1,2, Linli Xu†1,2 1School of Computer Science and Technology, University of Science and Technology of China 2State Key Laboratory of Cognitive Intelligence {bcli,gaozhujin}@mail.ustc.edu.cn ,linlixu@ustc.edu.cn Abstract Diffusion models have emerged as a promis- ing approach for text generation, with recent works falling into two main categories: dis- crete and continuous diffusion models. Dis- crete diffusion models apply token corruption independently using categorical distributions, allowing for different diffusion progress across tokens but lacking fine-grained control. Con- tinuous diffusion models map tokens to con- tinuous spaces and apply fine-grained noise, but the diffusion progress is uniform across to- kens, limiting their ability to capture semantic nuances. To address these limitations, we pro- pose Non-simultan eous C ontinuous Diffusion Models (NeoDiff), a novel diffusion model that integrates the strengths of both discrete and continuous approaches. NeoDiff introduces a Poisson diffusion process for the forward pro- cess, enabling a flexible and fine-grained nois- ing paradigm, and employs a time predictor for the reverse process to adaptively modulate the denoising progress based on token semantics. Furthermore, NeoDiff utilizes an optimized schedule for inference to ensure more precise noise control and improved performance. Our approach unifies the theories of discrete and continuous diffusion models, offering a more principled and effective framework for text gen- eration. Experimental results on several text generation tasks demonstrate NeoDiff’s supe- rior performance compared to baselines of non- autoregressive continuous and discrete diffu- sion models, iterative-based methods and au- toregressive diffusion-based methods. These results highlight NeoDiff’s potential as a pow- erful tool for generating high-quality text and advancing the field of diffusion-based text gen- eration. 1 Introduction Diffusion models have demonstrated remarkable success in generating high-quality samples in var- *Equal contribution. †Corresponding author.ious domains, including vision (Dhariwal and Nichol, 2021; Nichol and Dhariwal, 2021; Ho and Salimans, 2021; Rombach et al., 2022) and audio (Chen et al., 2020; Kong et al., 2020). Inspired by their achievements, there has been a growing inter- est in applying diffusion models to text generation tasks (Li et al., 2022; Gong et al., 2022; Gao et al., 2024; Zheng et al., 2023). The core idea behind diffusion models is to cor- rupt the data through a forward process and then learn to reverse this process to generate new sam- ples. In text generation, existing diffusion models can be broadly categorized into two classes: dis- crete and continuous diffusion models. Discrete diffusion models treat tokens as discrete random variables and perform state transitions indepen- dently for each token using a categorical distri- bution. While straightforward, this approach fails to capture the continuous and fine-grained nature of language, limiting the potential benefits of multi- step generation. Continuous diffusion models, on the other hand, operate in a continuous space by mapping tokens to continuous representations, en- abling more fine-grained perturbations. However, these models typically apply diffusion at the sen- tence level, resulting in uniform noise levels across all tokens within a sentence, restricting the model’s ability to leverage contextual information and re- cover tokens with varying | https://arxiv.org/abs/2505.22165v1 |
noise levels based on the surrounding context (Chen et al., 2023; Wu et al., 2024). To address these limitations, we propose inte- grating the complementary strengths of discrete and continuous diffusion approaches, enabling fine- grained noise control at the token level. This uni- fied approach aims to provide precise token-level control while maintaining continuous-valued noise distributions, which is absent in existing frame- works. While recent text diffusion models (Han et al., 2023; Gong et al., 2023; Wu et al., 2024) have made advances, they do not fully address 1 NicemeetDiscrete Diffusiontoyou!Nicemeettoyou!Nicemeettoyou!Nicemeettoyou!NicemeetContinuous Diffusiontoyou!Nicemeettoyou!Nicemeettoyou!Nicemeettoyou!NicemeetNon-simultaneous Continuous Diffusion toyou! Nicemeettoyou!Nicemeettoyou!Nicemeet toyou! 𝑡=100𝑡=66𝑡=33𝑡=0𝑡=99.86𝑡=67.96𝑡=33.26𝑡=0.05𝑡=100𝑡=66𝑡=33𝑡=0 Fine-grained flexible noisingFine-grained yet uniform noisingCoarse-grained noising controlFigure 1: Comparison of the noising paradigms employed by Non-simultaneous Continuous Diffusion and two other diffusion models. The color intensity on the text tokens represents the token-level noising progress (intrinsic timeτ). Discrete diffusion applies an independent but coarse-grained noising paradigm to each token within a sentence. In contrast, continuous diffusion utilizes a fine-grained noising schedule but applies it uniformly across all tokens. NeoDiff distinguishes itself by assigning an independent, fine-grained intrinsic time τto each token, with finer noising schedule in extrinsic time t. this requirement, necessitating a unified theoreti- cal framework that bridges discrete and continuous diffusion paradigms through a carefully designed forward process. Furthermore, we observe that existing ap- proaches primarily focus on enhancing the forward process, overlooking the inherent varying difficul- ties in denoising different tokens and the impact of generation context. In analyzing the reverse pro- cess, we recognize that tokens with lower noise levels can guide the recovery of more heavily cor- rupted tokens, thereby enhancing the overall text generation quality. In response to these challenges, we present Non-simultan eous C ontinuous Diffusion Models (NeoDiff), which unifies discrete and continuous diffusion models through a bi-temporal framework. The key insight is to generalize the time variable in previous diffusion models into extrinsic time t, representing the diffusion progress of the entire sen- tence, and intrinsic time τ, tracking the diffusion progress of each individual token. This generaliza- tion enables us to introduce a novel Poisson process as the forward process, seamlessly integrating the flexibility of discrete noise with the fine granularity of continuous noise. An overview of this noising paradigm is illustrated in Figure 1. To optimize the reverse process, we develop a context-aware time predictor that estimates the in- trinsic time τusing an adaptive modulation func- tion to guide the denoising process. The extrin- sic time schedule is further calibrated through Bayesian optimization, providing precise control over the noise distribution. NeoDiff achieves a fine-grained, improved diffu- sion process in both forward and reverse directions,naturally overcoming the constraints of previous discrete and continuous diffusion models, and ex- hibiting superior generation quality. We evaluate NeoDiff on a diverse set of NLP tasks, including machine translation, paraphrasing, text simplifica- tion, and question generation. NeoDiff consistently outperforms previous non-autoregressive diffusion- based and iteration-based methods, as well as au- toregressive diffusion baselines. Specifically, our contributions can be summarized as follows: •We introduce NeoDiff, a unified theoretical framework that combines the advantages of discrete and continuous noise, generalizing and | https://arxiv.org/abs/2505.22165v1 |
unifying existing text diffusion models. •We propose the Poisson diffusion process as the forward process, enabling fine-grained cor- ruption of text data, a context-aware time pre- dictor that adaptively modulates the reverse process based on semantic context, and an optimized extrinsic time schedule for precise noising control. •We conduct extensive experiments to eval- uate the effectiveness of NeoDiff and com- pare it to existing text diffusion models. Our results highlight the advantages of our uni- fied framework and suggest its potential to advance diffusion-based text generation. 2 Background 2.1 Diffusion Models Diffusion models assume a gradual noise injection process over time for data samples z0∈RN×d. The forward diffusion process forms a series of la- tent variables z1,z2,···,zTsatisfying the Markov 2 property, and finally become pure Gaussian noise zT∼ N (0,I): q(zt|zt−1) =N(zt;√αtzt−1, βtI),(1) where αt+βt= 1, determining the degree of noising at time tand consituting the noise schedule. The reverse process is parameterized as pθ(zt−1|zt) =N(zt−1;µθ(zt, t),Σθ(zt, t)), (2) Here, µθ(·)andΣθ(·)are the model’s estimates of the distribution mean and covariance matrix, respectively. The training objective is derived from the variational lower bound of the negative log- likelihood loss, and can be then simplified as an MSE loss (Ho et al., 2020; Li et al., 2022; Gao et al., 2024): LVLB=Eh ∥zθ(zt, t)−z0∥2−logp(z0|z1)i 2.2 Discrete Diffusion Models Discrete diffusion models directly model the noise on categorical distributions, discarding the assump- tion that the noise in latent variables follows a nor- mal distribution in continuous space. These mod- els typically represent data as sequences of one- hot vectors and employ a transition matrix to add noise to the data. Among them, Hoogeboom et al. (2021a) proposed a multinomial diffusion model that employs a uniform noising method. Austin et al. (2021) introduced D3PM, which employs a noising method with an absorbing state. Specif- ically, they added an absorbing state [MASK] to the vocabulary, which can only be entered but not exited. The remaining states, at each diffusion step, either stay in the current state or enter the absorb- ing state with a certain probability. Recently, Lou et al. (2024) made progress by developing score entropy, which extends score matching to discrete spaces and demonstrates substantial performance improvements. While effectively adapting diffu- sion models to discrete data, these methods have limitations. The discrete nature of the noise limits its expressiveness, making it difficult to capture the nuances of continuous transitions between states. This restricts the model’s ability to represent grad- ual semantic changes or finely adjust individual token features, potentially limiting the benefits of multi-step generation.2.3 Continuous Diffusion Models Continuous diffusion models map discrete tokens to a continuous vector space using a mapping func- tion, allowing the application of standard continu- ous diffusion processes. Analog Bits (Chen et al., 2022) uses a binary encoding scheme ( int2bit : Z→0,1⌈log2V⌉) to represent token indices as bi- nary sequences. After the reverse diffusion process, a quantization operation followed by binary decod- ing ( bit2int : 0,1⌈log2V⌉→Z) recovers the token indices. Han et al. (2023) proposed a mapping function logits-generation :Z→RV, which trans- forms token indices into a probability | https://arxiv.org/abs/2505.22165v1 |
simplex. Li et al. (2022) proposed Diffusion-LM, where the token sequence yis first mapped to a random repre- sentation z0using a word embedding as the mean. After the reverse diffusion process, the generated vectors are rounded back to discrete tokens. Gong et al. (2022) extended this approach to sequence-to- sequence generation with DiffuSeq, which concate- nates the source and target sentences and utilizes an attention mechanism to leverage source informa- tion during generation. However, a key limitation of continuous diffusion models is the uniform noise injection applied to all tokens during the forward process. This uniform noise injection hinders the model’s ability to effectively leverage contextual information. Ideally, varying noise levels across tokens would allow the model to utilize less noisy tokens as context for restoring more corrupted ones, facilitating better contextual modeling. 2.4 Improvements over Previous Diffusion Models Recent studies have explored various methods to address the limitations discussed above. Han et al. (2023) introduced a semi-autoregressive generation strategy that generates fixed-length blocks autore- gressively while employing non-autoregressive iter- ative denoising within each block. Wu et al. (2024) proposed a hierarchical noise addition method, where noise levels increase monotonically from left to right within a sentence, enabling autoregres- sive generation. Gong et al. (2023) presented a hy- brid approach that combines standard continuous noise with the probabilistic replacement of tokens with [MASK], integrating discrete and continuous noise. Although these studies have contributed to enhancing the forward diffusion process, their im- provements did not fully achieve fine-grained noise at the token level, thus not completely addressing 3 the limitations of both continuous and discrete dif- fusion models. Also, these approaches typically employ a fixed reverse process that mirrors the for- ward diffusion process, without considering the varying difficulties in denoising different tokens and the impact of the actual generation context. 3 Non-simultaeous Continuous Diffusion Models To address these limitations, we propose a uni- fied diffusion framework called Non-simultaneous Continuous Diffusion Models (NeoDiff). Figure 2 presents an overview of NeoDiff, illustrating its architecture and key components. NeoDiff em- ploys an Encoder-Decoder Transformer architec- ture(Vaswani et al., 2017), with the decoder serving as the primary component for denoising, and the encoder provides the embedding of the condition sentence xto a transformer-decoder-based time pre- dictor. In the following sections, we will provide a detailed formulation of NeoDiff and demonstrate how it addresses the limitations of previous ap- proaches. 3.1 Unified Formulation and Training Objective We present a unified framework for diffusion mod- els by introducing two time dimensions: extrinsic timetand intrinsic time τ. The extrinsic time t represents the global diffusion progress of the en- tire sentence, while the intrinsic time τcaptures the diffusion progress of individual tokens. This formulation generalizes existing ap- proaches. We can easily derive discrete diffu- sion models by modeling τas a monotonically increasing random function of t, with τt∈ {0,1}, where τt= 0 andτt= 1 signify original and fully corrupted tokens, respectively. And con- tinuous diffusion can be obtained by setting τ as a deterministic function that typically equals t(τt=t). Furthermore, recent hybrid diffu- sion models, such as DiffuSeq-V2(Gong et | https://arxiv.org/abs/2505.22165v1 |
al., 2023), can also be formalized under this frame- work by setting τt= max( t+τmask(t),1), where τmask(t)∼Bernoulli (γ,¯β(t))andγis the ratio of tokens replaced by [MASK] when t= 1. NeoDiff defines τt∈[0,1]as a continuous ran- dom function of extrinsic time t∈[0,1], enabling fine-grained control over the diffusion process. We impose boundary conditions τ0= 0andτ1= 1to guarantee token preservation at initialization andcomplete corruption at termination of the diffusion process. Letz∈Rddenote a token embedding and ztits latent representation at time t, with initial and final conditions z0=zandz1∼ N(0,I). The forward process defines the joint distribution as: q(z>0, τ>0|z0) :=Y t>0q(zt, τt|z0) =Y t>0q(zt|z0, τt)q(τt), where q(zt|z0, τt) :=N zt;p ¯α(τt)z0,¯β(τt)I , and¯α(·)and¯β(·)denote noise schedules with their domains scaled to [0,1]. Given t′=t−∆t, the reverse process is defined as pθ(z0:1, τ0:1) :=pθ(z1, τ1)Y t′<1pθ(zt′, τt′|zt, τt) =pθ(z1, τ1)Y t′<1pθ(zt′|zt, τt, τt′)pθ(τt′|zt, τt). We further parameterize the distribution of zt′ as pθ(zt′|zt, τt, τt′) =q(zt′|ˆz0(zt, τt, t), τt′), where ˆz0is the model prediction of z0. Following Ho et al. (2020) and Li et al. (2022), we derive NeoDiff’s training objective from the variational lower-bound LVLB, and with the sim- plified Lzand an anchor loss Lanchor (Gao et al., 2024) as a regularization term to avoid collapse of the embedding space, the training objective of Neodiff can be written as L=Lz+Lτ+Lanchor (3) =Eq ∥ˆz0(zt, τt, t)−z0∥2 | {z } Lz(4) +X 0<t′<1KL(q(τt′)∥pθ(τt′|zt, τt))| {z } Lτ(5) +−logpθ(y|ˆz0(zt, τt, t))| {z } Lanchor . (6) A detailed derivation can be found in Appendix A. 4 Optimized 𝑡Schedule ForwardNicemeettoyou!DenoisingTransformer𝑊Embedding𝑞!𝑧"|𝑤Rounding𝑝!𝑤|𝑧"Nicemeettoyou!Nicemeet toyou! Nicemeettoyou!𝜏!Nicemeet toyou! 𝒛"𝒛#!𝒛#𝒛! 𝜏# 𝜏#!𝑞𝑧#𝑧",τ#=𝒩𝑧#;α-τ#𝑧",β/τ#𝐼𝑝$𝑧#!𝑧#,τ#,τ#!=𝑞𝑧#!𝑧"1𝑧#,τ#,𝑡,τ#!Reverse 𝜏"Text NoiseEmbedding𝑝$τ#!𝑧#,τ#=𝜏$𝑧$𝑧#,τ#,𝑡,𝑡%,𝒙Time Predictor𝒙TransformerDecoderTransformerEncoderFigure 2: An overview of NeoDiff. 3.2 Fine-Grained Forward Process Using Poisson Diffusion After establishing the unified formulation, we define a fine-grained forward diffusion process through intrinsic time τ. To quantify the diffusion progression within a single token, we introduce a discrete state function st∈ {0,1,2,···, smax}, where uniformly divided states represent distinct levels of the diffusion process from st= 0(noise- less) to st=smax(maximum noise). For an in- finitesimal time interval ∆t, the transition dynam- ics follow a Poisson process characterized by: P[st=st′+ 1] = γ(t)∆t+o(∆t) P[st=st′] = 1−γ(t)∆t+o(∆t), where γ(·)is a hyperparameter function termed the transition schedule. This formulation yields a tractable distribution for st: st∼PoissonZt 0γ(t) dt = Poisson ( λ(t)). To ensure compatibility with the continuous-time framework of NeoDiff, we normalize the state func- tion to [0,1]through normalization and clipping: τt= Clipst smax,1 = Clip s′ t,1 , where Clip (·,·)denotes the truncation operation to maintain bounded noise levels. We choose smax sufficiently large to achieve fine-grained transitions between noise states, and set λ(t) =ksmaxtto maintain E[st] =λ(t)∝smax. This design en- sures that τtremains independent of smaxand re- duces the process to a homogeneous Poisson pro- cess with constant transition schedule γ(t). However, a critical limitation of this basic for- mulation emerges when examining the coefficientof variation (CV) of the normalized state function s′(t): CV =p V[s′ t] E[s′ t]=1p λ(t)∝1√smax, which indicates that as smaxincreases, the relative variation between token states diminishes propor- tionally to1√smax. Consequently, when smaxbe- comes sufficiently large, the discreteness of the process is lost as | https://arxiv.org/abs/2505.22165v1 |
all tokens effectively share nearly identical τvalues, causing NeoDiff to degenerate into a continuous diffusion model. To address this limitation, we further introduce a variance-controlled rescaling transformation: τt=Clip Round st−λ(t)√ λ(t)σ(t) +λ(t) , smax smax(7) Under this transformation, the variables within Clip (·,·)follow a distribution centered at λ(t) with variance σ(t). To ensure that the discrete characteristics of our process remain invariant to the choice of smax, we set σ(t) =λ(t). Since the choice of λ(t)andσ(t)may result in τ1̸= 1, we truncate τtto 1 for t > t max, where tmaxis a predefined threshold. 3.3 Context-aware Reverse Process with Time Predictor We propose a context-aware reverse process that explicitly models the conditional distribution pθ(τt′|zt, τt), in contrast to previous approaches that simply mirror the forward process by assum- ingpθ(τt′|zt, τt) =q(τt′). This explicit modeling enables adaptive denoising based on both semantic context and noise states. 5 Time Predictor Design. When modeling a known distribution, researchers typically employ reparameterization tricks to model its parameters. However, in our case, the Poisson distribution’s sole parameter λ(t)is a deterministic function of tthat measures the overall noise progress of the sample and is equivalent to t. To obtain the noise progression τfor each basic token, we directly treat bothpθ(τt′|zt, τt)andq(τt′)as standard discrete distributions and learn them using cross-entropy loss, without using reparameterization tricks. Model Input Design. While ztcould serve as an input to τθ, this choice would enable the model to predict noise levels through direct comparison with all other embedding vectors. Such an approach would result in a reverse process that merely re- traces the forward process, providing little value for generation quality control. To address this limi- tation, we propose using the generated sample zθ as input to τθ. This design choice increases the modeling complexity of the prediction task while enabling τθto serve dual purposes: noise level prediction and semantic quality assessment of the generated output. To provide temporal context, we incorporate t′=t−∆tas an additional in- put, ensuring the model’s awareness of the target time distribution. The complete formulation of pθ(τt′|zt, τt)is expressed as τθ(zθ(zt, τt, t), t′,x), where xrepresents the conditioning sentence em- bedding. Pseudo Label for Training the Time Predictor. The naive approach of using τt′as the direct train- ing label for the time predictor can introduce sys- tematic bias in the learning process. While τt′is de- rived from zθ, this predicted quality measure may not accurately reflect the actual generation quality after the complete denoising process. For example, tokens initially assigned high noise levels might still produce high-quality outputs after denoising, making their initial τt′assignment suboptimal. In- stead, we propose a pseudo-labeling strategy for training the time predictor. More specifically, we first compute a confidence score for each gener- ated output using the combined loss Lz+Lanchor from the denoised prediction zθ. To ensure these confidence scores follow a distribution compatible withτt′, we apply inverse transform sampling. To accomplish this, we compute the normalized rank rfor each token’s loss within the single sample and map these ranks through the inverse cumula- tive distribution function (ICDF) of the Poissondistribution: ˜s(t) =F−1(r;λ(t)), where Fde- | https://arxiv.org/abs/2505.22165v1 |
notes the Poisson cumulative distribution function. The resulting ˜s(t)values are then transformed via Eq. (7) to obtain the final pseudo labels. 3.4 Optimized Extrinsic Time Schedule The choice of time schedule in diffusion models significantly impacts both generation quality and computational efficiency. While previous works such as Dhariwal and Nichol (2021); Chen (2023) focus on optimizing the noise schedule function with fixed extrinsic time steps, we propose to per- form direct optimization on the schedule of extrin- sic time t. Our method builds upon Li et al. (2024), who introduced post-training Bayesian optimiza- tion to select optimal subsets of time steps for infer- ence acceleration. However, where they treat time steps as discrete variables and optimize for subset selection, we formulate the problem as continu- ous optimization over the complete time schedule {t1, t2, . . . , t K}, where Kdenotes the total num- ber of diffusion steps. This continuous formulation enables more precise calibration through Bayesian optimization, effectively exploring the full space of possible time schedules. We evaluate candidate schedules using a trained model on the validation set via Bayesian optimization, optimizing for the BLEU score as our objective metric. This approach yields task-specific optimal time schedules that fur- ther enhances generation quality. The detailed opti- mization procedure is presented in Appendix B.4. 4 Experiments 4.1 Experimental Setup Datasets and Metrics We evaluate our approach on several NLP tasks, including machine transla- tion (WMT14 En-De (Bojar et al., 2014), WMT16 En-Ro (Bojar et al., 2016), IWSLT14 De-En (Cet- tolo et al., 2014)), paraphrasing (QQP), text simpli- fication (Wiki-Auto (Jiang et al., 2020)), and ques- tion generation (Quasar-T (Dhingra et al., 2017)). Dataset splits are detailed in Table 16. We use BLEU score (Papineni et al., 2002) as the evalu- ation metric across all tasks, supplemented with SacreBLEU (Post, 2018) for translation tasks. For comprehensive evaluation, we employ LLM-based evaluation using DeepSeek-V3 685B (DeepSeek- AI, 2024) with specialized prompts, assessing ac- curacy, fluency, completeness, and task-specific criteria such as creativity for translation and phras- ing diversity for paraphrasing. The evaluation pro- 6 T Model b IWSLT WMT14 WMT16 DAbsorbing 5 28 .32∗21.62∗30.41∗ Multinomial 5 21 .28∗6.94∗25.25∗ CAR-Diffusion 1 26 .78 - - AR-Diffusion 10 30 .64 - - SeqDiffuSeq 1 28 .65†23.63†23.98† SeqDiffuSeq 10 30 .03†24.24†26.17† Difformer 1 30 .94 22 .32 30 .74 Difformer 10 32 .09 23 .80 30 .93 NeoDiff 1 32.39⇑24.41⇑30.87⇑ HNeoDiff 10 33.14⇑25.28⇑32.31⇑ Table 1: Machine translation BLEU scores for NeoDiff and baseline methods. T: Model type (AR: Autore- gressive, D: Discrete, C: Continuous, H: Hybrid). ⇑: NeoDiff outperforms baselines with beam size ≤b; bold: best result. *: Results from Zheng et al. (2023); †: Results from Yuan et al. (2024); remaining data repro- duced. T Model b IWSLT WMT14 WMT16 DCMLM 5 29 .41∗23.22∗31.26∗ CMLM(MBR) 5 29 .32∗23.09∗30.92∗ CDiffusionLM 5 26 .61∗15.33∗27.01∗ DiffusionLM 50 29 .11∗17.41∗29.39∗ SeqDiffuSeq 1 30 .16†19.16†- SeqDiffuSeq 10 30 .45†19.76†- DiNoiSer 5 31 .29∗24.25∗30.93∗ DiNoiSer 50 31 .61∗24.26∗31.08∗ Difformer 1 30 .06 22 .13 30 .52 Difformer 10 31 .08 23 .26 30 .75 NeoDiff 1 31.50⇑24.09 31.59⇑ HNeoDiff 10 32.20⇑24.64⇑32.21⇑ | https://arxiv.org/abs/2505.22165v1 |
Table 2: Comparison on SacreBLEU for machine trans- lation tasks. *: Results from Ye et al. (2023); †: Results from Yuan et al. (2024); remaining data are reproduced. ⇑: NeoDiff outperforms baselines with beam size ≤b. cess involved providing the LLM with source text, generated text from different models, and specific instructions tailored to each task. Figure 3 shows the prompt templates used. To rigorously assess the diversity of outputs of the model, We also included Inter-Sentence Div-4 as Gong et al. (2022), which measure diversity at the set-of-outputs-per-source level. Baselines We compared NeoDiff against sev- eral strong baselines across multiple diffusion model categories. For discrete diffusion mod- els, we included Absorbing Diffusion (Austin et al., 2021), Multinomial Diffusion (Hoogeboom et al., 2021b), and CMLM (Ghazvininejad et al., 2019). For continuous diffusion models, we bench- marked against DiffusionLM (Li et al., 2022), Dif-T Model b QQP QT WA ARTransformer 1 29 .65⋆16.83⋆41.68⋆ Transformer 5 30 .83⋆16.45⋆43.86⋆ GPT2-base FT - 19.80⋄7.41⋄- GPT2-large FT - 20.59⋄11.10⋄- DCMLM 1 24 .02 - - CMLM 10 26 .32 - - Absorbing 10 23 .82∗17.38∗- Multinomial 10 20 .70∗16.96∗- CSeqDiffuSeq 1 23 .28†17.20†37.09† SeqDiffuSeq 10 24 .34†17.46†37.12† Difformer 1 28 .52 16 .03 40 .37 Difformer 10 30 .43 16 .66 40 .77 Meta-DiffuBDθ1 -25.52¶18.20¶38.77¶ Meta-DiffuBDθ2 -26.32¶-39.57¶ Meta-DiffuBDθ3 -22.71¶-24.71¶ TESS - 30.20‡19.50‡- TEncDM(BERT) - 30.20◦- 41.60◦ TEncDM(T5) - 30.20◦- 41.60◦ TEncDM(RoBERTa) - 30.00◦- 40.50◦ H DiffuSeq-V2 1 22 .10§- - NeoDiff 129.47⇑20.44⇑41.57HNeoDiff 10 31.32⇑20.03⇑41.86 Table 3: BLEU scores on QQP, QT, and WA(Wiki- Auto). *: Results from Zheng et al. (2023); †: Results from Yuan et al. (2024); §: Results from Gong et al. (2023); ⋆: Results from Gao et al. (2024); ⋄: Results from Gong et al. (2022); ¶: Results from Chuang et al. (2024). Dθ1= DiffuSeq. Dθ2= SeqDiffuSeq. Dθ3= Di- noiser; ‡: Results from Karimi Mahabadi et al. (2024); ◦: Results from Shabalin et al. (2025). Remaining re- sults reproduced. ⇑: NeoDiff outperforms baselines with beam size ≤b. former (Gao et al., 2024), SeqDiffuSeq (Yuan et al., 2024), AR-Diffusion (Wu et al., 2024), Di- NoiSer (Ye et al., 2024), Meta-DiffuB (Chuang et al., 2024), TESS (Karimi Mahabadi et al., 2024) and TEncDM (Shabalin et al., 2025). For hybrid approaches, we compared with DiffuSeq-V2 (Gong et al., 2023). We also included Transformer and fine-tuned GPT2 models as autoregressive base- lines. Implementation Details We set the maximum noise state smaxto 100 for all tasks and datasets, incorporating self-conditioning (Chen et al., 2022) and noise rescaling with DGS MAX = 0.2(Gao et al., 2024). We used byte pair encoding (Sennrich et al., 2016) without knowledge distillation (Kim and Rush, 2016) to evaluate under challenging con- ditions. During decoding, we employed 2D parallel decoding (Gao et al., 2024) and selected the best candidate sentence using the minimum Bayes risk (MBR) method (Kumar and Byrne, 2004) based 7 Task Models b Semantic Faithfulness Fluency Completeness Phrasing Diversity QQPCMLM 10 72.86 81.99 75.60 55.86 Transformer 1 83.64 92.56 84.96 57.19 Transformer 5 83.70 94.73 86.05 54.52 Transformer 10 83.93 94.64 86.02 54.55 NeoDiff 10 87.42 91.87 88.79 45.83 Models b Accuracy | https://arxiv.org/abs/2505.22165v1 |
Fluency Completeness Creativity WMT14Difformer 10 79.72 80.31 85.24 75.12 Transformer 5 85.66 86.35 90.81 80.07 NeoDiff 10 80.30 80.81 85.61 76.20 Table 4: LLM evaluation of text generation tasks using DeepSeek-v3 685B. We evaluate Paraphrasing (QQP Dataset) and Machine Translation (WMT14 En-De Dataset). (1) We access QQP on Semantic Faithfulness, Fluency, Completeness, and Phrasing Diversity. (2) We access WMT14 En-De on Accuracy, Fluency, Completeness, and Creativity. Detailed prompts are provided in Fig 3. on the BLEU score. We also used post-training Bayesian optimization (Li et al., 2024) to calibrate the extrinsic time schedule, limiting the optimiza- tion to 100 rounds for all tasks. Details of the experimental settings are provided in Appendix B. 4.2 Results Our experimental evaluation demonstrates NeoD- iff’s effectiveness across multiple generation tasks. On machine translation benchmarks (Table 1 and 2), NeoDiff consistently outperforms exist- ing non-autoregressive diffusion-based, iteration- based, and autoregressive diffusion approaches. As shown in Table 3, these improvements extend be- yond translation to diverse generation tasks. Un- like baselines such as AR-Diffusion that rely heav- ily on MBR and show performance drops with single samples ( b= 1), NeoDiff maintains ro- bust performance even in this constrained setting. NeoDiff also demonstrates strong performance in LLM-based evaluations (Table 4, prompts in Fig- ure 3). Notably, on the QQP task (Table 4), NeoDiff achieves superior scores in semantic faithfulness and completeness. For the WMT14 task, NeoD- iff achieves performance comparable to Difformer across multiple aspects. NeoDiff also demonstrates strong inter-sentence diversity (Inter-Sentence Div- 4) when generating multiple candidates. Detailed comparisons against AR model on QQP dataset can be found in Appendix C (Table 8). Our results show that NeoDiff can balance the quality-diversity trade-off more effectively than autoregressive mod- els like Transformer as the output space scales (i.e., with increasing b), a characteristic also observed in Gong et al. (2022). The Bayesian Optimization component introduces a manageable overhead (Ap- pendix D.1, approximately 6% of training time onWMT14). Also, NeoDiff’s inference speed and memory usage are competitive with similar models (Appendix D.2). #Poisson Diffusion ProcessTime PredictorOptimized t ScheduleBLEU Base 32.09 +P ✓ 32.75 +PT ✓ ✓ 32.97 Full ✓ ✓ ✓ 33.14 Table 5: Ablation study on the impact of proposed com- ponents on IWSLT14 De-En dataset with a b= 10 . 4.3 Analysis Our ablation studies (Table 5) demonstrate clear improvements from each component, with the full model achieving a substantial +1.05 BLEU im- provement over the baseline. We further analyze each component’s impact on generation quality: Poisson Process for Multi-token Coherence The Poisson diffusion process enables more fine- grained control over multiple tokens by precise inter-token coordination. This advantage yields a substantial performance gain over standard con- tinuous diffusion ( τt=t). As evidenced in Ta- ble 6A, this improved control manifests itself in better phrase-level coherence. Time Predictor for Guided Denoising By lever- aging information from less-noised tokens to guide the denoising trajectory of noisier ones, the time predictor enhances the model’s ability of more con- textually informed token generation. Table 6B demonstrates this through more natural word se- lections and verb choices that better preserve the | https://arxiv.org/abs/2505.22165v1 |
original meaning. 8 ASrc: das zeigt die enorm große rolle , die ein meeress- chutzgebiet spielen kann. Ref: and hence , the enormous role that a marine protected area can play. Base: and this shows the enormously big role that a area can play with a sea protected +P:so this shows the enormously big role that a marine protected area can play BSrc: er ist ganz glücklich darüber, weil er sie getäuscht hat. Ref: he’ll be very happy because he’s deceived you. +P:he’s very happy about it because he decaked her. +PT: he’s very happy about it because he deceived her. C Src: die korrelation ist also gering . Ref: so the correlation is low . +PT: soit’s a small of the correlation . Full: sothe correlation is small . Table 6: Example outputs illustrating three key mecha- nisms of NeoDiff: (A) improved phrase-level coherence with Poisson process, (B) enhanced token-level refine- ments with time predictor, and (C) better sentence-level organization with optimized schedule. Optimized Schedule for Global Coherence The optimized extrinsic time schedule enables dynamic adjustments to the diffusion trajectory, facilitat- ing escape from sub-optimal samples where se- quence order or overall structure significantly devi- ates from the target distribution. This global refine- ment allows for more substantial rewriting when needed, as demonstrated in Table 6C where entire phrases are better reorganized. Additional examples demonstrating the impact of these components are provided in Table 11, 12, and 13. In Appendix E, we track the step-wise gen- eration processes, demonstrating superior conver- gence speed and accuracy for NeoDiff compared to continuous diffusion baselines. We also com- pared NeoDiff against continuous diffusion base- lines on token-level controlled generation, demon- strating its unique ability to perform targeted modi- fications while maintaining semantic consistency across translations (Appendix F). 5 Conclusion In this work, we introduce Non-simultaeous Con- tinuous Diffusion Models (NeoDiff), a novel diffusion-based text generation framework that unifies discrete and continuous diffusion models. NeoDiff generalizes the time variable, incorporates the Poisson diffusion process, adaptively modulates the reverse process based on semantic context, and uses an optimized extrinsic time schedule for infer- ence. This unified framework enables fine-grainedcontrol and achieves superior performance across diverse natural language processing tasks. Our ex- tensive experiments demonstrate the effectiveness of this unified framework, opening up new avenues for advancing diffusion-based text generation. Limitations While NeoDiff demonstrates strong performance across various Seq2Seq-based conditional gener- ation tasks (e.g., machine translation, paraphras- ing, text simplification, and question generation), we note some implementation considerations. The post-training optimization of extrinsic time sched- ules requires additional sampling iterations, though this overhead is negligible compared to the train- ing time. The time predictor introduces a modest parameter increase to the backbone network. Acknowledgements This research was supported by the National Nat- ural Science Foundation of China (Grant No. 62276245). References Jacob Austin, Daniel D Johnson, Jonathan Ho, Daniel Tarlow, and Rianne van den Berg. 2021. Structured denoising diffusion models in discrete state-spaces. Advances in Neural Information Processing Systems , 34:17981–17993. Ondˇrej Bojar, Christian Buck, Christian Federmann, Barry Haddow, Philipp Koehn, Johannes Leveling, Christof Monz, Pavel Pecina, | https://arxiv.org/abs/2505.22165v1 |
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zt′ as pθ(zt′|zt, τt, τt′) =q(zt′|ˆz0(zt, τt, t), τt′), where ˆz0is the model prediction of z0. Following Ho et al. (2020), the training objective is derived from the variational lower-bound LVLB=Eq −logpθ(z0:1, τ0:1) q(z>0, τ>0|z0) (12) =Eq" −logpθ(z1, τ1) q(z1, τ1|z0)(13) +X 0<t′<1−logq(zt′|ˆz0(zt, τt, t), τt′) q(zt′|z0, τt′) (14) +X 0<t′<1−logpθ(τt′|zt, τt) q(τt′)(15) −logpθ(z0, τ0|z∆t, τ∆t)# (16) =Eq" KL(q(z1, τ1|z0)∥pθ(z1, τ1))| {z } L1(17) +X 0<t′<1KL(q(zt′|z0, τt′)∥q(zt′|ˆz0, τt′))| {z } Lz (18) +X 0<t′<1KL(q(τt′)∥pθ(τt′|zt, τt))| {z } Lτ(19) −logpθ(z0, τ0|z∆t, τ∆t)| {z } L0# . (20) Note that L1is a constant and can be ignored, and L0also becomes negligible when ∆t→0. Ac- cording to prior works (Ho et al., 2020; Li et al., 2022), the term Lzcan be simplified as Lz=∥ˆz0(zt, τt, t)−z0∥2.We also add an anchor loss (Gao et al., 2024) Lanchor =Eq[−logpθ(y|ˆz0(zt, τt, t))]as a regu- larization term to avoid collapse of the embedding space. Finally, the training objective of the pro- posed NeoDiff can be written as L=Lz+Lτ+Lanchor . 12 B Experimental Settings B.1 Data Preprocessing We used byte pair encoding (BPE) (Sennrich et al., 2016) for tokenization. Unlike previous work, we did not employ knowledge distillation (Kim and Rush, 2016) for preprocessing to evaluate our model’s performance under more challenging con- ditions. B.2 Model Configuration For our experiments, we set the maximum noise state smax to 100 and used the sqrt schedule for training, optimized schedule for inference. To enhance model performance, we applied self- conditioning (Chen et al., 2022). The transi- tion schedule coefficient kwas set to 2, and the maximum truncation time tmaxwas set to 0.99. Following Gao et al. (2024), we also employed noise rescaling with a degradation score threshold DGS MAX of 0.2. Regarding the model architecture, we adopted the configuration from Gao et al. (2024) for the IWSLT14 De-En, WMT14 En-De, and WMT16 En-Ro datasets. For the QQP, Wiki-Auto, and QT datasets, we used the configuration from Gong et al. (2022) to enable a fair comparison with these mod- els. Detailed settings are presented in Table 9. B.3 Training and Generation We trained our models using NVIDIA RTX 3090 24G GPUs on Ubuntu 18.04 with FairSeq 0.12(Ott et al., 2019)(MIT-licensed). For the WMT14 En- De and WMT16 En-Ro datasets, training took nearly 4 days and 2 days, respectively, using 4 GPUs. For the IWSLT14 De-En dataset, training took approximately 1 day using a single GPU. The QQP, Wiki-Auto, and QT datasets each required around 8 hours of training on a single GPU. The training data splits are presented in Table 16. During generation, we used 20 iteration steps (K= 20 ) without early stopping for the IWSLT14 De-En dataset. For the other datasets, we employed 10 iteration steps without early stopping, which is faster than the 20 steps ( k= 20 ) used by Gao et al. (2024) across all datasets. We utilized 2D parallel decoding and selected the best sentence using the minimum Bayes risk (MBR) (Kumar and Byrne, 2004) method based on the BLEU score. The reported results are averaged over 3 runs. The random seed is set to 7.B.4 Optimized | https://arxiv.org/abs/2505.22165v1 |
Extrinsic Time Schedule We propose a systematic approach to optimize the extrinsic time schedule S={t1, t2, ..., t K}, where Kdenotes the number of diffusion steps and ti∈ [0,1]witht1< t2< ... < t K. While previous works (Dhariwal and Nichol, 2021; Chen, 2023) focus on optimizing noise schedules with fixed time steps, we directly optimize the time sched- ule through Bayesian optimization. Our method extends Li et al. (2024)’s framework from discrete subset selection to continuous optimization over the complete schedule. At its core, our approach is straightforward: we sample text using different time schedules on the validation set and select the schedule that achieves the highest BLEU score for inference. The opti- mization process (Algorithm 1) employs Gaussian Process-based Bayesian optimization with the GP- Hedge acquisition function (Brochu et al., 2011). Starting from a uniform time schedule, we itera- tively propose candidate schedules using Limited- memory BFGS (Liu and Nocedal, 1989) and evalu- ate them using BLEU scores on the validation set. This approach enables precise calibration of the time schedule while maintaining the ordering con- straint t1< t2< ... < t K. Following Li et al. (2024), we limit optimization to 100 iterations, keeping the computational overhead negligible compared to model training time(Li et al., 2024). The resulting task-specific schedules demonstrate improved generation quality while maintaining computational efficiency. C Additional Diversity Analysis on QQP dataset In this section, we provide a detailed comparison of NeoDiff and Transformer on the QQP task, specifi- cally focusing on multi-candidate generation and inter-sentence diversity. The results presented in Table 8 complement the main paper’s Table 4 by offering a deeper look into how diversity metrics evolve with an increasing number of generated sam- ples ( b). D Efficiency Analysis D.1 Bayesian Optimization Overhead (WMT14 En-De): • Training: 505.88 RTX3090 GPU Hours •Bayesian Optimization: 28.1 RTX3090 GPU Hours (approximately 6% of training time) 13 Models K Speed (sentences/second) Memory Cost (MB) Transformer* n 6.05 - CMLM* 10 11.80 - DiffuSeq* 2000 0.06 - SeqDiffuSeq* 2000 0.05 - Difformer 20 6.49 2034 NeoDiff 20 5.12 2080 Table 7: Runtime Comparison on IWSLT14 De-En. *: Results from Gao et al. (2024). Others are reproduced. Model b Semantic Faithfulness Fluency Completeness Phrasing Diversity Inter-Sentence div-4 Transformer 1 83.64 92.56 84.96 57.19 1.000 Transformer 5 83.70 94.73 86.05 54.52 0.686 Transformer 10 83.93 94.64 86.02 54.55 0.561 NeoDiff 1 84.24 88.95 87.83 39.18 1.000 NeoDiff 5 85.63 90.69 88.39 41.62 0.684 NeoDiff 10 87.42 91.87 88.79 45.83 0.631 Table 8: Detailed comparison of NeoDiff and Transformer on the QQP task. Metrics include Semantic Faithfulness, Fluency, Completeness, Phrasing Diversity (single-sample), and Inter-Sentence Diversity (Inter-Sentence Div-4, multi-candidate). Note: The cost of Bayesian optimization is di- rectly proportional to the amount of data sampled in each iteration. While we used the entire WMT14 validation set, significantly reducing the sample size (e.g., to 20 samples) can drastically lower this overhead to less than 0.1 GPU Hours (Li et al., 2024). D.2 Runtime Comparison (IWSLT14 De-En) Table 7 presents a runtime comparison of NeoD- iff and several baselines on the IWSLT14 De- En dataset. | https://arxiv.org/abs/2505.22165v1 |
We measured inference speed (sen- tences/second) and memory cost (MB). NeoDiff demonstrates competitive inference speed, process- ing 5.12 sentences per second, which is comparable to Difformer’s 6.49 sentences per second. While significantly faster than diffusion-based models like DiffuSeq and SeqDiffuSeq, NeoDiff’s speed is lower than the highly optimized Transformer and CMLM models. In terms of memory usage, NeoDiff’s 2080 MB consumption is similar to Dif- former’s 2034 MB. E Step-wise Generation Examples on IWSLT14 De-En for NeoDiff and Difformer Table 14 and 15 present a detailed compar- ison of the translation generation process on IWSLT14 De-En dataset between NeoDiff and Difformer(continuous diffusion model). Afterincorporating the three aforementioned compo- nents(Poisson process, time predictor and opti- mized schedule), NeoDiff demonstrates more ac- curate and faster convergence in translation on some sentences compared to Continuous Diffu- sion Model(Difformer), as illustrated by the step- by-step generation process. Specifically, NeoDiff avoids some of the common pitfalls of diffusion models, such as getting stuck in local optima or generating repetitive phrases. F Fine-grained Controlled Generation through Token Manipulation We demonstrate NeoDiff’s capability for token- level controlled generation while preserving seman- tic consistency across translations. Given a source sentence xsrcand its latent representation z0, we re- place a single token to obtain a modified source x′ src. For translation, we initialize the process with z0 and set τ= 1only for the modified token position, maximizing noise specifically at that location. This targeted noise application enables precise semantic modifications in the output translation x′ tgtwhile preserving the remaining content. As shown in Ta- ble 10, NeoDiff achieves localized modifications, whereas baseline methods like Difformer tend to alter substantial portions of the output sentence. This controlled generation capability stems from our fine-grained noise paradigm, enabling token- specific manipulation of the generation process. 14 Algorithm 1 Extrinsic Time Schedule Calibration via Bayesian Optimization Require: Trained Diffusion Model M, Initial Extrinsic Time Schedule Sinit={t1, t2, ..., t K}, where ti∈Rand0≤t1< t2< ... < t K≤ 1, Optimization iterations niter, Domain for elements in Extrinsic Time Schedule D ⊂[0,1], Source Text Tsrc, Target Text Ttgt Ensure: Optimized Extrinsic Time Schedule Sopt={t′ 1, t′ 2, ..., t′ K}, where t′ i∈Rand0≤t′ 1< t′ 2< ... < t′ K≤1 1:Initialize Sinit={t1, t2, ..., t K}such that tiare uniformly spaced in [0,1]. 2:Perform a sampling on Tsrcusing diffusion model Mand extrinsic time schedule Sinit, yielding predicted text Tpred. 3:Compute the BLEU score BLEU( Ttgt, Tpred)using TtgtandTpred. 4:Initialize the observation set for Bayesian optimization: O← {(Sinit,BLEU( Ttgt, Tpred))}. 5:fori= 1toniterdo 6: Update the Gaussian Process posterior given observations O. 7: Generate a candidate set D′={S′ 1,S′ 2, ...,S′ N}, where each S′ j={t′ j1, t′ j2, ..., t′ jK}represents a candidate extrinsic time schedule with t′ jk∈ D and0≤t′ j1< t′ j2< ... < t′ jK≤1. The candidate setD′is generated by performing 20 iterations of Limited-memory BFGS (Liu and Nocedal, 1989) with 5 random initial points within DK. 8: Compute the acquisition function value αGP-Hedge (S′ j)(Brochu et al., 2011) for all S′ j∈ D′. 9: Select the next observation point Si= arg max S′ j∈D′αGP-Hedge (S′ j). | https://arxiv.org/abs/2505.22165v1 |
10: Perform a sampling on Tsrcusing MandSi, yielding predicted text T′ pred. 11: Compute the BLEU score BLEU( Ttgt, T′ pred)using TtgtandT′ pred. 12: Update the observation set: O←O∪ {(Si,BLEU( Ttgt, T′ pred))}. 13:end for 14:Sopt= arg max (S,BLEU) ∈OBLEU 15 Hyper-parametersWMT14 En-DeWMT16 En-RoIWSLT14 De-EnQQP Wiki-Auto QT Architecture dmodel 512 512 512 768 768 768 demb 128 128 128 128 128 128 dffn 2048 2048 1024 3072 3072 3072 Heads 8 8 4 12 12 12 Encoder Layers 6 6 6 6 6 6 Decoder Layers 6 6 6 6 6 6 Time Predictor Layers 3 1 1 1 1 1 Activation ReLU ReLU ReLU ReLU ReLU ReLU Diffusion Steps 10 10 20 10 10 10 Training Schedule sqrt sqrt sqrt sqrt sqrt sqrt Inference Schedule Optimized Optimized Optimized Optimized Optimized Optimized DGS MAX 0.2 0 .2 0 .2 0 .2 0 .2 0 .2 Self-Conditioning ✓ ✓ ✓ ✓ ✓ ✓ Training Steps 600K 400K 300K 50K 100K 100K Batch Size (Tokens) 32K 24K 8K 8K 12K 16K Optimizer AdamW AdamW AdamW AdamW AdamW AdamW Adam β (0.9,0.98) (0 .9,0.98) (0 .9,0.98) (0 .9,0.98) (0 .9,0.98) (0 .9,0.98) Weight Decay 0.01 0 .01 0 .01 0 .01 0 .01 0 .01 Learning Rate 5×10−45×10−45×10−45×10−42.3×10−42×10−4 Warmup 10K 10K 10K 10K 10K 10K Clip Gradient 1.0 1 .0 1 .0 1 .0 1 .0 1 .0 Dropout 0.1 0 .1 0 .3 0 .1 0 .1 0 .1 Length Predict Factor 0.1 0 .1 0 .1 0 .1 0 .1 0 .1 Label Smoothing 0.1 0 .1 0 .1 0 .1 0 .1 0 .1 Inference Steps 10 10 20 10 10 10 Bayesian Optimization Rounds 100 100 100 100 100 100 Table 9: The model architectures and hyper-parameters used in our experiments. 16 Evaluate this translation from {src_lang} to {tgt_lang} (0-100 score): [Source] {source} [Reference] {reference} [Translation] {translation} Score these aspects STRICTLY IN THIS ORDER: 1. **Accuracy**: Faithfulness to source meaning 2. **Fluency**: Naturalness in target language 3. **Completeness**: Information retention 4. **Creativity**: Handling of ambiguous or open-ended source content Return ONLY 4 numbers separated by commas, NO text. Evaluate this paraphrase generation (0-100 score): [Original] {source} [Reference] {reference} [Paraphrase] {paraphrase} Score these aspects STRICTLY IN THIS ORDER: 1. **Semantic Faithfulness**: Meaning preservation from original 2. **Fluency**: Naturalness in language 3. **Completeness**: Retention of all information 4. **Phrasing Diversity**: Variation in wording/structure while preserving meaning Return ONLY 4 numbers separated by commas, NO text. Figure 3: Prompt templates used for LLM-based evaluation. Top: Translation evaluation prompt. Bottom: Paraphrase evaluation prompt. 17 <src> und die welt in der wir jetzt leben sieht so aus . <tgt> and the world we now live in looks like this . <src’> und die welt in der wir jetzt leben sieht anders aus . <tgt’> and the world we now live in looks different . Model Generated Content NeoDiff <tgt_pred> and the world we live in now , looks like this . NeoDiff <tgt’_pred> and the world we live in now , looks different . Difformer <tgt_pred> and the world we’re living in now looks like this | https://arxiv.org/abs/2505.22165v1 |
. Difformer <tgt’_pred> and the world that we’re living in right now , it looks different . <src> sein ganzer arbeitsprozess hat sich danach geändert . <tgt> and his whole work process changed after that . <src’> sein ganzer arbeitsprozess hat sich davon geändert. <tgt’> His whole work process changed because of that. Model Generated Content NeoDiff <tgt_pred> his whole work process has changed after that . NeoDiff <tgt’_pred> his whole work process has changed from that . Difformer <tgt_pred> and his whole work process has changed after that . Difformer <tgt’_pred> and his whole work process has changed from this . <src> der zweite faktor sind die dienste , die wir nutzen . <tgt> the second factor is the services we use . <src’> der zweite faktor sind die dienste , die wir kennen . <tgt’> The second factor is the services we know . Model Generated Content NeoDiff <tgt_pred> the second factor is the services that we use . NeoDiff <tgt’_pred> the second factor is the services we know . Difformer <tgt_pred> the second factor is the services that we use . Difformer <tgt’_pred> the second factor is really the services that we meet . Table 10: Token manipulation example. 18 Source Reference Base Translation +P Translation nein war nie eine möglichkeit gewesen .no had never been an option .no one has never been a pos- sibility.no had never been an oppor- tunity. weiß jemand , was drei sekunden sind ?does anyone know what three seconds are ?does anybody know what three seconds?does anyone know what three seconds are? sie hatten ein konzept von blauem blut .they had a concept of blue blood .they had a idea of blue blood.they had a concept of blue blood. und raten sie was wir in dem angriffscode gefunden haben ?and guess what we found in the attack code ?and do you guess what we’ve found in the code of attack?and guess what we found in the code of attack? jetzt sehen sie den dal- matiner .now you see the dalmatian . now this is the dalmatinan. now you see the dalmatiner. denn die kategorien sagen mir , wie ich sie auseinan- der halten kann .because the categories tell me how to tell them apart .because the categories are telling me how i can keep it apart.because the categories tell me how to keep them apart. wie konnte es möglich sein , dass wir dies tun ?how could it be possible that we would do this ?so how could it possible for us to do this?how could it be possible that we could do this? aber es gab immer einen lebenszyklus in ihren präsentationen .but there was always a life cycle to their presentations .but there has always been a life cycle in your presenta- tions.but there was always a life cycle in their presentations. wir reden zwiespältig davon .we talk about it ambiva- lently .we’re talking about it in elessly.we talk about it continally. was geschah also jahre danach ?so what happened years af- terward ?so for years after that, what happened?so what | https://arxiv.org/abs/2505.22165v1 |
happened years af- ter that? Table 11: Additional examples showing the improvements from introducing the Poisson diffusion process on IWSLT14 De-En dataset. The Base model often produces unnatural word ordering and incorrect lexical choices, while +P shows better handling of complex phrases and more natural English constructions. Source Reference +P Translation +PT Translation sie haben ihr telefon gemietet . sie haben es nicht gekauft .you rented your phone . you didn’t buy it .they’ve rtended your phone. they didn’t buy it.you rented your phone. you didn’t buy it. ihre familie versammelte sich .and the family gathered . her family. her family gathered. dunkler urin . dunkel . dark urine . dark . dark up. dark. dark urine. dark. diese leute verdienen geld . these guys make money . these people are earking money.these people make money. er ist ganz glücklich darüber , weil er sie getäuscht hat .he’ll be very happy because he’s deceived you .he’s very happy about it be- cause he decaked her.he’s very happy about it be- cause he deceied her. er hatte 20 minuten her- rlicher musik gehabt .he had had 20 minutes of glorious music .he’d had 20 minutes of god. he’d had 20 minutes of glo- rious music. ... es dem crowdsourcing beachtung schenkt .... paying attention to crowd- sourcing .... it’s adghting to the crowd- sourcing.... it gives attention to the crowdsourcing. er zeigte immer hier hin . he kept pointing here . he always showed here. he always pointed over here. wenn man es verallgemein- ert , passiert folgendes .if you generalize this , some- thing like this happens .when you generate it, this is what happens.when you generalize it, this is what happens. man konnte manhattan se- hen .you could see manhattan . see manhattan. you could see manhattan. Table 12: Additional examples demonstrating the impact of the time predictor module on IWSLT14 De-En dataset. The examples show how the time predictor enables finer-grained control primarily through word substitutions and better token-level refinements by leveraging information from less-noised tokens to guide the denoising process. 19 Source Reference +PT Translation Full Translation so wie es früher eben entsprechend auf dem dorf passierte .just like it used to happen in the village .in the same way that hap- pened in the village, it just happened.just as it used to happen in the village. darum helfen sie da mit , fra- gen sie bei den leuten mal nach .so you’re helping out there , just ask the people .so you can help with there, ask about people.that’s why they help there with, ask people to ask. noch immer sind wir dem storytelling als informa- tionsvermittlung sehr , sehr stark verhaftet .what’s left is storytelling . but we’re still very sted to storytelling as an informa- tion reation, very vivily ar- rested.we’re still very to story- telling as an information me- diation, very, very arrested. wir wählen jedes jahr einige fellows aus und wir lassen sie mit stadtverwaltungen ar- beiten .we select a few fellows ev- ery year and we have them work with city governments .we | https://arxiv.org/abs/2505.22165v1 |
choose some fellows ev- ery year, and we have them work with city adminicies.we choose some fellows ev- ery year, and we let them work with urban manage- ment. also bot ich einen 10000 $ preis an software für die gewinner .so i offered a 10,000 dollar prize of software to the win- ning team .so i offered a $10,000 price for the winner software.so i offered a 100,000 price of software for the winners. dies ist in unserem ganzen land der zweitgrösste ab- fallfluss amerikas .this , all over the country , is the second largest waste stream in america .this is in our entire country, the two-largest waste river of america.this is the second est waste flow in america’s land in our entire country. wir haben eine art gle- ichgewicht erreicht .we have reached a kind of equipoise .we’ve reachved some kind of equilibrium.we’ve reached some kind of balance. und das ist aber ganz im an- fang .and that’s just the beginning .and that’s just at the very be- ginning.and that’s at the very begin- ning. ich bin überzeugt , dass man irgendwie zur nostalgie , zu wunschdenken hingezogen ist .i’m convinced that there’s some sort of pull to nostal- gia , to wishful thinking .i’m convinced you’ve been drawn to nostalgia, sort of wokkthinking.i’m believe that there’s kind of moved to nostalgia, you’re moved to thinking. wir haben uns daran gewöhnt , dass dinge linear passieren .we no longer imagine the thing in images things in images , but codify them through language .we were used to make things happen to linear.so we’ve been used to linear that things happen. Table 13: Additional examples showing the impact of the optimized schedule on IWSLT14 De-En dataset. These examples demonstrate how the schedule primarily influences the overall sampling trajectory at the sentence level, leading to more natural sentence constructions and better semantic coherence. 20 Time Step Difformer Translation NeoDiff Translation (Ours) Source: ihr problem ist , dass sie zu wenig haben . Reference: their problem is that they have too little . 0 dete@@ social tious falsche foot ere secu- rity madeupword0000 sorry fold says write chri@@28 lar@@ electricity terms surface ting madeupword0001 madeupword0000 ®gen 1 your problem is that is that they have too little .their problem is they they have too little . 2 the problem of that is that they have too little .their problem is that they have too little . 3 the problem of that is that they have too little .their problem is that they have too little . 4 the problem your you is that they have too little .their problem is that they have too little . 5 the problem your problem is that they have too little .their problem is that they have too little . 6 and , your problem is that they have too little .their problem is that they have too little . 7 now , your problem is that they have too little .their problem is that they have too little . 8 now , your problem is | https://arxiv.org/abs/2505.22165v1 |
, they have too little . their problem is that they have too little . 9 now , your problem is , you have too little . their problem is that they have too little . ... ... ... 20 now , your problem is , you have too little . their problem is that they have too little . Final now , your problem is , you have too little . their problem is that they have too little . Source: denn die kategorien sagen mir , wie ich sie auseinander halten kann . Reference: because the categories tell me how to tell them apart . 0 es clay madeupword0002 ahead jobs in- volved line madeupword0001 fold <pad> <unk> giving bu@@ <unk> ers sa@@market@@ madeupword0003 made- upword0001 van mas price gun ba madeupword0000 <unk> 3 ator anima@@ once 1 because the categ@@ ories tell tell me how can can hold them apart .because the categ@@ ories tell me how to keep them apart . 2 because the categ@@ ories tell tell me how can can hold them apart .because the categ@@ ories tell me how to keep them apart . 3 because the categ@@ ories are tell me how i can hold it apart .because the categ@@ ories tell me how to keep them apart . 4 because the categ@@ ories are telling me how i can hold it apart .because the categ@@ ories tell me how to keep them apart . 5 because the categ@@ ories are telling me how i can hold it apart .because the categ@@ ories tell me how to keep them apart . 6 because the categ@@ ories are telling me how i can hold it apart .because the categ@@ ories tell me how to keep them apart . 7 because the categ@@ ories are telling me how i can hold it apart .because the categ@@ ories tell me how to keep them apart . 8 because the categ@@ ories are telling me how i can hold it apart .because the categ@@ ories tell me how to keep them apart . 9 because the categ@@ ories are telling me how i can hold it apart .because the categ@@ ories tell me how to keep them apart . ... ... ... 20 because the categ@@ ories are telling me how i can keep it apart .because the categ@@ ories tell me how to keep them apart . Final because the categories are telling me how i can keep it apart .because the categories tell me how to keep them apart . Table 14: Step-by-step generation process of Difformer(continuous diffusion model) and NeoDiff on IWSLT14 De-En dataset(Part 1/2). NeoDiff converges to the correct translation more quickly and accurately. 21 Time Step Difformer Translation NeoDiff Translation (Ours) Source: ich sprach also einige monate später bei einer konferenz . Reference: so i spoke at a conference a couple months after that . 0 complex positive which affect o went at- tac@@ care@@ <pad> gers <pad> david fri@@ levelmadeupword0001 le@@ leaders news made- upword0003 sta@@ cannot än@@ made- upword0001 spin@@ published | https://arxiv.org/abs/2505.22165v1 |
mes@@ exhi@@ 1 so i i to a few months later at a conference . so i spoke at at a a few months later . 2 so i i to a few months later at a conference . so i spoke at a conference a few months later . 3 so i i about a few months later at a confer- ence .so i spoke at a conference a few months later . 4 so i i about a few months later at a confer- ence .so i spoke at a conference a few months later . 5 so i i talking a few months later at a confer- ence .so i spoke at a conference a few months later . 6 so i i talking a few months later at a confer- ence .so i spoke at a conference a few months later . 7 so i i talking a few months later at a confer- ence .so i spoke at a conference a few months later . 8 so i was talking a few months later at a con- ference .so i spoke at a conference a few months later . 9 so i was talking a few months later at a con- ference .so i spoke at a conference a few months later . ... ... ... 20 so i was talking a few months later at a con- ference .so i spoke at a conference a few months later . Final so i was talking a few months later at a con- ference .so i spoke at a conference a few months later . Table 15: Step-by-step generation process of Difformer(continuous diffusion model) and NeoDiff on IWSLT14 De-En dataset(Part 2/2). NeoDiff converges to the correct translation more quickly and accurately. SplitsWMT14 En-DeWMT16 En-RoIWSLT14 De-EnQQPWiki- AutoQT Training 4,500,966 608 ,319 160 ,215 144 ,715 677 ,751 116 ,953 Validation 3,000 1 ,999 7 ,282 2 ,048 2 ,048 2 ,048 Test 3,003 1 ,999 6 ,750 2 ,500 5 ,000 10 ,000 Table 16: The dataset splits used in our experiments. 22 | https://arxiv.org/abs/2505.22165v1 |
arXiv:2505.22174v1 [cs.GT] 28 May 2025Online Fair Division for Personalized 2-Value Instances Georgios Amanatidis1,2, Alexandros Lolos1,2, Evangelos Markakis1,2,3, and Victor Turmel4 1Department of Informatics, Athens University of Economics and Business, Athens, Greece. 2Archimedes, Athena Research Center, Athens, Greece. 3Input Output Global (IOG), Athens, Greece. 4Institut de Math ´ematique, Universit ´e Paris-Saclay, Orsay, France. Abstract We study an online fair division setting, where goods arrive one at a time and there is a fixed set of n agents, each of whom has an additive valuation function over the goods. Once a good appears, the value each agent has for it is revealed and it must be allocated immediately and irrevocably to one of the agents. It is known that without any assumptions about the values being severely restricted or coming from a distribution, very strong impossibility results hold in this setting [He et al., 2019, Zhou et al., 2023]. To bypass the latter, we turn our attention to instances where the valuation functions are restricted. In particular, we study personalized 2-value instances , where there are only two possible values each agent may have for each good, possibly different across agents, and we show how to obtain worst case guarantees with respect to well-known fairness notions, such as maximin share fairness andenvy-freeness up to one (or two) good(s) . We suggest a deterministic algorithm that maintains a 1/(2n−1)-MMS allocation at every time step and show that this is the best possible any deterministic algorithm can achieve if one cares about every single time step; nevertheless, eventually the allocation constructed by our algorithm becomes a 1/4-MMS allocation. To achieve this, the algorithm implicitly maintains a fragile system of priority levels for all agents. Further, we show that, by allowing some limited access to future information, it is possible to have stronger results with less involved approaches. In particular, by knowing the values of goods for n−1time steps into the future, we design a matching-based algorithm that achieves an EF1allocation every ntime steps, while always maintaining an EF2allocation. Finally, we show that our results allow us to get the first nontrivial guarantees for additive instances in which the ratio of the maximum over the minimum value an agent has for a good is bounded. 1 Introduction The problem of fairly allocating a set of resources to a set of agents without monetary exchanges was mathematically formalized only in the 1940’s by Steinhaus [1949]—along with his students Banach and Knaster. Nevertheless, fair division is such a fundamental concept that the celebrated Cut & Choose protocol already appears in ancient literature. For this reason, since the formal introduction of the problem, numerous variants, each under different assumptions and constraints, have been studied in mathematics, economics, political science and, more recently, in computer science. Specifically, as far as the latter is concerned, the algorithmic nature of most questions one may ask about fair division problems has lead to a flourishing literature on the topic, often on variants that deal with discrete, indivisible resources. In the most standard variants of the problem a set of resources and a set of | https://arxiv.org/abs/2505.22174v1 |
agents, each equipped with a valuation function, are given as an input and one would like to produce a, complete or partial, partition of the resources to the agents, so that some predetermined fairness criterion is satisfied. Here we study a version of the problem where the set of nagents is indeed given, but the indivisible goods arrive in an online fashion. That is, at each time step a new good appears, its value for the agents is revealed, and the good must be irrevocably assigned to a single agent before the next good arrives. Online fair division problems of similar nature appear in many real-world scenarios, like, e.g., the operation of food banks, donation distribution in disaster relief, limited resource allocation within an organization like a hospital or a university, or memory and CPU management in cloud computing. Despite their wide applicability, however, online fair division settings have not been studied nearly as much as their offline counterparts. And admittedly, there is a good reason for the relative scarcity of such works. Namely, it is relatively easy to show strong impossibility worst-case results, even if one aims for rather modest fairness guarantees. For example, when the agents have additive valuation functions and the items are goods (i.e., they do not have a negative value for any agent) that arrive adversarially—as is the case here—there is no deterministic algorithm that achieves anypositive approximation factor with respect to maximin share fairness (MMS ) [Zhou et al., 2023] or to envy-freeness up to one good (EF1) [He et al., 2019]. Even if one allows some item reallocations in order to fix that, the number of reallocations cannot be bounded by any function of n[He et al., 2019]. Of course, there are ways to bypass these results, like assuming that the item values are drawn from distributions [Benade et al., 2018, Zeng and Psomas, 2020], restricting the valuation functions [Aleksandrov et al., 2015] or the number of distinct agent types [Kulkarni et al., 2025], allowing items to be reallocated [He et al., 2019], relaxing the requirement for fairness guarantees at the intermediate time steps [Cookson et al., 2025], augmenting the input with predictions [Banerjee et al., 2022] or even with full knowledge of the whole instance [Elkind et al., 2024, Cookson et al., 2025]. Here the main approach we take is to focus on designing deterministic algorithms for instances with restricted valuation functions, although we also explore how information about the future allows for stronger results with simpler algorithms. In particular, we mostly consider instances where each agent ihas two values for the goods: a low value βi and a high value αi. These instances, which we call personalized 2-value instances , generalize the binary case and the setting where there are only two types of goods; until now, these were the only cases where positive worst-case results were known (by Aleksandrov et al. [2015] and Elkind et al. [2024], respectively). Consequently, personalized 2-value instances capture the natural dichotomy between highly desirable and less desirable goods in a more nuanced way which also varies across | https://arxiv.org/abs/2505.22174v1 |
agents. Moreover, there is a natural way of approximating any additive instance via a 2-value instance by just setting a threshold for each agent iand rounding everything up (to the “high value” αi) or down (to the “low value” βi) accordingly. Although this approximation is not always meaningful, it does give nontrivial guarantees for agents whose values are bounded by lower and upper bounds that do not differ by too much, as we argue in Section 6. 1 It should be noted that this type of restriction has drawn significant attention in the fair division literature recently. The study of k-value instances (often called k-valued or, in the case where k= 2,bi-valued ), where there are at most kpossible common values that can represent the value an agent has for a good, was introduced by Amanatidis et al. [2021] as a way to study the existence of EFX allocations under a restriction that made the question easier, yet not straightforward. Indeed, it turns out that, in many cases, k-value instances seem to strike a good balance between maintaining the flavor and many of the challenges of the general additive version of the problem and allowing nontrivial positive results, even for k= 2or3. Consequently, there is a recent line of work studying fair division questions under such restrictions (e.g., [Akrami et al., 2022, Garg et al., 2022, Aziz et al., 2023, Amanatidis et al., 2024, Fitzsimmons et al., 2024]), often allowing for the kvalues to be different per agent (hence, personalized ), first studied by Murhekar and Garg [2021] for general k, under the name k-ary instances . Finally, as we mentioned above, we also explore another relaxation of the problem which is somewhat orthogonal to that of restricting the valuation functions, and aims to alleviate the lack of information due to the online arrival of the goods. Several recent works augment their online algorithms with additional information about the future in the form of (possibly erroneous) predictions [Lykouris and Vassilvitskii, 2021, Banerjee et al., 2022, 2023, Balkanski et al., 2024, Benomar and Perchet, 2024]. Here we follow an approach closer to He et al. [2019], Cookson et al. [2025], and Elkind et al. [2024], who assume that the whole instance can be known in advance and what truly remains online is the allocation process itself. As the number of goods could be much larger than the number of agents in many scenarios, we feel that knowing the whole instance is a rather strong assumption; instead, we allow some of our algorithms to see into the future only for a number of time steps that is comparable to the number of agents. In our setting this turns out to be enough for getting simpler algorithms with solid guarantees. Contribution and Technical Considerations. Following the aforementioned line of work on restricting the valuation functions, we explore what is possible for (personalized) 2-value instances in deterministic online fair division. We obtain positive and negative results, which we then extend beyond our main setting. Specifically: •The impossibility results one has for general additive instances persist here | https://arxiv.org/abs/2505.22174v1 |
as well, albeit milder. We show that, for any ε >0, no algorithm can guarantee (1/2 +ε)-EF1at every time step, even for two agents (Theorem 3.1), or (1/(2n−1) +ε)-MMS at every time step for general n(Theorem 3.2). •We present an algorithm with tight approximation guarantee with respect to MMS . In particular, our Algorithm 1 guarantees 1/(2n−1)-MMS at every time step, which improves to Ω(1) at every time step, assuming that mis large enough (Theorem 4.1 and Corollary 4.5). •We demonstrate that even very limited knowledge of the future may help significantly, as our Algorithm 2 guarantees EF1at every other time step for two agents just by looking one step ahead into the future (Theorem 5.2). •More generally, we show that having a foresight of n−1goods into the future suffices in order to guarantee EF2at every time step and EF1at every ntime steps for nagents (Algorithm 3 and Theorem 5.4). The latter also implies a 1/n-MMS approximation at every ntime steps, whereas achieving (1/n+ε)-MMS at every time step is impossible, even if the whole instance is fully known in advance. •We provide a simple reduction that allows us to translate our results to general additive instances at the expense of a multiplicative factor that depends on the largest ratio between the values of any two goods. To the best of our knowledge, these are the first positive results in this setting (Theorem 6.1 and Corollaries 6.4 and 6.5). 2 While the main ideas in all of our results are very intuitive at a high level, turning them into tight or best-possible statements is far from trivial. Our impossibility results (especially Theorem 3.2) cannot rely on boosting values as needed (as it is commonly done in the literature), given the nature of the valuation functions we study. Instead, we need a family of instances fully adjusted to what anyalgorithm could do in order to determine the allocation with enough precision to get what turns out to be a tight bound. Algorithm 1, despite its several cases and careful book-keeping, is based on the simple idea that throughout the allocation process an agent should gain higher priority the more value they lose to others. Although our parametrization of the indices that imply this priority (one out of every 2n−1goods in general; one out of 3n−2high-valued goods) may seem rather loose, not only is it tight but it requires an involved analysis that includes distinct inductive proofs (that differ drastically) for the initial and later phases of the algorithm, respectively. Algorithms 2 and 3 are simpler, especially the former, but there are subtleties here as well. Ideally, what we would like to achieve between time steps knand(k+ 1)nwould be to get an EF1 allocation by taking the union of two EF1allocations. However, in general, this fails trivially, even for agents with binary valuation functions. Instead, we take advantage of the structure of our instances and built the second EF1allocation (the one with the current and predicted goods) so that its envy graph “cancels out” the problematic edges of the envy graph of the current allocation, | https://arxiv.org/abs/2505.22174v1 |
via carefully defined auxiliary valuation functions. Further Related Work. There is a vast literature on fair division, both with divisible and with indivisible resources. For a recent survey on the latter, see Amanatidis et al. [2023]. Here we focus on online fair division settings, mostly with indivisible items. Aleksandrov et al. [2015] introduced the setting we study here. Their work focused more on binary instances and on welfare guarantees. Subsequent works studied the mechanism design version of the problem mostly for binary instances [Aleksandrov and Walsh, 2017] and explored the limitations of achieving fairness notions like EF1with general additive or even with non-additive binary valuations functions [Aleksandrov and Walsh, 2019]. The results of these early works are summarized in the survey of Aleksandrov and Walsh [2020]. A viable direction in order to bypass the existing strong negative results is to assume that there are underlying distributions with a bounded support. In such a setting Benade et al. [2018] show that it is possible to achieve envy that grows sublinearly with time with a very simple algorithm and that this is asymptotically best possible, whereas Zeng and Psomas [2020] study the compatibility of efficiency and envy in variants of the setting. However, the works that are closer to ours, one way or another, are those of He et al. [2019], Zhou et al. [2023], Cookson et al. [2025], and Elkind et al. [2024]. He et al. [2019] show that it is impossible to achieve temporal- EF1(or even any nontrivial approximation of it) unless a linear fraction of the goods are reallocated. Then they design algorithms (occasionally augmented with full knowledge of the future) that achieve temporal- EF1with a bounded number of reallocations. Zhou et al. [2023] focus on temporal- MMS and show that no approximation is possible for goods beyond the case of two agents. Then they turn their attention to chores (i.e., items that no agent values positively) where they present an algorithm with constant approximation guarantee. It should be noted here that although their 0.5-temporal- MMS algorithm for n= 2seems to contradict our Theorem 3.2, this is made possible by the additional assumption that vi(M)is known for all i∈N, which we do not make here. The very recent works of Elkind et al. [2024] and Cookson et al. [2025] formalize the notion of temporal fairness (although in slightly different ways) and assume that the full information about an instance is known upfront. However, even under this strong assumption, there still are impossibility results and, hence, both papers mostly focus on further restrictions. Cookson et al. [2025] obtain positive results at every step of the allocation for the case of two agents, and for the overall allocation for instances where the agents agree on the ordering over the goods or for the variant of the problem where the same set of goods arrives at every time step. 3 Elkind et al. [2024], on the other hand, show temporal EF1guarantees for two agents, two types of items, generalized binary valuations, and for unimodal preferences. There is also a line of work where the | https://arxiv.org/abs/2505.22174v1 |
setting is very similar to ours but the items that arrive online are divisible [Gkatzelis et al., 2021, Barman et al., 2022, Banerjee et al., 2022, 2023]. The similarities, however, are only superficial, as the fractional assignment of even a few goods allows us to bypass most strong impossibility results. Finally, online fair division with indivisible items has been very recently studied in the context of bandit learning [Yamada et al., 2024, Procaccia et al., 2024]. Of course, besides the online setting where the items arrive over time, a different scenario is to have a known set of resources and assume that agents arrive in an online fashion. Indeed, there is a significant amount of work in this direction, with an emphasis on indivisible resources [Kalinowski et al., 2013, Kash et al., 2014, Sinclair et al., 2021, Vardi et al., 2022, Banerjee et al., 2024]. For indivisible resources, the very recent work of Kulkarni et al. [2025] obtains the first solid maximin share fairness guarantees for this setting assuming that the agents fit into a limited number of different types. Nevertheless, these settings have very different flavor than ours. 2 Preliminaries LetN= [n] ={1,2, . . . , n }be set of agents and M={g1, g2, . . . , g m}be a set of indivisible goods, where n, m∈N. The high-level goal is to assign the goods of Mto the agents of Nin a way that is considered fair according to a well-defined fairness criterion. We usually call this assignment of goods to agents an allocation , which can be partial (if not all goods of Mhave been assigned) or complete (i.e., it defines a partition of M). Formally, an allocation (A1, . . . , A n)is an ordered tuple of disjoint subsets of M; we often call the set Aithebundle of agent i. Each agent iis associated with an additive set function vi: 2M→R≥0, where vi(S) =P g∈Svi({g}) represents the value of agent ifor the subset Sof goods. When Sis a singleton, we usually write vi(g) rather than vi({g}). Of course, valuation functions of the agents can be more general but in this work we only study special cases of additive instances of the problem, i.e., instances where all agents have additive valuation functions that are restricted in some way. In particular, we focus on personalized 2-value and personalized interval-restricted instances. Definition 2.1 (Personalized 2-Value Instances) .We say that an instance of the problem is a personalized 2-value instance , if for any i∈Nthe function viis additive and there are αi≥βi≥0, such that for any g∈M, it holds that vi(g)∈ {αi, βi}. When αi=α,βi=β, for all i∈N, we call this a 2-value instance . One could reasonably claim here that the interesting case is when αi> βi>0in Definition 2.1; indeed we say that such agents are of type 1 . Similarly, if αi=βi>0, agent iis of type 2 and if αi> βi= 0, agent i is of type 3 . The remaining agents (i.e., αi=βi= 0) are of type 0 and are completely irrelevant, as they are trivially satisfied | https://arxiv.org/abs/2505.22174v1 |
(namely, they see the allocation as being temporal- EF) by an empty bundle. So, without loss of generality, we may assume that the instances we consider only contain agents of types 1, 2, and 3. Of course, agents of type 2 and 3 are easier to satisfy (see, e.g., Corollary 4.4). As a final observation about agent types, we note that for the fairness notions we introduce below and study in this work, scaling the valuation functions has no effect on the fairness guarantees of any given allocation. Thus, it is also without loss of generality, to assume that βi= 1if agent iis of type 1 or 2, or that αi= 1if it is of type 3. The next definition aims to capture the continuous analog of 2-value instances, i.e., we would like all the values agent imay have for a good to be in the interval [βi, αi]. However, as we just mentioned above, scaling the valuation functions is irrelevant for the fairness notions we consider. That is, any general additive instance can be trivially transformed to an equivalent instance where, for any i∈Nandg∈M, 4 it holds that vi(g)∈[0,1]. Thus, in order for the restriction to be meaningful in our context, it should hold thatβi>0for all i∈N. By scaling appropriately, this is equivalent to asking that βi= 1for all i∈N. Definition 2.2 (Personalized Interval-Restricted Instances) .We say that an instance of the problem is a personalized interval-restricted instance , if for any i∈Nthe function viis additive and there is αi>1, such that for any g∈M, it holds that vi(g)∈[1, αi]. When αi=α, for all i∈N, we call this an interval-restricted instance . By maintaining a bound on max i∈[n]√αi, i.e., the ratio of the highest over the lowest value an agent has for a good in Definition 2.2, we can reduce the problem of dealing with personalized interval-restricted instances to dealing with personalized 2-value proxy instances instead (albeit at the expense of the approximation ratio guarantees; see Section 6). In standard—offline—fair division settings, all the goods of Mare known and available to be used as an input to an allocation algorithm. Here we consider an online fair division setting, where the goods arrive one at a time; the set M(or even its cardinality, m) is not known a priori. When a good garrives, its value for each agent is revealed and it needs to be added to the bundle of some agent immediately and irrevocably. In general, we associate a distinct time step with each good and it is often (although not always) convenient to implicitly rename the goods so that gkis the k-th good in order of arrival and arrived at (or rather triggered ) time step t=k. Also, we often use nh(i, t) =|{g∈M|vi(g) =αiandgis one of the first tgoods that arrived }| to denote the number of high-valued goods among the first tgoods from the perspective of agent i. As we mentioned in our introduction, there are no distributional assumptions about the arrival of the goods and in our results we follow a worst-case analysis. In Section 5, | https://arxiv.org/abs/2505.22174v1 |
however, we assume that our instances can be augmented with limited information about the future. We say that an online instance is augmented with foresight of length ℓif every time a good garrives (and still needs to be allocated immediately and irrevocably), we also get a preview of the ℓnext goods. We mentioned above that we want to produce allocations that are fairin some manner. We formalize this by introducing our main fairness notions, approximate EF kandapproximate MMS . Envy-freeness up to kgoods (EF k) is a relaxation of envy-freeness, introduced by Lipton et al. [2004] and formally defined by Budish [2011] for k= 1. For instance, according to EF1some envy is acceptable, as long as it can be eliminated by the hypothetical removal of a single good. Definition 2.3 (ρ-Envy-Freeness, ρ-EFk).Given a partial allocation A= (A1, A2, . . . , A n), constants ρ∈(0,1]andk∈Z>0, and two agents i, j∈N, we say that - agent iisρ-envy-free ( ρ-EF) towards agent j, ifvi(Ai)≥ρ·vi(Aj); -agent iisρ-EFktowards agent j, if there is a set S⊆Ajwith|S| ≤k, such that vi(Ai)≥ρ·vi(Aj\S). The allocation is called ρ-EF(resp. α-EFk) if every agent i∈Nisρ-envy-free (resp. ρ-EFk) towards any other agent j∈N. When ρ= 1, we drop the prefix and write EF krather than 1-EFk. Maximin share fairness, introduced by Budish [2011], is a share-based notion, like proportionality, and can be interpreted via a thought experiment inspired by the famous cut-and-choose protocol. The idea is to give to each agent at least as much value as it could get if it partitioned the goods into disjoint sets (as many as the agents) and kept the worst among them. Definition 2.4 (ρ-PROP ,ρ-MMS ).For a partial allocation A= (A1, A2, . . . , A n), letΠ(n,A)be the set of possible partitions of the set S=Sn j=1Ajintonsubsets. Given Aabove, a constant ρ >0, and an agent i∈N, we say that -Aisρ-proportional for agent i, ifvi(Ai)≥ρ·vi(S)/n; 5 -Aisρ-MMS for agent i, ifvi(Ai)≥ρ·µn i(S), where µn i(S) = max B∈Π(n,A)min Bj∈Bvi(S)is the maximin share of agent i; when nis clear and Sis a function of time t, we write µi(t)instead µn i(S). The allocation is called ρ-PROP (resp. ρ-MMS ) if it is ρ-proportional (resp. ρ-MMS ) for every agent i∈N. When ρ= 1, we write MMS rather than 1-MMS. It is known, and very easy to derive from the definitions, that ρ-envy-freeness implies not only ρ-envy- freeness up to kgoods, for any k >0, but also ρ-proportionality, which itself implies ρ-maximin share fairness. As our problem is online, we do not only care about the final (complete) allocation. As a result, we will make statements of the form “the allocation at time step tisρ1-EFk(resp. ρ2-MMS )” meaning that we consider and evaluate the allocation that has been constructed up to time step tas if the complete set of goods was only what we have seen so far. If each one of the partial allocations produced by an algorithm satisfies the same fairness guarantee, then one talks about temporal fairness , as this was formalized by Elkind et al. [2024] and Cookson et | https://arxiv.org/abs/2505.22174v1 |
al. [2025]. Definition 2.5 (Temporal Fairness) .Consider a sequence of partial allocations At= (At 1, At 2, . . . , At n), for t∈Z≥0, such that At i⊆At+1 ifor any i∈Nand any t≥0. IfAtisρ1-EFk(resp. ρ2-MMS ) for all t∈Z≥0, then we say that the sequence of allocations (At)t≥0isρ1-temporal -EFk(resp. ρ2-temporal -MMS). When referring to the allocation iteratively built by an algorithm, we may abuse the terminology and say that the algorithm computes a ρ1-temporal-EF k(resp. ρ2-temporal- MMS ) allocation, rather than talking about a sequence of allocations. Remark 2.6. Suppose that we have a personalized interval-restricted instance or a personalized 2-value instance with type 1 agents. Let α∗= max i∈[n]√αi. It should be noted that here one can get a trivial 1/α∗approximation with respect to temporal EF1orMMS . Indeed, this is done by completely ignoring the values and allocating the goods in a round-robin fashion. Although, in general, 1/α∗can be arbitrarily worse than the approximation factors we achieve throughout this work, in the special case where α∗is a small constant (e.g., between 1and2), it would be preferable to follow this trivial approach instead. 3 Impossibility Results Persist Even for 2-Value Instances The strong impossibility results in the literature [He et al., 2019, Zhou et al., 2023] typically exploit the following pattern: a bad decision is made about the very first good, due to lack of information, and this propagates throughout a linear number of goods; then, right about when the value of the allocated goods starts to balance out, this “bad” sequence is replicated but with all values scaled up by a large factor, and this cycle is repeated as necessary. One could hope that for 2-value instances there is not enough flexibility for creating instances that force any algorithm to perform poorly. While there is some truth in this, in the sense that things cannot go arbitrarily bad, we still get nontrivial impossibility results. Theorem 3.1. Letε >0be a constant. There is no deterministic algorithm that always builds an allocation which is (1/2 +ε)-temporal-EF1, even for 2-value instances with only two agents. Proof. Suppose we have a deterministic algorithm Afor the problem. We begin with the simple observation that when the very first good has the same value for both agents, it is without loss of generality to assume thatAassigns it to agent 1; if not, we just rename the agents in the following argument. So, consider a stream of goods that begins with g1, g2, such that v1(g1) =v2(g1) = 1 whereas v1(g2) = 5 andv2(g2) = 1 . Given that g1is added to A1, either g2is also added to A1and the resulting allocation ({g1, g2},∅)is not (1/2 +ε)-EF1from the point of view of agent 2, or g2is added to A2; we assume the 6 latter. Next, consider a good g3, such that v1(g3) =v2(g3) = 1 . There are two cases here, depending on whatAdoes with g3, both illustrated below: g1g2g3g4. . . g1g2g3g4g5. . . agent 1: 1 5 1 5 . . . or 1 5 1 1 5 . . . agent 2: 1 1 | https://arxiv.org/abs/2505.22174v1 |
1 5 . . . 1 1 1 5 5 . . . C C C C Ca a a a as s s s se e e e e1 1 1 1 1:g3g3g3g3g3i i i i is s s s sg g g g gi i i i iv v v v ve e e e en n n n nt t t t to o o o oa a a a ag g g g ge e e e en n n n nt t t t t1 1 1 1 1. . . . .In this case, consider a next good g4, such that v1(g4) = v2(g4) = 5 . Whoever gets g4, the resulting allocation, ({g1, g3, g4},{g2})or({g1, g3},{g2, g4}), is not (1/2 +ε)-EF1. Indeed, ({g1, g3, g4},{g2})is not (1/2 +ε)-EF1 from the point of view of agent 2, since 1 =v2(A2)<(1/2 +ε) min g∈A1(v2(A1)−v2(g)) = 1 + 2 ε . Similarly, ({g1, g3},{g2, g4})is not EF1 from the point of view of agent 1, since now we have 2 =v1(A1)<(1/2 +ε) min g∈A2(v1(A2)−v1(g)) = 2 .5 + 5 ε . C C C C Ca a a a as s s s se e e e e2:g3g3g3g3g3i i i i is s s s sg g g g gi i i i iv v v v ve e e e en n n n nt t t t to o o o oa a a a ag g g g ge e e e en n n n nt t t t t2 2 2 2 2. . . . .Here we first have an intermediate good g4, such that v1(g4) = 1 and v2(g4) = 5 . It is straightforward to see that if Aassigns g4to agent 2, then the resulting allocation ({g1},{g2, g3, g4})is not (1/2 +ε)-EF1from the point of view of agent 1. Hence, we assume that g4is added to A1instead. The last good we need is g5withv1(g5) =v2(g5) = 5 . Now, no matter who gets g5, the resulting allocation, ({g1, g4, g5},{g2, g3})or({g1, g4},{g2, g3, g5}), is not (1/2 +ε)-EF1. To see this, first consider ({g1, g4, g5},{g2, g3})and notice that from the point of view of agent 2, we have 2 =v2(A2)<(1/2 +ε) min g∈A1(v2(A1)−v2(g)) = 3 + 6 ε . Finally, consider ({g1, g4},{g2, g3, g5}). From the point of view of agent 1, we have a similar situation: 2 =v1(A1)<(1/2 +ε) min g∈A2(v1(A2)−v1(g)) = 3 + 6 ε . In any case, algorithm Afails to maintain a (1/2 +ε)-EF1 allocation within the first 5 time steps. By carefully inspecting the proof of Theorem 3.1, one could notice that the same construction can be used to show that no algorithm can build an allocation that is (1/3 +ε)-MMS at every time step. In fact, the latter would imply Theorem 3.1 since it is known that any (1/2 +δ)-EF1 allocation for two agents is also a(1/3 +δ/9)-MMS allocation (see, e.g., Proposition 3.6 of Amanatidis et al. [2018]). Nevertheless, for temporal maximin share fairness we can show a much stronger impossibility result | https://arxiv.org/abs/2505.22174v1 |
that degrades with the number of agents. Theorem 3.2. Letε >0be a constant. There is no deterministic algorithm that, given a 2-value instance with nagents, always builds a (1/(2n−1) +ε)-temporal-MMS allocation. Proof. Suppose we have a deterministic algorithm Afor the problem. We are going to consider a 2-value instance with nagents, such that, for all i∈N,βi= 1 andαi=α= 2n2+ 2n. Like in the proof of Theorem 3.1, we begin with some straightforward, yet crucial observations. First, if at any point during the first ntime steps an agent receives a second good, by the time the first ngoods are fully allocated, there is at least one agent jthat has received value 0(because it got no goods) despite having a positive maximin share value µj(n)≥1. So, for what follows, we may assume that algorithm Aassigns to each agent exactly one of the first ngoods. A second observation is that, given that ℓ < n goods have been 7 already allocated, if the (ℓ+ 1) -th good has the same value for all the agents who have not yet received a good, then it is without loss of generality to assume that Aassigns it to the agent with the smallest index; this is just a matter of renaming the agents. Given these two observations above, suppose that goods g1, . . . , g narrive, in this order, so that for any i∈N vi(gr) =( 1,ifi≤r α , otherwise. Then, algorithm Aassigns them exactly as shown below, i.e., giis given to agent i, for all i∈N. g1g2g3. . .gn−1gngn+1gn+2 . . .g2n−1 ag. 1: 1 α α . . . α α 1 1 . . . 1 ag. 2: 1 1 α . . . α α 1 1 . . . 1 ag. 3: 1 1 1 . . . α α 1 1 . . . 1 ................................. ag.n−1: 1 1 1 . . . 1 α 1 1 . . . 1 ag.n: 1 1 1 . . . 1 1 1 1 . . . 1 At this point it is not hard to see that the next claim about goods that are low-valued for all holds. Notice that1/2n <1/(2n−1) +ε. Claim 3.3. Assume that the allocation of the first ngoods is as shown above. If at any point t > n no agent i≤λhas yet received total value more than n+ 1and, furthermore, λgoods, g′ 1, . . . , g′ λ, which are high-valued for all agents arrive in that order, then either agent jwill get good g′ j, for all j∈[λ], or algorithm Abuilds at some point an allocation that is not 1/2n-MMS. Proof of Claim 3.3. This is a simple proof by induction on λ. For λ= 1, after g′ 1appears, say at time t′, we have nh(1, t′) =n, i.e., agent 1 has already seen nhigh-valued goods (from its own perspective). So, we have µ1(t′)≥α. On the other hand, if Adoes not add g′ 1toA1, we have v1(A1)≤n+ 1≤ (n+ 1)µ1(t′)/α=µ1(t′)/2n. Next, for λ >1, assume that the claim is true for λ−1agents and high-valued | https://arxiv.org/abs/2505.22174v1 |
goods. Now suppose that no agent i≤λhas received value more than n+ 1yet and that λgoods, g′ 1, . . . , g′ λ, which are high-valued for all agents arrive in that order. By this induction hypothesis, either algorithm Abuilds at some point an allocation that is not 1/2n-MMS or, for all j∈[λ−1], agent jwill get good g′ j. Given that, agent λ has now seen nhigh-valued goods in total and we can repeat the argument we made for agent 1 in the base case. Let t′′be the time step when g′ λarrives. We have nh(λ, t′′) =nand, thus, µλ(t′′)≥α, whereas vλ(Aλ)≤n+ 1≤µλ(t′′)/2n, unless Aaddsg′ λtoAλ. This completes the induction step. Cl. 3.3 ⊡ The next n−1goods, gn+1, . . . , g 2n−1, are low-valued for all agents. Clearly, no matter how Aassigns these goods, there is at least one agent, say k∈N, who does not get any of those by the end of time step 2n−1. We will distinguish two cases, depending on whether agent kis the last agent or not. C C C C Ca a a a as s s s se e e e e1 1 1 1 1:k=n. . . . .In this case, the next n−1goods, g2n, . . . , g 3n−2, are high-valued for all agents. Given that no agent has received total value more than nby the end of time step 2n−1, Claim 3.3 applies, forcing the algorithm Ato either fail or allocate g2n, g2n+1, . . . , g 3n−2to agents 1, 2, .. ., n−1, in that order. We assume the latter. After this happens we notice that µn(3n−2) = 2 n−1, since agent nhas already seen n−1high-valued and 2n−1low-valued goods, and α >(2n−1)·1. On the other hand, An={gn}, i.e., we have vn(An) = 1 = µn(3n−2)/(2n−1)<(1/(2n−1) +ε)µn(3n−2). 8 C C C C Ca a a a as s s s se e e e e2:k < n k < n k < n k < n k < n . . . . . In this case, the next n−kgoods, g2n, . . . , g 3n−k−1, are such that, for ℓ∈[n−k]and for i∈N, vi(2n−1 +ℓ) =( α , ifi=n−ℓ+ 1 1,otherwise i.e.,g2nis high-valued only for agent n,g2n+1only for agent n−1, and so on, as is shown in the left part of the table below. Note, however, that we may not see the whole subsequence g2n, . . . , g 3n−k−1, depending on the behavior of algorithm A. g2ng2n+1 . . .g3n−k−1g3n−kg3n−k+1 . . .g3n−2 ag. 1: 1 1 . . . 1 α α . . . α ag. 2: 1 1 . . . 1 α α . . . α ........................... ag.k−1: 1 1 . . . 1 α α . . . α ag.k: 1 1 . . . 1 α α . . . α ag.k+ 1: 1 1 . . . α α α . . . α ........................... ag.n−1: 1 α . . . 1 α α . . . α ag.n: α 1 . . . 1 α α | https://arxiv.org/abs/2505.22174v1 |
. . . α We claim that either algorithm Afails to maintain a (1/(2n−1) + ε)-MMS allocation or it allocates g2n, g2n+1, . . . , g 3n−k−1to agents n,n−1, ...,k+ 1, in that order. Towards a contradiction, suppose that this is not the case, i.e., Amaintains a good approximation to temporal maximin share fairness, yet some of these goods are not allocated to the agent who values them highly. In particular, let 2n−1 +j∈ {2n,2n+ 1, . . . , 3n−k−1}be the lowest index of a good for which this happens. That is, g2nis given to agent n,g2n+1to agent n−1, and so on, g2n−2+jis given to agent n−j+2,butg2n−1+jis not given to agent n−j+1. As a result, no agent i∈[n−j+1]has received total value more than n+1by the end of time step 2n−1+j. So, at this point we may change the stream of goods, forget about g2n+j, . . . , g 3n−k−1above, and replace them by n−jgoods, ˆg2n+j, . . . , ˆg3n−1, which are high-valued for all agents. As we argued already, Claim 3.3 applies here, forcing Ato either fail or allocate ˆg2n+j,ˆg2n+j+1, . . . , ˆg3n−1to agents 1, 2, . . . , n−j, respectively, in that order. We assume the latter. After this happens we notice that µ2n−1+j(3n−1)≥α, since agent 2n−1+jhas already seen nhigh-valued goods ( j−1among the first ngoods, the good g2n−1+j itself, and all of the last n−jgoods). On the other hand, A2n−1+jonly contains at most n+ 1low-valued goods. That is, we have v2n−1+j(A2n−1+j)≤n+ 1≤(n+ 1)µ2n−1+j(3n−1)/α <µn(3n−1)/2n. At this point we may assume that algorithm Aallocates g2n, g2n+1, . . . , g 3n−k−1to agents n,n−1, ..., k+ 1, in that order, as shown in the corresponding table. However, this means that none of the first k agents has received more than nlow-valued goods so far. In particular, Ak={gk}. As a result we can consider k−1additional goods, g3n−k, . . . , g 3n−2, which are high-valued for all agents and apply Claim 3.3. The claim guarantees that either algorithm Adoes not maintain a sufficiently high approximation guarantee throughout, or that the allocation will be completed as shown in the last part of the second table, i.e.,g3n−k, g3n−k+1, . . . , g 3n−2are given to agents 1, 2, . . . , k−1, respectively, in that order. This is a very poor allocation from the point of view of agent k. We notice that µk(3n−2) = 2 n−1, since agent khas already seen n−1high-valued goods ( n−kamong the first ngoods and all of the last k−1goods) and 2n−1low-valued goods ( kamong the first ngoods and all the (n−1) + ( n−k)goods right after the first ngoods). On the other hand, recall that Ak={gk}. Thus, we have vk(Ak) = 1 = µn(3n−2)/(2n−1)< (1/(2n−1) +ε)µn(3n−2). 9 By inspecting the proof, it is not hard to see that Theorem 3.2 could have been stated with respect to the largest ratio of the high over the low value of an agent. This is particularly relevant for Section 6, where this term will appear in the fairness guarantees we obtain | https://arxiv.org/abs/2505.22174v1 |
for personalized interval-restricted instances. Corollary 3.4. Letε >0be a constant. There is no algorithm that, given a 2-value instance with nagents and values α >1 =β, always builds a (1/√ 2α+ε)-temporal-MMS allocation. Proof. Notice that in the proof of Theorem 3.2 we have α= 2n2+ 2n. Also, by standard calculus we get limn→∞ 1 2n−1−1√ 4n2+4n = 0. That is, for large enough n, it holds that1√ 4n2+4n+ε >1 2n−1+ε 2and the impossibility follows directly by Theorem 3.2. 4 A Tight Algorithm In this section we present the main result of this work, an algorithm with tight temporal maximin share fairness guarantees. Given how nuanced the construction of the example in the proof of Theorem 3.2 was, it is not particularly surprising that matching this 1/(2n−1)factor requires a fairly elaborate algorithm that performs careful book-keeping of who should get the next contested high-valued goods. Before discussing any details of our Deferred-Priority algorithm (Algorithm 1), we revisit the observation that we made at the beginning of the proof of Theorem 3.2. When agents have positive values, if we aim for a nonzero temporal maximin share fairness guarantee, Algorithm 1 should assign to each agent exactly one of the first ngoods. Indeed, if this is not the case, there would be at least one agent jwho receives value 0(because jgot no goods) at the end of time step n, despite having a positive maximin share value µj(n)≥βi= 1. So, no matter how our algorithm works in general, the allocation of the first ngoods is “special” in the sense that it allows for extremely little flexibility. We call these first ntime steps Phase 0. More generally, Algorithm 1 will operate in phases; we want to ensure that during each phase every agent gets at least one good and the phases do not last for too long. As a second general design goal, however, we want to allocate goods to agents who consider them high- valued as frequently and as uniformly as possible. Note that this is rather incompatible with the aforemen- tioned goal of frequently giving goods to everyone. Our solution to that is to have two sets of counters (the entries of the vectors HandL, introduced in line 3) that keep track of how many high- or low-valued goods, respectively, an agent affords to lose to others before we are in a situation where a “bad” sequence of goods inevitably destroys the temporal maximin share fairness guarantee. So, the general idea is that throughout the allocation the agents should gain higher priority the more value they lose to others; the corresponding priority levels are implicitly described by the entries of HandL (and later on explicitly defined for high-valued goods as Hℓ(t)). Indeed, every time a good arrives and is allocated, the entries of HandLare updated accordingly. The way these indices are updated enforces that one out of every 2n−1goods, in general, and one out of n—that later becomes 3n−2—high-valued goods is allocated to each agent. These quantities may seem somewhat loose, but as it shown in our analysis, asking for more frequent allocations per | https://arxiv.org/abs/2505.22174v1 |
agent, would not leave enough room for our competing goals to work simultaneously. In our proofs and statements we often need to refer to the agents’ bundles at different time steps. For clarity, we write At i(rather than just Ai) to denote the bundle of agent iat the end of time step t. In fact, we use this notation to the statement of the main theorem of this section as well. Recall also that nh(i, t) =|{g∈M|vi(g) =αiandgis one of the first tgoods that arrived }|is the number of high- valued goods agent ihas seen up to (and incuding) time t. 10 Algorithm 1 Deferred-Priority (v1, . . . , v n;M) (The valuation functions, vi,i∈[n], are given via oracles; Mis given in an online fashion, one good at a time.) 1:phase ←0;low←0;high←0;t←0// We initialize all of our counters. 2:fori∈Ndo 3: Ai← ∅;H[i]←n;L[i]←2n−1;χi←0// Ag. itolerates the loss of less than nhigh-valued goods. 4:whenever a new good garrives: 5:t←t+ 1 6:fori∈Ndo 7: ifvi(g) =αi>0then 8: H[i]←H[i]−1// Potential loss of a high-valued good; i’s priority for high-valued goods is increased. 9: else 10: L[i]←L[i]−1// Potential loss of a low-valued good; i’s priority for low-valued goods is increased. 11:Nh(g, t)← {i∈N|vi(g) =αiandχi= 0}// Potential recipients of gas a high-valued good. 12:Nℓ(g, t)← {i∈N|vi(g) =βiandχi= 0}// Potential recipients of gas a low-valued good. 13:ifNh(g, t)̸=∅then 14: high←high + 1 //gis allocated as a high-valued good. 15: j= arg mini∈Nh(g,t)H[i]// Ag. jhas the highest priority for g; ties are broken lexicographically. 16: Aj←Aj∪ {g}// The good is added to ag. j’s bundle. 17: H[j]←H[j] + 3n−2// Now ag. jtolerates the loss of at most 3n−2high-valued goods. 18: χj←1// Ag. jwill not get any more goods during the current phase. 19:else 20: low←low+ 1 //gis allocated as a low-valued good. 21: j= arg mini∈Nℓ(g,t)L[i]// Ag. jhas the highest priority for g; ties are broken lexicographically. 22: Aj←Aj∪ {g}// The good is added to ag. j’s bundle. 23: L[j]←2n+t// Ag. jnow has the lowest priority among active agents for the rest of the phase. 24: ifphase = 0then 25: χi←1// If this is phase 0, ag.jwill not get any more goods. 26:if(phase = 0andlow+high =n)or(phase >0andmax{low,high}=n)then 27: phase ←phase + 1 // The conditions to conclude this phase were met and we move to the next one. 28: low←0;high←0// We reset our counters. 29: fori∈Ndo 30: L[i]←2n−1// We reset the priority for low-valued goods. Theorem 4.1. Algorithm 1 builds an allocation such that, for every i∈N: 1.|An i|= 1, i.e., agent igets one of the first ngoods (assuming m≥n; otherwise |Am i| ≤1). 2.At (the end of) any time step t,At icontains at least ⌊nh(i, t)/(3n−2)⌋high-valued goods. Moreover, if agent igets to see at least nhigh-valued goods, at (the end of) time step ti0= min {t|nh(i, t)≥n},Ati0 i contains at least 1high-valued good. 3.At (the end of) any time step t≥n,|At i| ≥ ⌊(t−n)/(2n−1)⌋+ 1, i.e., agent ihas received at least one out of every 2n−1goods they have seen after the first ngoods. Proof of parts 1. and 3. We are going to show parts 1., 2., and 3. | https://arxiv.org/abs/2505.22174v1 |
separately. While parts 1. and 3. are relatively straightforward, part 2. requires an elaborate analysis using delicate inductive arguments. During any phase, an agent iis called active ifχi= 0andinactive otherwise. As it is clear by the definition of the sets 11 Nh(g, t)andNℓ(g, t)(lines 11 and 12), no agent can receive any more goods during the current phase once it becomes inactive. We begin with part 1. of the theorem. Lines 18 and 24-25 update χito1for any agent who receives any good during Phase 0, ensuring that no one gets more than 1good during this phase. Further, if m≥n, lines 14 and 20, combined with the first part of the condition in line 26, ensure that exactly ngoods are allocated during Phase 0, as either low-valued or high-valued goods. Therefore, |An i|= 1, i.e., each agent gets exactly 1of the first ngoods. Moving to part 3., let q∈Z+. We observe that, during Phase q, no more than 2n−1goods are allocated, as it is enforced by lines 14 and 20, combined with the second part of the condition in line 26. Next, note that, like in Phase 0, if an agent ireceives a high-valued good (which triggers χito become 1in line 18), becomes inactive and never receives another good during Phase q. However, unlike in Phase 0, when agent ireceives a low-valued good at time t, it now stays active. Nevertheless, in such a case, agent ibecomes lastin the implicit priority list of active agents for low-valued goods, as now L[i] = 2n+t(line 23) and L[j]≤2n+t′with t′< t, forj∈N\ {i}. The important observation here is that once this happens, agent icannot receive another good before every other agent is inactive or receives a low-valued good at a timet′′> t. We are now ready for the main argument that implies part 3. If Phase qterminates because high =n, then every agent received exactly one high-valued good. Otherwise, if Phase qterminates because low=n, we distinguish two simple cases. If no agent got more than one low-valued good in Phase q, then everyone received exactly one low-valued good (and at least one good, in general). So, assume that there is some agent iwho received at least 2low-valued goods during Phase q. By the preceding discussion, this means that every other agent became inactive at some point (by receiving a high-valued good) or received a low-valued good between the times when agent igot its first and second low-valued good. In any case, if Phase qterminates, then each agent has received at least 1out of at most 2n−1goods that were allocated during the phase. Combining this with the fact that every agent gets 1out of ngoods in Phase 0(by part 1.), we conclude that at the end of any time step t, |At i| ≥1 +⌊(t−n)/(2n−1)⌋. Most of the remaining section is dedicated to proving part 2. of Theorem 4.1. Thus, it would be useful to give some intuition behind both the statement and its proof. In doing so, we will establish some additional notation and terminology. The obvious way to show that each agent gets at least | https://arxiv.org/abs/2505.22174v1 |
1out of the first n high-valued goods they see and 1out of every 3n−2high-valued goods overall, is to show that it is always possible to allocate the goods in such a way so that H[i]>0, for all i∈N, at the end of each time step t(i.e., right before the (t+ 1) -th good arrives). The reason, of course, is that H[i]has been defined to contain the number of high-valued goods that agent ican afford to lose before the desideratum of part 2. of Theorem 4.1 is violated. A straightforward necessary condition for H[i]>0, i∈N, to hold at the end of each time step tis to have at most one agent j, such that H[j] = 1 before a new good arrives. To see this, assume that there are distinct j, j′such that H[j] =H[j′] = 1 and that the next good gthat arrives is high-valued for everyone. Then, no matter how gis allocated, at least one of H[j], H[j′]will hit 0, meaning that enough high-valued goods were lost for one of the two agents for part 2. of Theorem 4.1 to fail. In fact, we can extend this necessary condition to having at most kagents j1, . . . , j k, such that H[jℓ]≤kbefore a new good arrives, for any k∈[n]. Indeed, if there are at least k+ 1such agents and the next kgoods are high-valued for everyone, it is impossible to allocate them without making some coordinate(s) of vector Hequal to 0. The interesting thing here, is that this simple necessary condition always allows us to “legally” allocate at least one next good g. Roughly speaking, if we consider the agents who see gas high-valued and give it to the agent with the smallest Hentry among them, then it is not very hard to see (we will prove it formally 12 in Claim 4.3) that it is not possible to end up with any agent ihaving H[i] = 0 (recall that after receiving a good, an agent’s Hentry increases significantly). So, if one could allocate each good and, at the same time, maintain the condition that Hcontains at most kentries that are kor below, for all k∈[n], then part 2. of Theorem 4.1 would follow. The tricky part is to make sure that this condition still holds after allocating any good and this is the core of the technical difficulty of proving the theorem, mainly because we often need to allocate goods that are not low-valued for everyone to agents who see them as low-valued. In fact, the latter is absolutely necessary for part 3. of the theorem shown above. In order to formalize things, we introduce the following notation for the level sets that contain all agents with the same priority according to Hat any given time: Hℓ(t) ={i∈N|H[i] =ℓat the end of time step t}. With this notation, the above necessary and sufficient condition for being able to legally extend the allocation at time t+ 1becomes k[ ℓ=0Hℓ(t) ≤k,for all 0≤k≤n . (1) In the discussion above, we imply that maintaining (1)is easier if, whenever a good is | https://arxiv.org/abs/2505.22174v1 |
viewed as high-valued by someone, it is always allocated as a high-valued good. Indeed, this is the case: if we only allocated goods so as to maximize the social welfare, then we would be able to maintain (1)for every t, even if in line 17 we only added nrather than 3n−2. The technical reason why will become clear in the proofs of Claims 4.6 and 4.7, but the issue with this is that it would mean that agents who mostly see low-valued goods might have to wait arbitrarily long before getting anything. Hence, we add 3n−2in line 17, to give us some extra room to keep every agent content. From a technical point of view, within our proofs we typically need to decouple the two cases that cause changes to entries of H(allocating a good that is not globally low-valued as high-valued versus as low-valued), as they are qualitatively very different. The following lemma states the fact that condition (1)holds for all time steps t≥0. As its proof is fairly long and complicated, it is deferred to the Section 4.1. Lemma 4.2. At the end of any time step t≥0, we have Sk ℓ=0Hℓ(t) ≤k, for all 0≤k≤n. At this point we are ready to prove part 2. of Theorem 4.1. Proof of part 2. of Theorem 4.1. In the discussion preceding this proof, we claimed that condition (1)is sufficient, and we briefly argued about it in a rather hand-wavy way. Here we begin by formalizing this fact. It should be noted that Claim 4.3 does not say that by allocating the (t+ 1) -th good gwe ensure that the condition holds for time step t+ 1; this is shown separately in Lemma 4.2. The claim barely states that whenever condition (1)holds and things have gone well in the past, Algorithm 1 can allocate the next good without violating the guarantees we aim to show. Claim 4.3. Lettbe any time step in {0,1, . . . , m −1}and let gbe the (t+ 1)-th good. Assuming that all entries of Hhave remained positive at the end of all time steps up to t, condition (1)guarantees that Algorithm 1 can allocate gwithout any entry of Hbecoming 0at the end of time step t+ 1. Proof of Claim 4.3. For the sake of clarity, here we make the dependency of Hontexplicit and write Ht[i] to denote H[i]at the end of time step t. Assume that Sk ℓ=0Hℓ(t) ≤k, for all k∈[n]. In particular, |H1(t)| ≤1. If|H1(t)|= 0, then Ht[i]≥2for all i∈Nand clearly, no matter how good gis allocated, Ht+1[i]≥1for all i∈N. Similarly, if |H1(t)|= 1 andj∈Nis the unique agent such that Ht[j] = 1 13 butvj(g) =βj, we have Ht+1[j] =Ht[j] = 1 as well as Ht+1[i]≥Ht[i]−1≥1for all i∈N\ {j}, like before, no matter how gis allocated. The interesting case here is when |H1(t)|= 1,j∈Nis the unique agent such that Ht[j] = 1 and vj(g) =αj. Again, for i∈N\ {j},Ht+1[i]≥1but now we must show that Algorithm 1 will add gto Aj. Given that jhas the highest priority (i.e., lowest entry in H, which has | https://arxiv.org/abs/2505.22174v1 |
temporarily dropped to 0), for jto get git suffices to show that χj= 0. Towards a contradiction, assume this is not the case, i.e., jhas received a high-valued good at time t′which belongs to the same phase as t. Since each phase has at most 2n−1time steps (see the proof of part 3. of Theorem 4.1), t−t′≤2n−2. But then, Ht[j]≥Ht′[j]−(2n−2) = Ht′−1[j]−1 + 3 n−2−2n+ 2≥n , contradicting the choice of j. We conclude that an agent jwithHt[j] = 1 cannot have received a high- valued good in the phase that includes t, thus χj= 0. Therefore, jis the agent who gets gin line 15, and Ht+1[j] =Ht[j]−1 + 3 n−2 = 3 n−2>0, completing the proof. Cl. 4.3 ⊡ Recall that, by the design of the priority vector H, in order to show that each agent gets at least 1out of the first nhigh-valued goods they see and 1out of every 3n−2high-valued goods overall, it suffices to show that it is always possible to allocate the goods so that H[i]>0, for all i∈N, at the end of each time stept. By combining Claim 4.3 with Lemma 4.2, which shows that condition (1)is maintained throughout the execution of Algorithm 1, we have exactly that. The algorithm allocates all goods without any entry of Hbecoming 0at the end of any time step. Equivalently, by the definition of how His updated, agent i receives at least 1high-valued good by time ti0= min {t|nh(i, t)≥n}and at least 1out of every 3n−2 high-valued goods they see after that, for a total of at least ⌊nh(i, τ)/(3n−2)⌋high-valued goods by the end of a time step τ≥0. Now, using Theorem 4.1, we can argue about the temporal maximin share guarantees of the Deferred- Priority algorithm (Algorithm 1). Recall from Section 2 that an agent iis of type 1 when αi> βi= 1, it is of type 2 when αi=βi= 1and it is of type 3 when 1 =αi> βi= 0. As the arguments needed from different types are somewhat different, we will state the corresponding guarantees separately. Corollary 4.4. Any agent iof type k∈ {2,3}receives at least a constant fraction of its temporal maximin share by Algorithm 1. In particular, vi(At i)≥µi(t)/k, for any time step t≥0. Proof. First let agent ibe of type 2. We will bound the value igets during the allocation sequence induced by Algorithm 1. At the end of any time step t < n , we have µi(t) = 0 , so the statement trivially holds. If, instead, n≤t < n + (2n−1), then by part 1. of Theorem 4.1 we have |At i| ≥1and so, vi(At i)≥1, whereas µi(t)≤ ⌊(3n−1)/n⌋= 2. Finally, we may assume that n+k(2n−1)≤t < n +(k+1)(2 n−1), for some k∈Z>0. Then, by part 3. of Theorem 4.1, agent ihas received at least one out of every 2n−1 goods they have seen after Phase 0and so, vi(At i) =|At i| ≥ ⌊(t−n)/(2n−1)⌋+ 1 = k+ 1, whereas, by the definition of maximin share, µi(t) =t n <n+ (k+ 1)(2 n−1) n =2(k+ 1)n+ (n−k−1) n ≤2(k+ 1). Next, assume | https://arxiv.org/abs/2505.22174v1 |
that agent iis of type 3. Given that low-valued goods are completely irrelevant to agent i, we can consider a time alternative τthat starts at 0, like t, but only increases when a high-valued good for i arrives. That is, while treflects how many goods have arrived in general, τreflects how many high-valued goods with respect to agent ihave arrived instead. We can repeat the exact same analysis we did for agents of type 2, but using τinstead of t,3instead of 2for the factor, 3n−2whenever the quantity 2n−1was used, and by invoking part 2. of Theorem 4.1 instead of part 3. 14 Corollary 4.5. Any agent iof type 1 receives at least a 1/(2n−1)fraction of its temporal maximin share by Algorithm 1., i.e., vi(At i)≥µi(t)/(2n−1), for any time step t≥0, which improves to Ω(1) from time ti0 onward (recall that ti0= min {t|nh(i, t)≥n}). Proof. Letibe a type 1 agent and consider any time step t. Let κ, λbe the number of high-valued and low-valued goods in At i, respectively, where κ, λ∈Z≥0. C C C C Ca a a a as s s s se e e e e1 1 1 1 1:κ= 0. . . . .At the end of any time step t < n , we have µi(t) = 0 and the statement trivially holds. So assume that t≥nAsκ= 0implies that nh(i, t)≤n−1, or equivalently, t < t i0(by part 2. of Theorem 4.1), we have n≤t < t i0. On one hand, we know that vi(At i) =λ≥1, where the second inequality follows by part 1. of Theorem 4.1. On the other hand, by considering a hypothetical allocation where each one of the nh(i, t)high-valued goods for agent iis a whole bundle and the low-valued goods for agent i are split as equally as possible into n−nh(i, t)bundles, we see that i’s maximin share is at most the value of the worst bundle among the ones filled with low-valued goods, i.e., µi(t)≤t−nh(i, t) n−nh(i, t) ≤t−nh(i, t)−(n−1−nh(i, t)) n−nh(i, t)−(n−1−nh(i, t)) ≤t−(n−1) 1 =t−(n−1), where the second inequality follows from nh(i, t)≤n−1and the simple fact thata b≤a−c b−cfor any a≥b > c≥0. Note however that there is a straightforward upper bound on t, implied by parts 1. and 3. of Theorem 4.1: t≤n+ (λ−1)(2n−1) + (2 n−2). Thus, µi(t)≤n+ (λ−1)(2n−1) + (2 n−2)−(n−1) = λ(2n−1) = (2 n−1)vi(At i). For the remaining two cases, we are going to show an approximation factor of at least 1/4with respect to proportionality, which then implies the same guarantee for maximin share fairness. Let Sbe the set containing the first tgoods for some t. Note that necessarily one of these two cases holds if t≥ti0(but possibly even earlier than that). C C C C Ca a a a as s s s se e e e e2:κ⩾1κ⩾1κ⩾1κ⩾1κ⩾1a a a a an n n n nd d d d dλ= 0. . . . .In this easier case we have vi(At i) =καiandimight have seen at most κ(2n− 1) + (2 n−2)goods (by part 3. of Theorem 4.1) of total value vi(S)≤(κ(2n−1) | https://arxiv.org/abs/2505.22174v1 |
+ (2 n−2))αi. We have µi(t)≤vi(S) n≤κ(2n−1) + (2 n−2) nαi≤2nκi+ 2n nαi= 2(κ+ 1)αi≤4καi= 4vi(At i). C C C C Ca a a a as s s s se e e e e3 3 3 3 3:κ⩾1κ⩾1κ⩾1κ⩾1κ⩾1a a a a an n n n nd d d d dλ⩾1λ⩾1λ⩾1λ⩾1λ⩾1. . . . .Similarly to Case 2, vi(At i) =καi+λandimight have seen at most n+ (κ+ λ−1)(2n−1) + (2 n−2)goods (by parts 1. and 3. of Theorem 4.1), out of which at most n+ (κ−1)(3n− 2) + (3 n−3)can be high-valued for i(by parts 1. and 2. of Theorem 4.1). Then it is a matter of simple calculations to show that for the total value vi(S)to be maximized, the low-valued goods are at most (λ−1)(2n−1) + (2 n−2). That is, vi(S)≤[n+ (κ−1)(3n−2) + (3 n−3)]αi+ [(λ−1)(2n−1) + (2 n−2)] ≤[n+ 3n(κ−1) + 3 n]αi+ 2n(λ−1) + 2 n= 3n κ+1 3 αi+ 2nλ . Therefore, we have µi(t)≤vi(S)/n≤(3κ+ 1)αi+ 2λ≤4καi+ 4λ= 4vi(At i). 4.1 Proving Lemma 4.2 Proof of Lemma 4.2. As we did in the proof of Claim 4.3, for clarity, we write Ht[i]to denote H[i]at the end of time step t. The proof will be broken down into two proofs, one for t≤nand one for t≥n; for the sake of presentation, the corresponding cases of the lemma are stated as Claims 4.6 and 4.7 below. 15 Claim 4.6. At the end of any time step t≤n, we have Sk ℓ=0Hℓ(t) ≤k, for all 0≤k≤n. Proof of Claim 4.6. We will use induction on the number of agents nfor a slightly more general version of the algorithm that takes the initialization of H=H0as part of the input, where it must be that H0[i]≥n for all i∈N. Then the statement of the claim follows by fixing this part of the input to be H0[i] =nfor all i∈N. For a single agent, it is straightforward that, for H0[1]≥1, initially H0(0) = 0and H1(0) ≤1, whereas after the first good is allocated, either H1[1]remains unchanged and, thus, at least 1(if the good was low-valued) or it is updated to H1[1]−1 + 3·1−2≥1(if the good was high-valued). Either way, H0(1) = 0and H1(1) ≤1, completing our base case. Now assume that the statement of the claim is true for a certain n′≥1, and consider any instance with n=n′+1agents and any H0∈Zn ≥n. For this particular instance, let g1, g2, . . . , g mbe the goods that arrive, in this order. Because of how goods are allocated in Phase 0, i.e., no agent gets a second good, once an agent jreceives g1at time t= 1, the remaining of Phase 0is indistinguishable from (the complete) Phase 0of an instance with the agents of N\ {j}, an appropriate initial priority vector (defined by H0[i]−1Nh(g1,1)(i) orH0[i]−1Nh(g1,1)(i)−1; see Cases 1 and 2 below), and the sequence of goods being g2, g3, . . . , g m. We are going to invoke the induction hypothesis on this sub-instance but we distinguish two cases, depending on whether g1is allocated as a high-valued or a low-valued good. Notice that it | https://arxiv.org/abs/2505.22174v1 |
is without loss of generality to assume that the agent who gets good g1is agent 1, as it is a matter of renaming the agents, if needed, and is consistent with our lexicographic tie-breaking. For the sub-instance that only involves agents 2through n, has a properly defined initial priority vector H′ 0(see within the cases for the corresponding description), and the sequence of goods is g2, . . . , g m, we use a prime to distinguish the corresponding quantities. That is, we use t′to denote time, rather than t that we reserve for the original instance; in general t′=t−1, e.g., the 5th time step in the sub-instance corresponds to the 6th time step of the original problem. Thus, we will write H′ t′for the priority vector of the sub-instance at the end of time step t′of that instance andH′ ℓ(t′)for the level sets that it induces. Assuming that H′ 0is such that H′ 0[i]≥n−1fori∈ {2, . . . , n }, by the induction hypothesis, we have for this sub-instance: At the end of any t′≤n−1, k[ ℓ=0H′ ℓ(t′) ≤k, for all 0≤k≤n−1. (2) C C C C Ca a a a as s s s se e e e e1 1 1 1 1:v1(g1) =α1 v1(g1) =α1 v1(g1) =α1 v1(g1) =α1 v1(g1) =α1. . . . .In this case, the initial vector H′ 0given as input for the sub-instance is defined by H′ 0[i] =H0[i]−1Nh(g1,1)(i)≥n−1fori∈ {2, . . . , n }, where 1S(i)is the indicator function of whether i∈S. Notice that this way, the priority among agents remains exactly the same as in the original instance andH′ t′[i] =Ht′+1[i]for all t′∈ {0, . . . , n −1}andi∈ {2, . . . , n }; of course, H′ t′[1]is not defined. Further, because agent 1gets a high-valued good at time 1,H1[1] = H0[1]−1+3n−2≥4n−3and, thus, throughout Phase 0 of the original instance (i.e., t∈ {0,1, . . . , n }) we have Ht[1]≥4n−3−(t−1)≥3n−2≥n. So, agent 1may only appear in Hn(t)for any t∈[n]. By the discussion about the correspondence between H′ andHabove, this means that H′ ℓ(t′) =Hℓ(t′+ 1) forℓ∈ {0,1, . . . , n −1}. Therefore, by the induction hypothesis, at the end of any t∈ {1, . . . , n }, Sk ℓ=0Hℓ(t) = Sk ℓ=0H′ ℓ(t−1) ≤k, for all 0≤k≤n−1. For the missing cases, namely t= 0and0≤k≤nor0≤t≤nandk=n, we note that they are both trivial: (i) for t= 0, anyHℓ(0)is empty with the possible exception of Hn(0)that may contain up to n agents, and (ii) for k=n, Sn ℓ=0Hℓ(t) ≤ntrivially holds for any t. We conclude that at the end of any t≤n, we have Sk ℓ=0Hℓ(t) ≤k, for all 0≤k≤n. 16 C C C C Ca a a a as s s s se e e e e2:v1(g1) =β1< α1 v1(g1) =β1< α1 v1(g1) =β1< α1 v1(g1) =β1< α1 v1(g1) =β1< α1Since the very first good, g1, was allocated as a low-valued good despite χi= 0, for all i∈N, it must be the case that vi(g1) =βi. That is, H1[i] =H0[i]≥n, for all i∈N. | https://arxiv.org/abs/2505.22174v1 |
In this case, the initial vector H′ 0of the sub-instance given as input is defined by H′ 0[i] =H0[i]−1≥n−1 fori∈ {2, . . . , n }. Like in Case 1, the priority among agents remains exactly the same as in the original instance but here H′ t′[i] =Ht′+1[i]−1for all t′∈ {0, . . . , n −1}andi∈ {2, . . . , n }; again, H′ t′[1]is not defined. Unlike Case 1, however, here agent 1may appear in Hℓ(t)for several combinations of ℓandt. Nevertheless, this will not be an issue. First notice that, even if things went really wrong, there are not enough goods forH′ t′[i]to become negative for any i∈ {2, . . . , n }and any t′∈ {0, . . . , n −1}, and so H′ −1(t′) =∅ for all t′∈ {0, . . . , n −1}. By the correspondence between H′andHdiscussed above, we have that Hℓ(t)\ {1}=H′ ℓ−1(t−1)and, thus, Hℓ(t)⊆ H′ ℓ−1(t−1)∪ {1}, forℓ∈ {0,1, . . . , n }. Therefore, by the induction hypothesis, at the end of any t∈ {1, . . . , n }, k[ ℓ=0Hℓ(t) ≤ k[ ℓ=0H′ ℓ−1(t−1)∪ {1} ≤ k−1[ ℓ=0H′ ℓ(t−1) + 1≤k−1 + 1 = k , for all 0≤k≤n. The missing cases ( t= 0and0≤k≤n) are trivial exactly like their counterparts in Case 1. We conclude that at the end of any t≤n, we have Sk ℓ=0Hℓ(t) ≤k, for all 0≤k≤n.Cl. 4.6 ⊡ It is not hard to see that the proof of Claim 4.6 cannot be extended beyond t=nas it crucially relies on the fact that during Phase 0(which has fixed length and lasts up to t=n) once an agent gets a good, they are inactive for the rest of the phase. Interestingly enough, the proof of Claim 4.7 below (induction with respect to t) could not have been used for t < n as it crucially depends on H[i]not being relevant unless agent iactively competes for a high-valued good is allocated as high-valued (which, in turn, is the result of the “slack” we add to H[i]after an agent igets a high-valued good). Claim 4.7. At the end of any time step t≥n, we have Sk ℓ=0Hℓ(t) ≤k, for all 0≤k≤n. Proof of Claim 4.7. Given an arbitrary instance, we will prove the statement using strong induction on the time step t. Essentially, Claim 4.6 serves as the base case. So, assume that the statement of the claim is true for all time steps up to a certain t0≥n, and consider the next time step t=t0+ 1. Letgbe the t-th good andjbe the agent who eventually gets it. C C C C Ca a a a as s s s se e e e e1 1 1 1 1:vj(g) =βj< αj vj(g) =βj< αj vj(g) =βj< αj vj(g) =βj< αj vj(g) =βj< αj. . . . .That is, gis allocated as a low-valued good. We claim that for any agent i∈N, such that Ht[i]≤n−1, we have Ht[i] =Ht−1[i], i.e., the entries of Htmay have changed only for agents who are irrelevant with respect | https://arxiv.org/abs/2505.22174v1 |
to condition (1). Indeed, if Ht[i]has changed, then vi(g) =αi. Moreover, it must be χi= 0, as otherwise Nh(g, t)̸=∅andgwould be allocated as a high-valued good instead. But χi= 0means that agent ihas received a high-valued good, say at a time step th, during the current phase. Recall that each phase has at most 2n−1time steps (see the proof of part 3. of Theorem 4.1) and, thus, t−th≤2n−2. Also, by the induction hypothesis (for time step thandk= 0), we have Hth−1[i]≥1. Combining these, we get Ht[i]≥Hth[i]−(2n−2) = Hth−1[i]−1 + 3 n−2−2n+ 2≥n , (3) as claimed. This means that Hℓ(t) =Hℓ(t−1)forℓ∈ {0,1, . . . , n −1}. Therefore, by invoking the induction hypothesis, at the end of time step t, we have Sk ℓ=0Hℓ(t) = Sk ℓ=0Hℓ(t−1) ≤k, for all 0≤k≤n−1. For the missing case, namely k=n, it is trivial as Sn ℓ=0Hℓ(t) ≤nalways holds for any t. We conclude that at the end of time step t, we have Sk ℓ=0Hℓ(t) ≤k, for all 0≤k≤n. 17 C C C C Ca a a a as s s s se e e e e2:vj(g) =αj vj(g) =αj vj(g) =αj vj(g) =αj vj(g) =αj. . . . .That is, we assume next that gis allocated as a high-valued good. Let N+ h(g, t) ={i∈ N|vi(g) =αi}, i.e., the set of all agents who see the t-th good as high-valued. The agents of N+ h(g, t)are exactly those whose entries in Htare updated during the current time step and how relevant it is for our analysis. We carefully categorize agents according to what happens to their entry in Htduring time step t: •For any agent i∈N\N+ h(g, t),Ht[i] =Ht−1[i], as they see gas a low-valued good. •For any agent i∈Nh(g, t)\ {j},Ht[i] =Ht−1[i]−1, as they see gas a high-valued good and they miss it. •For any agent i∈N+ h(g, t)\Nh(g, t), again Ht[i] =Ht−1[i]−1, as they see gas a high-valued good and they miss it, but they are essentially irrelevant because Ht[i]≥n. This follows by the argument in Case 1 above, as χi= 0and the chain of (in)equalities of (3) applies exactly as is. •For agent jitself, we have Ht[j] =Ht−1[j]−1 + 3 n−2≥3n−2≥n, where Ht−1[j]≥1follows from the induction hypothesis for time step t−1andk= 0. We conclude that agent jis also essentially irrelevant. From the above, it becomes clear that agents in Nh(g, t)∩Sk ℓ=0Hℓ(t)should be treated carefully when we try to bound Sk ℓ=0Hℓ(t) , for some k∈ {0,1, . . . , n }. The easiest case, of course, is when Nh(g, t)∩Sk ℓ=0Hℓ(t) =∅. Then, Hℓ(t) =Hℓ(t−1)forℓ≤kand, thus, by the induction hypothesis, we have that at the end of time step t, Sk ℓ=0Hℓ(t) = Sk ℓ=0Hℓ(t−1) ≤k. Also, when k=n, it trivially holds Sn ℓ=0Hℓ(t) ≤n. Next, assume that k∈ {0,1, . . . , n −1}is such that Nh(g, t)∩Sk ℓ=0Hℓ(t) =S̸=∅. At this point, we need three simple observations. The first one (following from the last bullet above) is that j /∈Sk ℓ=0Hℓ(t) and, thus, j /∈S. The second one is thatSk ℓ=0Hℓ(t)⊆Sk+1 ℓ=0Hℓ(t−1), as no entry | https://arxiv.org/abs/2505.22174v1 |
of Hreduces by more than 1in a single time step. The last one is that j∈Sk+1 ℓ=0Hℓ(t−1); if not, we would have Ht−1[j]≥k+ 2and the j-th entry of Hright before gwas allocated to jwould be Ht−1[j]−1≥k+ 1> k≥mini∈SH[i]≥mini∈Nh(g,t)H[i], contradicting the choice of j. Using these observations, as well as the induction hypothesis, we have that at the end of time step t, k[ ℓ=0Hℓ(t) ≤ k+1[ ℓ=0Hℓ(t−1)\ {j} = k+1[ ℓ=0Hℓ(t−1) −1≤(k+ 1)−1 =k. This exhausts all possible cases for k∈ {0,1, . . . , n }and concludes Case 2. Cl. 4.7 ⊡ Clearly, combining the two claims completes the proof of the lemma. 5 The Power of Limited Foresight A natural question at this point is whether one can avoid the linear term in the MMS approximation guarantee by allowing some additional information about the future. Unfortunately, there is a very simple instance that illustrates that this is not possible. Proposition 5.1. Letε > 0be a constant. Even if the whole instance is known in advance, as long as it is required to irrevocably allocate each good right after it arrives, no algorithm can always compute a (1/n+ε)-temporal-MMS allocation, even on 2-value instances. 18 Proof. We consider a 2-value instance with nagents, where βi= 1andαi=α≥n, for all i∈N. There arenuniversally low-valued goods g1, . . . , g nandn−1universally high-valued goods gn+1, . . . , g 2n−1, as shown below, that arrive according to their indices. g1g2. . .gn−1gngn+1gn+2 . . .g2n−1 ag. 1: 1 1 . . . 1 1 α α . . . α ag. 2: 1 1 . . . 1 1 α α . . . α .............................. ag.n−1: 1 1 . . . 1 1 α α . . . α ag.n: 1 1 . . . 1 1 α α . . . α We may assume that this information is available even before the first good arrives. Like in the proof of Theorem 3.2, we note that if at any point during the first ntime steps an agent receives a second good, then at the end of time step t=nthere would be at least one agent that has received value 0despite having a positive maximin share value. So, we may assume that any algorithm with non-trivial guarantees assigns to each agent exactly one of the first ngoods. Without loss of generality, agent igets good gias shown. Given that, no matter how goods gn+1, . . . , g 2n−1are allocated, at least one agent, say agent 1, will not receive any of these. So, at the end of time step t= 2n−1we have vi(A1) =v1(g1) = 1 , while µ1(2n−1) = n. Despite Proposition 5.1, in this section we show that being able to see only a linear number of steps in the future leads to significantly simpler algorithms with EF1 and EF2 guarantees. 5.1 The Illustrative Case of Two Agents There is an easy algorithm augmented with foresight of length 1that achieves EF1in every even time step. Although Naive-Matching (Algorithm 2) is essentially subsumed by the main result of the | https://arxiv.org/abs/2505.22174v1 |
next section, it is simpler to state, much simpler to analyze and still illustrates the power of—even very limited—foresight. Theorem 5.2. For any two agent 2-value instance augmented with foresight of length 1, Algorithm 2 builds an allocation that is temporal-EF2, while it is EF1 for every even time step t≥0. Proof. We first observe that if an allocation is EF1at the end of some time step t≥0, then it will trivially beEF2at the end of time step t+ 1, no matter how the corresponding good is allocated. That is, it suffices to show that Algorithm 2 builds an allocation that is EF1for every even time step t, and the fact that it is also temporal- EF2immediately follows. We will show a somewhat stronger statement for even time steps: for any k∈Z≥0, at the end of time step t= 2k≤m,(i)both bundles contain kgoods each, and (ii)either ctr= 0and only agent 1may envy agent 2by at most α1−β1, orctr= 1and only agent 2may envy agent 1by at most α2−β2.Of course, (i) is straightforward because, for any k≥0, we always create a matching between the two agents and the goods g2k+1, g2k+2. We are going to show (ii) using induction on k. At time t= 0(i.e., for k= 0) the statement of (ii) trivially holds: ctr= 0and no agent envies the other. Now assume that (ii) holds for some t= 2k, such that k≥0andt= 2k+ 2≤m. At time t= 2k+ 1 the algorithm enters the ‘else’ in line 8. Note that whenever the condition of line 10 is true, i.e., goods g2k+1, g2k+2induce any pattern of Table 1 except of those of blocks I or II, each agent (weakly) prefers the good they receive according to the corresponding allocation shown in Table 1. That is, for agent 1we have v1(A2k+2 1)−v1(A2k 1)≥v1(A2k+2 2)−v1(A2k 2), where the added superscript indicates the time step at the end of which we consider the bundle, and similarly for agent 2. This observation that envy can only be 19 β β β α α β α α β β β β β β β β β β I β α α β α α β α β α β α β α β β β α II α β α α α β α β α β α β β β β α α β α α α α α α α α α α Table 1: When we are given foresight of length 1, achieving EF1on every other time step for two agents is very simple. We only need to keep track of who “wins” the next block of type I or II, i.e., who gets the contested high-valued good ( dashed: agent 1 wins, dotted: agent 2 wins). In every other case, we may allocate the goods in a predetermined way as shown for any other block here. Since we only care about the general value patterns, for the sake of readability, we omit the indices; e.g., we write αrather than α1in the first row and α2in the second row of each block. reduced in this case, | https://arxiv.org/abs/2505.22174v1 |
combined with the fact that ctrdoes not change here, implies that (ii) still holds for t= 2k+ 2. So, we may assume that the condition of line 10 is false, i.e., goods g2k+1, g2k+2induce the pattern of block I or block II of Table 1 with one universally high-valued and one universally low-valued good. Here we consider the two cases of the induction hypothesis. Case 1: att= 2k,ctr= 0,v1(A2k 1)≥v1(A2k 2)−(α1−β1)andv2(A2k 2)≥v2(A2k 1).According to lines 14-17, in this case, ctrbecomes 1and the algorithm commits to giving the high-valued good to agent 1. Thus, v1(A2k+2 1) =v1(A2k 1) +α1≥v1(A2k 2)−(α1−β1) +α1=v1(A2k+2 2), as well as v2(A2k+2 2) =v2(A2k 2) +β2≥v2(A2k 1) +β2=v2(A2k+2 1)−α2+β2, i.e., (ii) still holds for t= 2k+ 2. Case 2: att= 2k,ctr= 1,v1(A2k 1)≥v1(A2k 2)andv2(A2k 2)≥v2(A2k 1)−(α2−β2).This is completely symmetric to Case 1 above. According to lines 14-17, ctrbecomes 0and the algorithm commits to giving the high-valued good to agent 2. Thus, v1(A2k+2 1) =v1(A2k 1) +β1≥v1(A2k 2) +β1=v1(A2k+2 2)−α1+β1, and v2(A2k+2 2) =v2(A2k 2) +α2≥v2(A2k 1)−(α2−β2) +α2=v2(A2k+2 1), i.e., (ii) still holds for t= 2k+ 2. This concludes the induction and shows that Algorithm 2 builds an allocation that is EF1for every even time step tand, thus, temporal-EF2. 20 Algorithm 2 Naive-Matching (v1, v2;M) (The valuation functions, v1, v2, are given via oracles; Mis given in an online fashion, one good at a time, along with a preview of the next good after that.) 1:t←0;ctr←0// We initialize our counters and the allocation. 2:fori∈Ndo 3: Ai← ∅ 4:whenever a new good garrives along with a preview of the next good, g′: 5:t←t+ 1 6:ift= 0 mod 2 then 7: Allocate gaccording to the commitment of time step t−1. 8:else 9: B← v1(g)v1(g′) v2(g)v2(g′) 10: ifBfollows any pattern of Table 1 except of those of blocks I or II then 11: AddgtoA1orA2according to the allocation of the corresponding pattern in Table 1. 12: Commit to add g′toA1orA2at time step t+ 1according to the aforementioned allocation. 13: else 14: ctr←(ctr+ 1) mod 2 15: j= 2−ctr 16: AddgtoA1orA2according to the allocation of the corresponding pattern in Table 1 (i.e., I or II) which gives the contested high-valued good to agent j. 17: Commit to add g′toA1orA2at time step t+ 1according to the aforementioned allocation. Corollary 5.3. Assuming mis large enough, for any λ∈Z>0, after a sufficient number of steps, the allocation built by Algorithm 2 becomes and remains λ/(λ+ 2) -EF,λ/(λ+ 1) -EF1, and λ/(λ+ 1) -PROP. Proof. Clearly, if mis large enough (e.g., m≥2λmax i∈[2]⌈αi⌉) there is some t∗by which each agent ihas received value equal to at least λαi. At the end of any time step t≥t∗, we have for agent 1 v1(At 2)≤min g,g′∈At 2v1(At 2\ {g, g′}) + 2α1≤v1(At 1) + 2α1≤(1 + 2 /λ)v1(At 1), min g∈At 2v1(At 2\ {g})≤min g,g′∈At 2v1(At 2\ {g, g′}) +α1≤v1(At 1) +α1≤(1 + 1 /λ)v1(At 1),and 0.5(v1(At 1) +v1(At 2))≤0.5 v1(At 1) + (1 + 2 /λ)v1(At 1) = (1 + 1 /λ)v1(At 1), and similarly for agent 2. 5.2 Foresight of Length n−1Suffices As we mentioned in the beginning of | https://arxiv.org/abs/2505.22174v1 |
the previous section, we can generalize Theorem 5.2 to any number of agents, albeit with a more complicated algorithm. Similarly to Algorithm 2, what we would like to achieve between time steps knand(k+ 1)n(i.e., with goods gkn+1, . . . , g (k+1)n) is to obtain an EF1allocation by taking the union of the EF1allocation of time stepknand an appropriately chosen matching. However, it is easy to see that this fails if not done carefully, even for two agents, which is the reason why Algorithm 2 treats the patterns of blocks I and II with some care. In order to achieve a similar thing for general n, at time step kn+ 1we construct a matching M involving the current good and the n−1predicted goods, so that the matching’s envy graph “cancels 21 out” any problematic edges of the envy graph of the EF1allocation of time step kn. Then, for what we call the(k+ 1) -th round, we allocate these ngoods according to M. ForMto have the nice aforementioned behavior, however, we take it to be a maximum weight matching with respect to carefully defined auxiliary valuation functions. These auxiliary functions encode all the information about which goods are high or low for which agents, while giving additional weight to goods for agents who come earlier in the topological sorting. The latter captures the intuition that such agents should have a higher priority during this round. Algorithm 3 Priority-Matching (v1, . . . , v n;M) (The valuation functions, vi, i∈[n], are given via oracles; Mis given in an online fashion, one good at a time, along with a preview of the next n−1goods after that.) 1:t←0;ctr←0// We initialize our counters and the allocation. 2:fori∈Ndo 3: Ai← ∅ 4:whenever a new good garrives along with a preview of the next n−1goods 5:t←t+ 1 6:ift= 1 mod nthen // A new round begins, namely round number ⌈t/n⌉. 7: Construct the current envy-graph Gt= (Vt, Et).//Vt=Nand(i, j)∈Etifvi(Aj)> vi(Ai). 8: Find a permutation πthat induces a topological sorting of Gt.// If(i, j)∈Et, then π(i)< π(j). 9: fori∈[n]andh∈ {gt, gt+1, . . . , g t+n−1}do// Define auxiliary valuation functions consistent with π. 10: ˜vπ−1(i)(h)← 2(1 + 1 /2n)n−iifvπ−1(i)(h) =απ−1(i) (1 + 1 /2n)n−iotherwise//π−1(i)is the i-th agent in the ordering. 11: Find a maximum weight matching Mbetween the agents and goods in {gt, . . . , g t+n−1}with respect to these auxiliary functions. // The weight of a pair (j, h)is˜vj(h). The matching is perfect unless this is the last round; in this case, the matching is between Nand{gt, . . . , g m}. 12: Addg=gtto a bundle according to M, i.e., add gttoAjif and only if (j, gt)∈ M . 13: Commit to also allocate {gt+1, . . . , g t+n−1}according to Mat time steps t+ 1through t+n−1. 14:else // The allocation of gwas decided at the beginning of this round. 15: Allocate gaccording to the commitment of time step (⌈t/n⌉ −1)n+ 1(when this round begun). Theorem 5.4. For any personalized 2-value instance augmented with foresight of length n−1, Algorithm 3 builds an | https://arxiv.org/abs/2505.22174v1 |
allocation that is temporal- EF2, while it is EF1(and, thus, 1/n-MMS ) for every time step t=kn, k∈ Z≥0. Moreover, if at any step t0the allocation fails to be 1/2-EF1, then it remains 1/2-EF1at the end of every time step t≥ ⌈t0/n⌉n. Proof. The proof has similar structure to the proof of Theorem 5.2, but the induction is fairly more com- plicated due to the fact that the matching process is much less trivial here. In what follows, we refer to the execution of Algorithm 3 for time steps kn+ 1, kn+ 1, . . . , (k+ 1)nas the (k+ 1) -thround , for any k∈Z≥0. We begin with a generalization of the first observation made in the proof of Theorem 5.2: if an allocation is EF1at the end of some time step t≥0, then it will trivially be EF2at the end of time step t+ℓ, as long as no bundle receives more than one out of the last ℓgoods. For Algorithm 3, this means that if the allocation isEF1at the end of time step t=kn(i.e., right before round k+ 1begins), for every k∈Z≥0, then it remains EF2for every time step in between. This holds just because in every round each agent receives (at most) one good via the corresponding matching M. So, it suffices to show that Algorithm 3 builds an allocation that is EF1for every time step twhich is a multiple of n, and the fact that it is also temporal- EF2 immediately follows. Again we show a somewhat stronger statement: for any k∈Z≥0, at the end of time stept=kn≤m, 22 (i) all bundles contain kgoods each; (ii)the corresponding envy graph (i.e., Gkn+1constructed in the beginning of the next time step in line 7) is acyclic; (iii) any edge (i, j)in the above graph indicates envy which is at most αi−βi. Like before, (i) is straightforward, since the algorithm always creates a matching between the agents and (at most) ngoods in the beginning of a round and, as these goods arrive, it allocates them according to the matching. We are going to show (ii) and (iii) using induction on k. At time t= 0(i.e., for k= 0) the statements of (ii) and (iii) trivially hold, as no agent envies another agent. Now assume that (ii) and (iii) hold at the end of some time step t=kn, such that k≥0andt= (k+ 1)n≤m. At the beginning of time stept=kn+ 1the algorithm enters the ‘if’ in line 6 and constructs the envy graph G:=Gkn+1in line 7. ThisGis exactly the envy graph for which the two parts of the induction hypothesis hold. The fact that G is acyclic is what allows it to be topologically sorted (see, e.g., Cormen et al. [2022]) and, hence, makes line 8 well-defined. We use G′to denote the envy graph at the end of the current (i.e., the (k+ 1)-th) round; note that G′is constructed as G(k+1)n+1at the beginning of the next round. Claim 5.5. Any edge (i, j)ofG′indicates envy which is at most αi−βi. Proof of Claim 5.5. Consider any edge (i, j)ofG′and let hi, hjbe the | https://arxiv.org/abs/2505.22174v1 |
goods agents iandjreceived, respectively, in round k+ 1. This means that hi, hjwere matched with iandj, respectively, in M. By the induction hypothesis, we know that the envy from agent itowards agent jat the end of time step t=kn—if it existed at all—was upper bounded by αi−βi, i.e.,vi(Akn j)−vi(Akn i)≤αi−βi. Ifvi(hi)≥vi(hj), then vi(A(k+1)n j )−vi(A(k+1)n i ) =vi(Akn j) +vi(hj)−vi(Akn i)−vi(hi)≤αi−βi. We need to consider the case where βi=vi(hi)< vi(hj) =αi. If there was no edge (i, j)inG, i.e., if vi(Akn j)−vi(Akn i)≤0, then clearly vi(A(k+1)n j )−vi(A(k+1)n i ) =vi(Akn j) +αi−vi(Akn i)−βi≤αi−βi. So we may assume that (i, j)was an edge in G. Dy definition, this means that in any topological sorting agent imust come before agent j. In particular, ˆı=π(i)< π(j) = ˆȷ, i.e., in the sorting induced by the permutation π, agent iis the ˆı-th agent and agent jis the ˆȷ-th. By the definition of the auxiliary functions in line 10, we have ˜vi(hj) = 2 1 +1 2nn−ˆı and ˜vi(hi) = 1 +1 2nn−ˆı . LetM′be the matching that one gets from Mby switching hiandhj. That is, in M′agent iis matched withhj, agent jis matched with hi, and every other agent ℓis matched with hℓas inM. Now, if we use w(M)to denote the sum of weights of the pairs in M, we have w(M′)−w(M) = ˜vi(hj) + ˜vj(hi) +X ℓ∈N\{i,j}˜vℓ(hℓ) −X ℓ∈N˜vℓ(hℓ) = ˜vi(hj)−˜vi(hi) + ˜vj(hi)−˜vj(hj) ≥2 1 +1 2nn−ˆı − 1 +1 2nn−ˆı + 1 +1 2nn−ˆȷ −2 1 +1 2nn−ˆȷ = 1 +1 2nn−ˆı − 1 +1 2nn−ˆȷ = 1 +1 2nn−ˆȷh 1 +1 2nˆȷ−ˆı −1i ≥ 1 +1 2n0h 1 +1 2n1 −1i =1 2n>0, 23 where for the first inequality we used the exact values of ˜vi(hj),˜vi(hi)and lower and upper bounds for ˜vj(hi),˜vj(hj). This, however, contradicts the choice of Mas a maximum weight matching. Thus, under the assumption that βi=vi(hi)< vi(hj) =αi,(i, j)cannot be an edge in G. We conclude that, in any case, an edge (i, j)ofG′indicates envy which is no more than αi−βi. Cl. 5.5 ⊡ Claim 5.6. The graph G′is acyclic. Proof of Claim 5.6. Suppose, towards a contradiction, that G′contains a simple directed cycle C= (i1, i2, . . . , i s, i1). Also, let hi1, . . . , h isdenote the goods these agents received, respectively, in round k+ 1. Since Gwas acyclic, not all edges of Cexisted in G. Without loss of generality (as it is a matter of renaming the agents / vertices), we may assume that the (i1, i2)was not an edge in G. Our first observation is that the only way this could happened, is that vi1(hi1) =βi1butvi1(hi2) =αi1. We next note that it must be the case that vi2(hi2) =αi2, otherwise we could define a new matching M′ by only switching hi1andhi2inMand improve the maximum weight, similarly to what we did in the proof of Claim 5.5: w(M′)−w(M) = ˜vi1(hi2)−˜vi1(hi1) + ˜vi2(hi1)−˜vi2(hi2) ≥2 1 +1 2nn−π(i) − 1 +1 2nn−π(i) + 1 +1 2nn−π(j) − 1 +1 2nn−π(j) ≥ 1 +1 2n0 >0, where we used vi2(hi2) =βi2for the first inequality, | https://arxiv.org/abs/2505.22174v1 |
contradicting the choice of M. So, it must be vi2(hi2) =αi2. The third observation we need for the proof is that for any edge (ir, ir+1)inC, such that vir(hir) =αir, it must also be the case that vir(hir+1) =αir.1To see this, notice that vir(A(k+1)n ir+1)−vir(A(k+1)n ir) =vir(Akn ir+1)−vir(Akn ir) +vir(hir+1)−vir(hir) ≤αir−βir+vir(hir+1)−αir =vir(hir+1)−βir, where we used the induction hypothesis for the first inequality. Since this difference must be positive for (ir, ir+1)to be in C, we get that vir(hir+1) =αir. Finally, we distinguish two cases, depending on whether there is another agent, besides i1, who sees the good it received in this round as low-valued. Case 1: vir(hir) =αir, for all r∈ {2, . . . , s }.In this case, we can get a new matching M′by assigning each good among hi1, . . . , h isto the ‘previous’ agent with respect to the cycle, i.e., match hi1tois,hi2to i1, and so on, and keeping the rest of the matching the same as M. By the third observation above, we have that vir(hir+1) =αir, for all r∈ {2, . . . , s }, whereas by our first observation about agent i1’s envy, we also have vi1(hi2) =αi1. Note that no weight is decreased going from MtoM′and˜vi1(hi2)is now increased to 2 1 +1 2nn−π(i1)(from 1 +1 2nn−π(i1)that was in M). This contradicts the choice of Mas a maximum weight matching. Case 2: vir(hir) =βir, for some r∈ {2, . . . , s }.Letℓthe smallest such r. That is, vix(hix) =αix, for all x∈ {2, . . . , ℓ −1}, butviℓ(hiℓ) =βiℓ. In this case, we can get a new matching M′by assigning each good among hi1, . . . , h iℓto the ‘previous’ agent with respect to the cycle and , i.e., match hi2toi1,hi3toi2, 1We use the standard convention that is+1:= i1. 24 and so on, as well as hi1toiℓ, while keeping the rest of the matching the same as M. Now we can argue like in Case 1. We have vir(hir+1) =αir, for all r∈ {2, . . . , ℓ −1}, by our third observation, whereas vi1(hi2) =αi1like before and, of course, viℓ(hi1)≥βiℓ. Again, no weight is decreased going from Mto M′and˜vi1(hi2)is increased from 1 +1 2nn−π(i1)to2 1 +1 2nn−π(i1)(possibly ℓ’s weight increased as well). Like in Case 1, this contradicts the choice of Mas a maximum weight matching. We conclude that G′cannot contain any cycles. Cl. 5.6 ⊡ Claims 5.5 and 5.6 complete the induction. Thus, Algorithm 3 builds an allocation that is EF1for every time step which is a multiple of nand, because the allocation is balanced, it is also temporal-EF2. What remains to be shown is the last part of the theorem. Suppose that at the end of some step t0the allocation fails to be 1/2-EF1. Note that this happens during round k=⌈t0/n⌉and let i, jbe two agents, such that vi(At0 i)<0.5vi(At0 j\S)for any S⊆At0 jwith|S| ≤1. It is easy to see that this can only happen ifiwas envious of jalready at the end of round k−1; otherwise no single good added in j’s | https://arxiv.org/abs/2505.22174v1 |
bundle can violate (even exact) EF1from i’s perspective. Besides this, we also claim that, if hi, hjare the goods agents iandjreceive, respectively, in round k, then vi(hj) =αi. Indeed, using property (iii) we showed above, even if agent jreceives its good first in round k, we have vi(At0 j)−vi(At0 i)≤vi(A(k−1)n j ) +vi(hj)−vi(A(k−1)n i )≤αi−βi+vi(hj). So, if hjwas low-valued for agent i, we would be able to eliminate i’s envy towards jby just removing a high-valued good from At0 j(which must exist, otherwise vi(At0 j)andvi(At0 i)would only differ by a single low-valued good at most). Now, given that vi(hj) =αiand that iwas already envious of j, it must also be the case that vi(hi) =αi, or we could repeat the exact same argument as in the proof of Claim 5.5 (i.e., construct the matching M′by switching hiandhjand get a contradiction by showing it has larger weight thanM). Therefore, by the end of the round (at time step kn), we have vi(Akn i)≥αi. Since the allocation is temporal-EF2, at the end of any time step t≥kn, we have vi(At i)≥min S:|S|≤2vi(At j\S)≥min S:|S|≤1vi(At j\S)−αi≥min S:|S|≤1vi(At j\S)−vi(At i), and, thus, vi(At i)≥0.5 min S:|S|≤1vi(At j\S), i.e., the allocation at the end of time step tis1/2-EF1 Like in Section 5.1, we can get the analog of Corollary 5.3 but for any number of agents. Corollary 5.7. For any λ∈Z>0, ifmis large enough, after a sufficient number of steps the allocation built by Algorithm 3 becomes and remains λ/(λ+ 2) -EF,λ/(λ+ 1) -EF1, and λ/(λ+ 2) -PROP. Proof. The proof differs from that of Corollary 5.3 only in getting the guarantee for proportionality. As- suming that by time step t∗by each agent ihas received value equal to at least λαi, we have for agent 1 1 nnX i=1v1(At i)≤1 nh v1(At 1) + (n−1) 1 +2 λ v1(At 1)i = 1 +2(n−1) λn v1(At 1)≤ 1 +2 λ v1(At 1), and similarly for agents 2through n. 6 Going Beyond 2-Value Instances As we mentioned in the Introduction, there is a simple way of approximating any additive instance via a 2-value instance: we set a threshold for each agent and we round everything up to the maximum value this 25 agent has for any good or down to the minimum value this agent has for any good. Of course, this naive idea does not give any guarantees, in general, but it seems like it is a very natural approach for personalized interval-restricted instances. Indeed, we can transfer all of our positive results to personalized interval- restricted instances at the expense of an additional multiplicative factor that is equal to the harmonic mean of the upper and lower bounds of the values an agent may have for the goods. Theorem 6.1. For any personalized interval-restricted instance (augmented with foresight or not), there is a simple reduction to a personalized 2-value instance (which is equally augmented), so that any guarantee with respect to EF,EF1,EF2,PROP , orMMS we may obtain for the latter (e.g., via Algorithms 1, 2, or 3) can be translated to the same guarantee for the original | https://arxiv.org/abs/2505.22174v1 |
instance at the expense of an additional multiplicative factor of√αifor each agent i. Proof. The main idea is to use auxiliary valuation functions that aim to approximate the original valuation functions while taking only two values. Given the personalized interval-restricted valuation function viof an agent i, such that vi(g)∈[1, αi], for all g∈M, we define the personalized 2-value threshold function ˆvi as follows: ˆvi(g) =( αi, ifvi(g)>√αi√αi,otherwise for any g∈M. It is not hard to see how viandˆviare related. Claim 6.2. For any set of goods S⊆Mand any agent i∈N, it holds that ˆvi(S)/√αi≤vi(S)≤ˆvi(S). Proof of Claim 6.2. For any g∈M,vi(g)is rounded up in order to obtain ˆvi(g), so it is straightforward thatvi(g)≤ˆvi(g). On the other hand, vi(g)is always rounded up by a factor that is at most√αi, so√αivi(g)≥ˆvi(g). Since both functions are additive, these inequalities extend to any set of goods. Cl. 6.2 ⊡ We first argue about EF,EF1, and EF2. Suppose that at the end of some time step tthe allocation (At 1, . . . , At n) isρ-EFk(where EF0is just EF) with respect to the threshold functions ˆv1, . . . , ˆvn. Then, for any i, j∈N, we have vi(At i)≥1√αiˆvi(At i)≥1√αiρmin S:|S|≤kˆvi(At j\S)≥ρ√αimin S:|S|≤kvi(At j\S), where the first and the third inequalities follow from Claim 6.2. Thus, (At 1, . . . , At n)isρ/√αi-EFkwith respect to the original functions v1, . . . , v n. Next, suppose that at the end of time step tthe allocation (At 1, . . . , At n)isρ-PROP with respect to ˆv1, . . . , ˆvn. Then, similarly to the above, for any i∈N, we have vi(At i)≥1√αiˆvi(At i)≥1√αiρ nˆvinS j=1At j ≥ρ n√αiˆvinS j=1At j , where again the first and the third inequalities follow from Claim 6.2. Thus, (At 1, . . . , At n)isρ/√αi-PROP with respect to the original functions v1, . . . , v n. We last argue about MMS . Although the idea is the same, now it is not straightforward to get the last inequality by Claim 6.2 in the chain of inequalities needed. Instead, we are going to relate the maximin shares with respect to the original and to the threshold valuation functions. For a set of goods S⊆M, let µn i(S)andˆµn i(S)be the maximin shares of agent iwith respect to viand to ˆvi, respectively. Claim 6.3. For any set of goods S⊆Mand any agent i∈N, it holds that µn i(S)≤ˆµn i(S). 26 Proof of Claim 6.3. Suppose T= (T1, . . . , T n)is a maximin share defining partition of Sfor agent iwith respect to vi, i.e.,µn i(S) = min Tj∈Tvi(Tj). Now it is easy to relate the two maximin shares: ˆµn i(S)≥min Tj∈Tˆvi(Tj)≥min Tj∈Tvi(Tj) =µn i(S), where the first inequality follows by the definition of ˆµn i(S)(that takes the maximum over all such partitions) and the second inequality follows by Claim 6.2. Cl. 6.3 ⊡ Suppose that at the end of some time step tthe allocation (At 1, . . . , At n)isρ-MMS with respect to the threshold functions ˆv1, . . . , | https://arxiv.org/abs/2505.22174v1 |
ˆvn. Then, for any i∈N, we have vi(At i)≥1√αiˆvi(At i)≥1√αiρˆµn inS j=1At j ≥ρ√αiµn inS j=1At j , where, as usual the first and third inequalities follow from Claim 6.2. Thus, (At 1, . . . , At n)isρ/√αi-MMS with respect to the original functions v1, . . . , v n. For any personalized interval-restricted instance let α∗:= max i∈N√αi. Then Theorem 6.1, combined with Corollary 4.5 or Theorem 5.4, directly implies the following corollaries. Corollary 6.4. For any personalized interval-restricted instance, we can construct a 1/√a∗(2n−1)-temporal- MMS allocation. Moreover, for any agent iwho sees at least nhigh-valued goods with respect to ˆviof Theorem 6.1, this guarantee eventually improves to Ω(1/√ai). Corollary 6.5. For any personalized interval-restricted instance augmented with foresight of length n−1, we can construct a 1/√a∗-temporal- EF2allocation that is also 1/√a∗-EF1(and, thus, 1/n√a∗-MMS ) for every time step t=kn, k∈Z≥0. If at any step t0the allocation fails to be 1/2√a∗-EF1, then it remains 1/2√a∗-EF1 at the end of every time step t≥ ⌈t0/n⌉n. 7 Discussion and Open Questions In this paper we study a prior-free online fair division setting where the items are indivisible goods and we focus on the design of deterministic algorithms with solid worst-case fairness guarantees that hold frequently, if not at every time step. By restricting the input space to personalized 2-value (or interval- restricted) instances we are able to obtain nontrivial guarantees that are not possible for the general additive case. We see this as a main take-home message of this work; despite the existence of strong impossibility results, there are meaningful restrictions which can lead to technically interesting findings, broadening our understanding of online fair division. So, the most natural direction for future work is to identify such restrictions and push the boundaries of positive results accordingly. Another promising direction is to fully explore the power of knowing the future. Our work does not answer whether it is possible to efficiently utilize foresight which is sublinear innfor personalized 2-value instances. In fact, Algorithms 2 and 3 only use the (linear) information they have every nsteps and ignore it otherwise. We suspect that it is possible to design algorithms that build temporal- EF1allocations with linear foresight in our setting, although there are simple examples suggesting that this cannot be done via balanced allocations (like the ones our algorithms construct). Finally, given that personalized interval-restricted instances are already very expressive, it would be particularly interesting to get tight results directly for those. Although the dependency on α∗cannot be completely removed (since for large enough α∗the impossibility results of He et al. [2019] and Zhou et al. [2023] can be replicated), it is likely that the guarantees of Corollaries 6.4 and 6.5 are not tight. 27 Acknowledgments This work was partially supported by the project MIS 5154714 of the National Recovery and Resilience Plan Greece 2.0funded by the European Union under the NextGenerationEU Program. This work was partially supported by the framework of the H.F.R.I call “Basic research Financing (Horizontal support of all Sciences)” under the National Recovery and Resilience Plan “Greece 2.0” funded by | https://arxiv.org/abs/2505.22174v1 |
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arXiv:2505.22179v1 [cs.CL] 28 May 2025Speculative Decoding Meets Quantization: Compatibility Evaluation and Hierarchical Framework Design Yudi Zhang1, Weilin Zhao2, Xu Han2, Tiejun Zhao1*, Wang Xu2∗,Hailong Cao1,Conghui Zhu1 1Faculty of Computing, Harbin Institute of Technology, Harbin, China. 2Tsinghua University, Beijing, China. yudizhang@stu.hit.edu.cn, tjzhao@hit.edu.cn, xwjim812@gmail.com Abstract Speculative decoding and quantization effec- tively accelerate memory-bound inference of large language models. Speculative decoding mitigates the memory bandwidth bottleneck by verifying multiple tokens within a single forward pass, which increases computational effort. Quantization achieves this optimiza- tion by compressing weights and activations into lower bit-widths and also reduces compu- tations via low-bit matrix multiplications. To further leverage their strengths, we investigate the integration of these two techniques. Sur- prisingly, experiments applying the advanced speculative decoding method EAGLE-2 to var- ious quantized models reveal that the mem- ory benefits from 4-bit weight quantization are diminished by the computational load from speculative decoding. Specifically, verifying a tree-style draft incurs significantly more time overhead than a single-token forward pass on 4-bit weight quantized models. This finding led to our new speculative decoding design: a hierarchical framework that employs a small model as an intermediate stage to turn tree- style drafts into sequence drafts, leveraging the memory access benefits of the target quantized model. Experimental results show that our hi- erarchical approach achieves a 2.78 ×speedup across various tasks for the 4-bit weight Llama- 3-70B model on an A100 GPU, outperforming EAGLE-2 by 1.31 ×. Code available at https: //github.com/AI9Stars/SpecMQuant . 1 Introduction The excellent performance of Large Language Models (LLMs) across diverse domains has driven their widespread integration into everyday applica- tions (Brown et al., 2020; Grattafiori et al., 2024; Guo et al., 2025). However, the large-scale pa- rameters and auto-regressive decoding make in- ference memory-bound (Patterson, 2004; Shazeer, *indicates corresponding authors.2019), particularly on the single-batch inference of resource-constrained devices with model weights dominating memory bandwidth. To mitigate this memory bandwidth bottleneck, speculative decoding (Leviathan et al., 2023; Chen et al., 2023) and quantization (Frantar et al., 2022; Xiao et al., 2023) are commonly employed in LLM deployment. Speculative decoding generates mul- tiple tokens in a single forward pass by verify- ing draft outputs, reducing memory access fre- quency through increased computation. Among these methods, self-speculative decoding uses a draft model with the same architecture as the target model (Sun et al., 2024a; Sadhukhan et al., 2025). On the other hand, another approach employs a lightweight and independent draft model (Cai et al., 2024; Li et al., 2024b). Concurrently, quantiza- tionimproves LLM inference efficiency by reduc- ing memory and computation demands, includ- ing weight-only quantization (Frantar et al., 2022), weight-activation quantization (Xiao et al., 2023), and KV cache quantization (Hooper et al., 2024). Recently, several studies have explored integrat- ing quantization into self-speculative frameworks, whose draft model shares the architecture with the target model but differs in precision. QSpec (Zhao et al., 2024) drafts using 4-bit weights shared with the target model but employing lower-precision 4- bit activations. QuantSpec (Tiwari et al., 2025) drafts with 4-bit quantized weights and a 4-bit hier- archical KV cache. ML-SpecQD (Georganas et al., 2025) similarly | https://arxiv.org/abs/2505.22179v1 |
employs the 4-bit version of the target model as the draft and further introduces a smaller 4-bit model to enable a multi-level specula- tive decoding method. However, self-speculative decoding, which uses the same architecture for both draft and target mod- els, inherently limits speedup. In contrast, specu- lative decoding methods with a lightweight draft model achieve superior speedup, as demonstrated by the state-of-the-art approach EAGLE (Li et al., 1 2025). To further enhance acceleration and com- bine the benefits of speculative decoding and quan- tization, we integrate these two techniques by ap- plying the speculative decoding method with a lightweight draft model to a quantized target model. Given that speculative decoding and quantization mitigate memory bottlenecks from different per- spectives, it is necessary to study their compatibil- ity systematically. In this paper, we explore two key questions: (1) How does the integration of specu- lative decoding and quantization perform? Do these two techniques conflict in terms of mitigating memory bottlenecks? (2) Within the integrated framework, what are the dominant factors that affect the overall speedup? To achieve the objectives, we first integrated the advanced speculative method EAGLE-2 (Li et al., 2024a) and various optimized quantization kernels into a highly optimized native C and CUDA implementation. This implementation filters out non-algorithmic overheads from Python inefficien- cies (Zhao et al., 2025), thereby revealing each method’s true speedup. The results of reliable ex- periments across various integration schemes show that EAGLE-2 provides limited benefit for 4-bit weight quantized models (W4A16 and W4A8), in- dicating a potential conflict. Subsequently, we sys- tematically experimented with various draft tree sizes to identify the factors behind the integration conflict. We find that the increased computational load of tree-style draft verification undermines the memory access benefits from 4-bit weight quanti- zation, leading to limited compatibility. Motivated by this finding, we propose to design a hierarchical speculative decoding framework for W4A16 quantized models, which have low memory bandwidth demand and near-lossless performance. To fully leverage the memory advantages of such 4- bit quantized models, we introduce an intermediate stage between EAGLE-2 and the quantized target model, which employs a small model for tree-style verification, turning tree drafts into sequence drafts, enabling fast and accurate drafting without impos- ing significant verification computational overhead. Across various tasks with the W4A16 Llama-3- 70B model on a single A100 GPU, our hierarchical framework achieves 2.78 ×speedup, outperforming the advanced EAGLE-2 method by 1.31 ×.2 Preliminary In this section, we present the speculative decod- ing method EAGLE-2, the quantization schemes (W8A8, W4A16, and W4A8) investigated in this paper, and the performance of quantized models. 2.1 Speculative Decoding Speculative Decoding presents a draft-then-verify decoding paradigm. EAGLE-2 uses a lightweight module that consists of a single Transformer layer and uses the original LM Head to generate tree- style draft tokens auto-regressively. It dynamically adjusts the draft tree structure by adopting beam- search algorithm based on the softmax output of the draft model. During drafting, the draft model forwards dtimes and selects the top nprobability tokens from the beam search history as the draft, where dis the search depth | https://arxiv.org/abs/2505.22179v1 |
and nis the tree size. Letτ(n, d)denote the expected accepted length, defined as the expected number of tokens accepted by the target model after verifying the drafts. Td andTtdenote the decoding time of the draft model and target model, respectively. Tv(n)denotes the time taken by the target model to verify ntokens. Tsd avgrepresents the expected latency per token for speculative decoding. The speedup effect of specu- lative decoding can be understood with the follow- ing equation (Sadhukhan et al., 2025): Tsd avg Tt=1 τ(n, d)d·Td Tt+Tv(n) Tt (1) The impressive speedup achieved by EAGLE- 2 is mainly attributed to three factors:(1) a high excepted accepted length τ(n, d), (2) a low draft- to-target decoding time ratio Td/Ttclose to 0, (3) a low target verification-to-decoding time ratio Tv(n)/Ttclose to 1. In addition, these three fac- tors also guide our subsequent speed analysis when combined with quantization. 2.2 Quantization Quantization methods compress model weights and activations into low-bit representations. In this pa- per, we denote x-bit weight and y-bit activation quantization precision in LLM as WxAy. The fol- lowing is a brief introduction of quantization preci- sions and algorithms investigated in this paper: W8A8: For 8-bit weight-activation quantization, SmoothQuant (Xiao et al., 2023) shifts the quanti- zation difficulty from activations to weights. We adopted SmoothQuant with channel-wise scaling 2 Precision AlgorithmLlama-3-8B-Instruct Llama-3-70B-Instruct WikiText2 ↓GSM8K ↑ HumanEval ↑WikiText2 ↓GSM8K ↑ HumanEval ↑ FP16 - 8.28 76.95 61.59 5.32 91.05 78.65 W8A8 SmoothQuant 8.37 (+0.09) 77.33 (+0.38)58.54 (−3.05) 5.87 (+0.55) 90.60 (−0.45)74.39 (−4.26) W4A16GPTQ-g128 8.73 (+0.45) 73.18 (−3.77)53.05 (−8.54) 5.86 (+0.54) 89.31 (−1.74)75.00 (−3.65) GPTQ-g128+Rot 8.55 (+0.27) 73.69 (−3.26)57.93 (−3.66) 5.89 (+0.57) 90.22 (−0.83)76.22 (−2.43) W4A8QoQ 8.73 (+0.45) 73.39 (−3.56)54.27 (−7.32) 5.97 (+0.65) 88.86 (−2.19)73.78 (−4.87) QoQ-g128 8.63 (+0.35) 74.07 (−2.88)56.10 (−5.49) 5.76 (+0.44) 89.69 (−1.36)73.78 (−4.87) QQQ 8.84 (+0.56) 71.65 (−5.30)50.61 (−10.98)6.44 (+1.12) 87.57 (−3.48)73.78 (−4.87) QQQ-g128 8.76 (+0.48) 71.65 (−5.30)52.44 (−9.15) 6.10 (+0.78) 89.31 (−1.74)74.39 (−4.26) Table 1: WikiText2 perplexity with 2048 sequence length, 8-shot performance on GSM8K and zero-shot performance on HumanEval of different quantized method on Llama-3-8B-Instruct and Llama-3-70B-Instruct. in our W8A8 experiments. This method leverages INT8 Tensor Cores to lower computation costs. W4A16: GPTQ (Frantar et al., 2022) adopts second-order information to minimize precision loss for weight-only quantization. We adopted GPTQ to symmetrically quantize weights to 4-bit with a group size of 128 while keeping activation 16-bit. This method mitigates memory bottleneck. W4A8: To achieve 4-bit weights and 8-bit activa- tions, QoQ (Lin et al., 2024b) employs an asymmet- ric scheme with better accuracy performance and QQQ (Zhang et al., 2024) employs a symmetric quantization scheme for superior efficiency. They support 4-bit weight quantization with both per- channel and per-group granularity, while enabling matrix multiplications on INT8 Tensor Cores. In both W4A8 quantization methods, rotation with Hadamard transformation (Ashkboos et al., 2024) was introduced as a general offline quantiza- tion optimization. Inspired by Sun et al. (2024b) demonstrating that Hadamard transformation can improve the flatness of model weights, we also applied rotation optimization to the W4A16 quan- tization. Additionally, a subset of 128 sequences sampled from the Pile validation dataset (Gao | https://arxiv.org/abs/2505.22179v1 |
et al., 2020) was used for calibration. To evaluate the performance of different quan- tization precisions, we conduct experiments on Llama-3-8B-Instruct and Llama-3-70B-Instruct models (Grattafiori et al., 2024) quantized with various algorithms. Three benchmarks are evalu- ated: WikiText2 (Merity et al., 2016) for perplexity, GSM8K (Cobbe et al., 2021) for arithmetic rea- soning, and HumanEval (Chen et al., 2021) for code generation. As shown in Table 1, W8A8 and W4A16 achieve the best performance and maintain near-lossless performance, followed by the asym- metric W4A8 quantization method (QoQ), while the symmetric W4A8 algorithm (QQQ) exhibitsthe most significant degradation. 3 Experimental Study for Integration In this section, we first present the setup and our experimental results in Section 3.1 and Section 3.2, respectively. While integration improves overall speedup, applying EAGLE-2 on 4-bit weight quan- tized models (W4A16 and W4A8) yields limited additional speedups compared to higher precision settings. To investigate this limitation, Section 3.3 explores the underlying factors that hinder further memory bandwidth optimization through experi- ments with varying draft tree sizes. 3.1 Experimental Setup Models and Datasets. We conduct experiments on Llama-3 series models (Grattafiori et al., 2024), including Llama-3-8B-Insturct and Llama-3-70B- Instruct. And we evaluate the decoding speedup of different methods using the multi-turn conversation dataset MT-Bench (Zheng et al., 2023). Speculative Decoding. We adopt EAGLE-2 (Li et al., 2024a) as the speculative method and the implementation in native C and CUDA (Zhao et al., 2025). Following the original settings of EAGLE- 2, we set the search depth dto 6 and the tree size nto 60 for the Llama-3-8B-Instruct model, and a search depth dof 6 and a tree size nof 48 for the Llama-3-70B-Instruct model. And we keep the draft model FP16 precision due to the main bottleneck of drafting is the language model head and softmax operation, which are not quantizable (Zhao et al., 2025). Moreover, GPTQ quantization to the draft model leads to substantial degradation of the acceptance rate (Zhao et al., 2024). Quantization Methods. We evaluate sev- eral representative quantization methods: W8A8, W4A16, W4A8-QoQ, W4A8-QoQ-g128, W4A8- QQQ, and W4A8-QQQ-g128. For W4A16, we 3 FP16 W8A8 W4A16-g128 W4A8-QoQ W4A8-QoQ-g128 W4A8-QQQ W4A8-QQQ +EAGLE-2 ↑Relative Speedup (a) Llama-3-8B(A100) (b) Llama-3-8B(3090) (c) Llama-3-70B(A100)0123 ↑2.2 ↑2.3 1.3 1.2 1 1Speedup↑2.7↑1.8 ↑2 1.41.4 1.7 1.41.7 1↑1.9 ↑1 ↑1.3 0.51.5 2.12.6 1.7↑2.1 ↑1.3 ↑1.7 1.10.7 1.8 1.62.5 1.5↑2↑1.3 ↑1.7 1.10.8 1.3 1.62.4 1.3↑1.8 ↑1.2 ↑1.4 0.80.6 1.5 2.12.8 1.8↑1.8 ↑1.2 ↑1.3 0.60.5 1.4 22.6 1.7 Figure 1: Comparison of speedup ratios for Llama-3-8B (relative to FP16) and Llama-3-70B (relative to W8A8) under various quantization methods and with EAGLE-2 integration. Solid bars show speedup from quantization alone, dashed bars represent the additional speedup from EAGLE-2, and red arrows indicate the relative speedup achieved by EAGLE-2 across different quantized models. use advanced Marlin kernels (Frantar et al., 2025) adapted by vLLM (Kwon et al., 2023). For W8A8, W4A8-QoQ, and W4A8-QoQ-g128, we adopt the implementation from QServe (Lin et al., 2024b). For W4A8-QQQ and W4A8-QQQ-g128, we em- ploy novel kernels from QQQ (Zhang et al., 2024). Hardware. Experiments are conducted on a sin- gle NVIDIA 80G A100 and | https://arxiv.org/abs/2505.22179v1 |
a single RTX 3090, rep- resenting high-performance and consumer-grade GPUs, respectively. 3.2 Experimental Observation Figure 1 presents the speedup performance of spec- ulative decoding EAGLE-2, various quantization methods, and their integration. It also includes the relative speedup improvement contributed by EAGLE-2 while integrating. EAGLE-2 vs. quantization. We compare the speedup achieved by applying EAGLE-2 to FP16 models with the speedup obtained through vari- ous quantization methods across different hard- ware platforms. EAGLE -2 achieves a higher speedup on the high-performance device A100 due to its computation-intensive design as shown in Figure 1(a), while quantization of 4-bit weights (W4A16 and W4A8) yields higher gains on the customer-grad GPU RTX 3090 for its high re- duction of memory demands demonstrated in Fig- ure 1(b). Notably, EAGLE-2 is a lossless accel- eration method, while quantization may introduce performance degradation, though W4A16 is con- sidered to be nearly lossless. Integration and compatibility. The dashed bars in Figure 1 show that integrating EAGLE-2 with quantization yields additional speedup compared to using either technique alone, except when EAGLE- 2 is applied to W4A16 8B model on the RTX 3090. To evaluate the compatibility between EAGLE-2 and different quantization precisions, we presentthe relative speedup brought by EAGLE-2 across models with different precisions in Figure 1. We observe that, for models with 4-bit weight quantiza- tion (W4A16 and W4A8), the relative speedup pro- vided by EAGLE-2 drops significantly compared to FP16 and W8A8, indicating lower compatibil- ity, with W4A16 exhibiting the lowest compatibil- ity. This limited compatibility suggests a poten- tial conflict between 4-bit weight quantization and the EAGLE-2 method. A plausible hypothesis is thatthe increased computational overhead from EAGLE-2 diminishes the memory benefits of 4-bit weight optimization, resulting in limited speedup. 3.3 Factors behind Integration Conflict As indicated in Equation 1, the speedup of specu- lative decoding depends on three key factors: the average accepted length τ(n, d), the draft to target decoding time ratio Td/Tt, and the target verifi- cation to decoding time ratio Tv(n)/Tt. To fur- ther understand the conflict underlying the limited compatibility between EAGLE-2 and 4-bit weight quantized models, we analyze the impact of three factors on overall speedup when applying EAGLE- 2 to various quantization schemes. We perform experiments by varying the draft tree size nand corresponding draft forward passes d, the only variables in Equation 1 that are strongly correlated with the three factors. For Llama-3-8B and Llama-3-70B, we use n∈ {30,40,50,60} andn∈ {24,32,40,48}respectively, with corre- sponding draft forward passes d∈ {3,4,5,6}. For W4A8, we use the symmetric quantization method QQQ, as it offers higher decoding speedup. Experimental results are shown in Figures 2 and 3, with speedup in Figures 2c, 2f, and 3c. We find that the size of drafts significantly affects the overall speedup, with fewer drafts yielding a higher speedup in 4-bit weight models. 4 FP16 W8A8 W4A16 W4A8-QQQ W4A8-QQQ-g128 30 40 50 6033.54 Draft tree sizeAvg. accepted length (a) Accepted Length (8B, A100)30 40 50 6011.21.41.61.82 Draft tree sizeVerification/decoding (b) Verification Ratio (8B, A100)30 40 50 6022.533.5 Draft tree sizeSpeedup (c) Speedup (8B, A100) 30 40 50 6033.54 Draft | https://arxiv.org/abs/2505.22179v1 |
tree sizeAvg. accepted length (d) Accepted Length (8B, 3090)30 40 50 6011.522.5 Draft tree sizeVerification/decoding (e) Verification Ratio (8B, A100)30 40 50 6022.533.544.5 Draft tree sizeSpeedup (f) Speedup (8B, 3090) Figure 2: Comparison of average accepted length, verification-to-decoding ratio, and speedup for various quantiza- tion precisions (FP16, W8A8, W4A16, W4A8-QQQ, W4A8-QQQ-g128) on Llama-3-8B with EAGLE-2, evaluated on A100 and RTX 3090. Panels (a–c) show A100 results; (d–f) show RTX 3090 results. Quantization has minimal impact on average accepted length. For different quantization preci- sions in Figure 2a, 2d, and 3a, the decrease in the average accepted length τcaused by quantization is minimal compared to FP16, with W4A16 and W8A8 exhibiting nearly no degradation. However, although the average accepted length τincreases with draft tree size, this increase does not result in improved speedup for 4-bit weight models. It suggests that changes in average accepted length are not the primary factor affecting the integrated speedup for 4-bit weight models. Higher draft-to-target decoding time ratio from quantization partly contributes to the higher speedup of fewer drafts. The ratio of draft to tar- get decoding time Td/Ttis another factor affect- ing the speedup. The decoding speed of the 4-bit weight quantized model is improved compared to the FP16 model as shown in Figure 1, which in turn increases the ratio of draft to target decoding time Td/Tt. According to Leviathan et al. (2023), if the acceptance rate remains almost unchanged while this ratio increases, the optimal speedup is achieved with fewer draft forward passes. This indicates that the increased draft-to-target decoding ratio is one reason why fewer drafts yield higher speedup. Verification-to-decoding time ratio of 4-bit weight models favors fewer drafts and under- mines EAGLE-2 compatibility. As shown in Fig- ures 2b, 2e, and 3b, 4-bit weight quantized modelsexhibit a significantly greater increase trend in the verification-to-decoding time ratio Tv(n)/Ttwith draft tree size compared to FP16 and W8A8 mod- els. For example, the 8B model on the A100 with a draft tree size of 60 shows a ratio below 1.2 for FP16 and W8A8, while W4A16 reaches 1.8, signif- icantly exceeding the ideal value of 1. As a conse- quence, the increase in the verification-to-decoding ratio for 4-bit quantized models with growing draft tree size drives the decline in integrated speedup, despite gains from longer average accepted length. This clearly indicates the incompatibility between 4-bit weight models and the EAGLE-2 method, where the heavy computation time Tv(n)required during draft verification undermines the memory efficiency gained by the 4-bit weight quantization. To confirm that the verification-to-decoding ratio and draft-to-decoding ratio contribute to the nega- tive correlation between speedup and draft-tree size in 4-bit quantized models, we compare the speedup of FP16 and W4A16 models with EAGLE-2 un- der three draft configurations : (1) full-sized tree with 6 forward passes, (2) half-sized tree with 6 forward passes, and (3) half-sized tree with 3 for- ward passes. Figure 4 shows that reducing tree size decreases speedup for FP16 but improves it for W4A16, indicating that smaller trees better pre- serve memory efficiency under W4A16’s higher verification-to-decoding ratio. Further W4A16 | https://arxiv.org/abs/2505.22179v1 |
speedup from fewer draft forward passes also con- 5 W8A8 W4A16 W4A8-QQQ W4A8-QQQ-g128 24 32 40 4833.54 Draft tree sizeAvg. accepted length (a) Accepted Length (70B, A100)24 32 40 4811.21.41.61.8 Draft tree sizeVerification/decoding (b) Verification Ratio (70B, A100)24 32 40 482.533.544.5 Draft tree sizeSpeedup (c) Speedup (70B, A100) Figure 3: Comparison of average accepted length, verification-to-decoding ratio, and speedup for various quantiza- tion precisions (W8A8, W4A16, W4A8-QQQ, W4A8-QQQ-g128) on Llama-3-70B with EAGLE-2 on A100. firms the impact of the higher draft-to-decoding ratio, which is more pronounced in 8B models, since the gap between draft and target model sizes is smaller than that in 70B models. In addition, to demonstrate that the increased verification-to-decoding ratio primarily causes the conflict between 4-bit quantization and EAGLE-2 in memory optimization, we compare two methods on a W4A16 70B model: EAGLE-2 and the vanilla speculative decoding. For the latter, we employ a W4A16 8B draft model to generate sequence- style drafts. As shown in Figure 4, despite the increased draft overhead, the vanilla speculative decoding approach outperforms EAGLE-2 due to the low computational cost of its sequential draft verification. This result highlights that the heavy computational cost of tree-style draft verification in EAGLE-2 undermines the memory access ad- vantages of 4-bit weight quantization, explaining the conflict underlying their limited compatibility. 4 Hierarchical Framework For near-lossless W4A16 models, EAGLE-2 is mainly bottlenecked by tree-style draft verification, while vanilla speculative decoding is limited by drafting overhead. To overcome the limitations of both methods, we propose a hierarchical specula- tive decoding framework. We also evaluate the hierarchical framework across multiple tasks. 4.1 Methodology Building upon the EAGLE-2 framework, we intro- duce a hierarchical design for W4A16 models by introducing a small intermediate model between the draft and target stages. We adjust the original EAGLE-2 to perform tree-style speculation for the small intermediate model, whose outputs subse- quently serve as the draft inputs for the final target model. This hierarchical framework mainly con-FP16(8B) W4A16(8B) W4A16(70B)22.533.54 2.32.73.3 2.12.93.6 2.33.43.73.9SpeedupEG-2 (6,full) EG-2 (6,half) EG-2 (3,half) SP (6) Figure 4: Speedup comparison of different EAGLE- 2 configurations and vanilla speculative decoding on Llama-3 models. EG-2(6/3, full/half) uses 6 or 3 draft passes with full (60, 48) or half (30, 24) tree sizes; SP(6) denotes vanilla speculative decoding with 6 draft passes. sists of two levels: the compute-intensive drafting stage and the memory-efficient verification stage. Through the small model as a bridge, it achieves an effective combination of efficient tree-style drafting and sequential draft verification. Computation-intensive drafting stage. This stage accelerates the small model drafting using EAGLE-2. In each draft iteration, the lightweight EAGLE-2 draft model generates an initial draft of length d1, and the small model performs multi- token drafting through verification. This process repeats until the draft length of the small model exceeds draft length d. The small model under- takes the computation-intensive tree-based verifica- tion, converting the tree-based drafts into sequen- tial drafts, thus enabling high-speed drafting. Memory-efficient verification stage. In this verification stage, the target model verifies the se- quential draft with length dfrom the smaller model to generate multiple tokens | https://arxiv.org/abs/2505.22179v1 |
in one forward pass. Compared to directly applying tree-based draft ver- ification, sequential draft verification enables fur- ther memory access optimization while fully retain- ing the memory advantages of 4-bit weight models, achieving orthogonality between speculative decod- ing and 4-bit weight quantization. 6 Method dMT. Conv. RAG Math QA Summ. Avg. τ Tok/s τ Tok/s τ Tok/s τ Tok/s τ Tok/s τ Tok/s τ Tok/s Vanilla AR - 1.00 35.86 1.00 35.08 1.00 35.02 1.00 35.49 1.00 35.66 1.00 34.59 1.00 35.28 (1.0 ×) Vanilla SP64.45 78.54 4.71 79.77 4.81 86.55 5.57 92.07 4.28 73.64 4.66 75.40 4.72 81.33 (2.31 ×) 74.74 77.75 5.10 80.24 5.19 87.36 6.12 95.57 4.55 73.06 5.00 74.77 5.09 81.46 (2.31 ×) EAGLE-233.18 74.14 3.28 74.48 3.40 76.98 3.49 79.54 2.99 68.88 3.16 70.34 3.25 74.06 (2.10 ×) 43.44 74.27 3.63 76.51 3.77 79.03 3.92 82.41 3.19 67.61 3.42 69.78 3.57 74.93 (2.12 ×) 53.60 64.61 3.82 67.46 4.00 70.58 4.14 72.18 3.28 57.78 3.56 60.65 3.73 65.54 (1.86 ×) 63.65 62.41 3.93 66.22 4.09 68.06 4.23 70.21 3.31 55.45 3.59 58.29 3.81 63.44 (1.80 ×) HierSpec6(3) 4.84 92.12 5.30 98.23 5.35 105.65 6.58 120.29 4.62 83.61 5.16 89.21 5.28 98.19 (2.78 ×) 6(4) 4.82 86.02 5.17 90.45 5.28 98.76 6.36 112.12 4.61 77.13 5.14 82.34 5.19 91.14 (2.58 ×) 7(3) 4.99 91.06 5.47 97.79 5.56 106.09 6.82 120.89 4.80 83.39 5.37 88.76 5.46 98.00 (2.78 ×) 7(4) 5.12 86.29 5.64 93.27 5.75 102.75 7.18 116.70 4.89 76.94 5.48 82.45 5.62 93.06 (2.64 ×) Table 2: Average accepted length τand decoding speed Tok/s of different methods on W4A16 Llama-3-70B with different draft lengths d. The numbers in draft length parentheses (e.g., 3 in 6(3)) denote the draft length d1of EAGLE-2 for the small model in the first hierarchy. And the numbers in decoding speed parentheses (2.78 ×) represent the speedup compared to the W4A16 vanilla auto-regressive approach. Method MT. Conv. RAG Math QA Summ. Avg. Vanilla SP 60.9 71.6 91.8 63.3 57.3 95.3 73.4 EAGLE-2 3.8 6.4 9.4 4.8 3.7 10.3 6.4 HierSpec 63.7 75.1 94.1 65.3 59.9 100.2 76.4 Table 3: Draft latency (ms) of different methods on W4A16 Llama-3-70B on A100. 4.2 Experimental Setup Dataset. We use SpecBench (Xia et al., 2024) as the evaluation dataset for our method, which includes six types of text generation tasks: machine translation (MT.), multi-turn conversation (Conv.), retrieval-augmented generation (RAG), arithmetic reasoning (Math), question answering (QA), and document summarization (Summ.). Models and Evaluation. We adopted W4A16 Llama-3-70B as the target model. We evaluate our hierarchical speculative decoding framework (Hier- Spec) and three baselines: vanilla auto-regressive decoding (Vanilla AR), vanilla speculative decod- ing (Vanilla SP), and EAGLE-2 method. W4A16 Llama-3-8B serves as the small intermediate model in HierSpec and the draft model in Vanilla SP. Av- erage acceptance length τand decoding speed (to- kens/s) are reported. Given our hierarchical design, we also report the draft latency (the draft model’s prefilling time), excluded from decoding time. For a systematic comparison, we vary the draft length d: {6,7}for Vanilla SP, and d∈ {3,4,5,6}with cor- responding tree sizes {24,32,40,48}for EAGLE- 2, following Section | https://arxiv.org/abs/2505.22179v1 |
3.3. For HierSpec, the first- level draft length d1∈ {3,4}matches the opti- mal EAGLE-2 setting for 8B, and the second-level d∈ {6,7}follows Vanilla SP. All experiments areconducted on a single NVIDIA 80GB A100 GPU. EAGLE-2 Vanilla SP HierSpec0246810 2.75.2 3.8Draft Time (ms)Drafting time 01020304050 45.3 29.7 30.0 Verif. Time (ms)Verfication time Figure 5: Comparison of drafting time (per draft length) and verification time of three speculative decoding meth- ods applied on W4A16 Llama-3-70B on A100. 4.3 Main Results Table 2 compares the average acceptance length τ and decoding speed (tokens/s) of different meth- ods on W4A16 Llama-3-70B with different draft lengths. Experiment results demonstrate that our proposed HierSpec achieves the highest decoding speed on average and also consistently outperforms baselines across various tasks. Specifically, Hier- Spec with d1= 3 andd= 6 outperforms the other configurations, achieving an average decod- ing speed 98.19 tokens/s, which is 1.31 ×over the best of EAGLE-2 and 1.21 ×over the best of Vanilla SP. In addition, the average acceptance length of Hierspec shows some increase compared to Vanilla SP, as the tree-level drafting in the first hierarchical level generates additional tokens be- yond the draft length d. Table 3 reports the draft latency (ms) of different methods. Although our hi- erarchical framework has a larger prefill overhead, this cost is amortized over decoding steps, making it more suitable for long text generation. Figure 5 compares the drafting time per draft 7 length and verification time per iteration across speculative decoding methods at d= 6. Vanilla SP exhibits a drafting bottleneck with 1.93 ×longer drafting time than EAGLE-2, while EAGLE-2 faces a verification bottleneck with 1.53 ×longer verification time than Vanilla SP. Our HierSpec ap- proaches EAGLE-2 in drafting time and matches Vanilla SP in verification time. It combines efficient drafting with memory-efficient verification, leading to further speedup in the integration of speculative decoding with 4-bit weight quantization. 4.4 Integration with EAGLE-3 We further assess HierSpec by integrating with the EAGLE-3 (Li et al., 2025) method. Given that the training details of EAGLE-3 remain unpublished, we can only experiment with the publicly avail- able checkpoints. For the 70B model size, only Llama-3.3-70B has an available EAGLE-3 check- point, which we adopt as our baseline. For our HierSpec method, due to the absence of an 8B model in the Llama-3.3 series, we can only adopt an 8B model from the Llama-3.1 series as the inter- mediate model. Even under this unfair configura- tion, HierSpec still achieves a further speedup over EAGLE-3, as shown in Appendix A. 5 Related Work This section introduces the related work on acceler- ating LLM inference using quantization and specu- lative decoding. 5.1 Quantization of LLMs Quantization reduces LLM memory footprint and accelerates inference. Weight-only quan- tization only quantizes the model weights and is highly effective for memory-bound LLM in- ference. GPTQ (Frantar et al., 2022) uses second-order information to minimize rounding errors, while AWQ (Lin et al., 2024a) and SqueezeLLM (Kim et al., 2024) prioritize im- portant weights. QuIP# (Tseng et al., 2024) ap- plies pre-quantization transformations. Weight- activation quantization further reduces compu- tation | https://arxiv.org/abs/2505.22179v1 |
by quantizing both weights and activa- tions. To alleviate the impact of activation outliers, LLM.int8() (Dettmers et al., 2022) performs mixed- precision decomposition, SmoothQuant (Xiao et al., 2023) and OmniQuant (Shao et al., 2024) adopt per-channel scaling transformation. Further, QuaRot (Ashkboos et al., 2024) and FlatQuant (Sun et al., 2024b) apply Hadamard transformation andAffine Transformation, respectively. KV cache quantization (Liu et al., 2024; Hooper et al., 2024) quantizes key and value caches to mitigate the KV bottleneck in the long-context inference. However, long-context inference is not included in our work. 5.2 Speculative Decoding Speculative decoding (Leviathan et al., 2023; Chen et al., 2023) accelerates LLM inference by draft- ing multiple tokens and verifying them in parallel. One line of work uses external and lightweight components for draft generation to enable low-cost drafting. Medusa (Cai et al., 2024) and EAGLE se- ries works (Li et al., 2024b,a, 2025) employ an LM head and a single Transformer layer as draft models with tree-style drafting, respectively. EAGLE-3 (Li et al., 2025) further achieves state-of-the-art perfor- mance via Training-Time Test. Another line of work, self-speculative decoding, shares the model architecture between draft and target models for better alignment. Triforce (Sun et al., 2024a) and MagicDec (Sadhukhan et al., 2025) draft with sparse KV-cache to mitigate KV bottleneck. Recently, several studies also inte- grated quantization into self-speculative frame- works. QSpec (Zhao et al., 2024) accelerates batch inference for 4-bit weight-only models us- ing shared weights and 4-bit activations, but fails in single-batch settings. QuantSpec (Tiwari et al., 2025) improves long-context efficiency using 4- bit weights and KV caches, though limited gains on short contexts. ML-SpecQD (Georganas et al., 2025) adopts the 4-bit target model as the draft model and further introduces a tiny 4-bit model for multi-level speculative decoding, but the FP16 target model limits deployment. However, our work focuses on integrating specu- lative decoding with a lightweight draft model into quantized target models for further acceleration. 6 Conclusion In this work, we systematically study the compat- ibility of speculative decoding and quantization when applied jointly to LLMs under various preci- sions and speculative decoding configurations. Our study reveals that the substantial computation over- head from the tree-style verification of speculative decoding undermines the memory bandwidth bene- fits of 4-bit weight quantization. Motivated by this finding, we propose a hierarchical speculative de- coding framework for W4A16 models, leveraging 8 a small model as a bridge to enable both efficient drafting and memory-efficient verification. Exper- imental results show that our method achieves a 2.78×speedup over auto-regressive decoding and a 1.31×speedup over the EAGLE-2 approach, en- hancing the compatibility of these two techniques. Limitations Our current research focuses primarily on weight- only quantization and weight-activation quantiza- tion across several common tasks, while lacking assessments under some conditions, such as long- context tasks with KV cache quantization. Nev- ertheless, our study remains systematic and com- prehensive, providing a more effective speculative decoding framework based on our findings. In the future, we will explore the integration of these two techniques in more challenging tasks. References Saleh Ashkboos, Amirkeivan Mohtashami, Maximilian Croci, | https://arxiv.org/abs/2505.22179v1 |
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arXiv:2505.22184v1 [cs.CL] 28 May 2025Breaking the Cloak! Unveiling Chinese Cloaked Toxicity with Homophone Graph and Toxic Lexicon Xuchen Ma1, Jianxiang Yu1, Wenming Shao2, Bo Pang2, Xiang Li1∗ 1School of Data Science and Engineering, East China Normal University 2Shanghai EastWonder Info-tech Co., Ltd. {xuchenma, jianxiangyu}@stu.ecnu.edu.cn {simon, pangbo}@wdit.com.cn xiangli@dase.ecnu.edu.cn Abstract Social media platforms have experienced a significant rise in toxic content, includ- ing abusive language and discriminatory remarks, presenting growing challenges for content moderation. Some users evade censorship by deliberately disguising toxic words through homophonic cloak, which necessitates the task of unveiling cloaked toxicity. Existing methods are mostly designed for English texts, while Chinese cloaked toxicity unveiling has not been solved yet. To tackle the issue, we propose C2TU, a novel training-free and prompt-free method for Chinese cloaked toxic content unveiling. It first employs substring matching to identify candidate toxic words based on Chinese homo-graph and toxic lexicon. Then it filters those candidates that are non-toxic and corrects cloaks to be their corresponding toxi- cities. Specifically, we develop two model variants for filtering, which are based on BERT and LLMs, respectively. For LLMs, we address the auto-regressive limitation in computing word occurrence probability and utilize the full semantic contexts of a text sequence to reveal cloaked toxic words. Extensive experiments demonstrate that C2TU can achieve superior performance on two Chinese toxic datasets. In particular, our method outperforms the best competitor by up to 71% on the F1 score and 35% on accuracy, respectively. Disclaimer :The paper contains content that may be profane, vulgar, or offensive. 1 Introduction With the exponential growth of user-generated contents, online social media platforms have become an inevitable communication tool with massive users. Although the rapid information dissemination has clear benefits, a significant amount of toxic contents have emerged on social platforms over the past decade, such as abuse, discrimination, and cyberbullying [ 22]. Social networks are thus facing increasingly severe challenges in governing toxicity. Toxicity detection has recently attracted extensive attention, but most of the proposed methods [20,26,7,28] can only output binary predictions without clarifying the true toxicity the given content violates. Further, to evade censorship, some users on social platforms intentionally disguise toxic words by replacing parts or all of the words with homophonic characters or emojis [ 24,15]. For example, replace “ 操” (cào, means “f*ck”) with “ 草” (cˇao, means “grass”), or “ 垃圾 ” (l¯a j¯ı, means “rubbish”) with “ 辣鸡” (là j ¯ı, means “spicy chicken”) in Chinese. These deliberate cloaks significantly degrade the effectiveness of existing toxicity detection methods [ 24] and also necessitate the task ofunveiling cloaked toxicity , i.e., correct cloaked toxic words into proto toxic words . While there ∗Corresponding Author Preprint. Under review. have been some methods [ 12,9] proposed to solve the task, most of them are specially designed for English texts, which cannot be directly used for Chinese due to the different characteristics of the two languages. Further, although a recent work [ 24] points out the existence of Chinese cloak toxic content, they fail to offer a solution. Therefore, to bridge the gap, a research question | https://arxiv.org/abs/2505.22184v1 |
naturally arises: Can we develop a model to unveil Chinese cloaked toxic contents? We notice that Chinese Spelling Correction (CSC), which aims to correct misspelled Chinese char- acters, shares some similarities with our task. However, the two tasks are not entirely equivalent. While CSC deals with unintentional user typos, misspellings in toxic content are often deliberate. Moreover, the corpora used in CSC tasks typically consist of more standardized language, which differs significantly in distribution from the toxicity found on the internet. Therefore, although models like SCOPE [ 16] and Simple-CSC [ 29] have demonstrated strong performance on CSC tasks, directly applying them to reveal Chinese cloaked toxicity may not yield satisfied results (see Section 4). In this paper, we study the problem of Chinese Cloaked Toxicity Unveiling, and propose the C2TU model (see Figure 1). Specifically, we utilize a homo-graph and a toxic lexicon to identify potential toxic words within the input text, thereby transforming the problem into a candidate toxic words filtering task. This task aligns well with BERT model’s pretraining objective of masked language modeling. Hence, we first employ BERT model to compute the probability difference between tokens derived from raw and toxic words at specific positions in a sentence. If toxic tokens have larger probabilities, we then unveil the cloak. After that, we further explore using LLMs to address the filtering task. Similar to the BERT-based approach, we attempt to calculate word occurrence probabilities to decide whether a replacement should occur. Due to the auto-regressive nature of LLMs, the computation of word probabilities only conditions on the left-side context and thus lacks access to right-side semantics. Since LLMs are well-suited for computing sentence-level likelihoods, we mathematically reformulate the word-level probability difference into the sentence-level one based on Bayes’ theorem to leverage the full semantic context of a sentence. Finally, we summarize the main contributions of our paper as: •We propose a training-free and prompt-free method C2TU for Chinese cloaked toxicity unveiling. Different from most existing methods for English, to our best knowledge, we are the first to solve the problem under Chinese. •We leverage both BERT and LLMs to compute the occurrence probability of words for filtering candidate toxic words, respectively. In particular, we address the auto-regressive limitation of LLMs, allowing them to compute the word probability differences based on the full semantic contexts of a sentence. •We conduct extensive experiments to evaluate the model performance on two Chinese toxic datasets. The results show that our methods are more competitive than other baselines w.r.t. both F1 score and accuracy metrics. In particular, our method outperforms the best competitor by up to 71% on the F1 score and 35% on accuracy. 2 Related Work 2.1 Cloaked Toxicity Unveiling While extensive researches have been conducted on toxic content detection [ 6,13,2,11,20,26,7,19], most of them are not specially designed for cloaked toxic words [ 24]. Further, the reveal of deliberately cloaked toxic words emerges as a critical research challenge worthy of attention. Some recent efforts have been made to address the issue by handling intentional obfuscation through character scrambling and misspellings in | https://arxiv.org/abs/2505.22184v1 |
English. For example, some works [ 12] leverage contextual information to correct intentionally misspelled words, while others [ 9] focus on addressing word order disruptions by employing LLMs for robust comprehension of noisy text. However, these methods are designed for English. Unlike English—a word-based language where obfuscation typically alters intra-word characters—Chinese operates at the character level but cloaks content at the word level. This fundamental linguistic disparity prevents direct application of existing English-based methods to Chinese. However, research on unveiling Chinese cloaked toxic content remains largely unexplored. 2 BERT 这个班级真 干净 ，一个嘿辣鸡都没有。[mask] Masked Language Model 干净 clean杠精 troll 𝑃𝑤 𝑃𝑙 𝑑𝑃=log𝑃𝑤 𝑃𝑙Probability differenceReplace 𝑤to 𝑙for the minimal 𝑑𝑃 satisfying 𝑑𝑃<0, and do filtering iteratively 𝑑𝑃≥0 for all 𝑑𝑃 or no more (𝑤,𝑙)pair any morezhè gè bān jí zhēn gān jì ng， yí gè hēi lā jī dōu méi yǒu。 这个班级真 干净，一个黑垃圾都没有。 What a clean classroom, without any black trash. LLMs 这个班级真 干净，一个嘿辣鸡都没有。 这个班级真 杠精，一个嘿辣鸡都没有。𝑃𝑤 𝑃𝑙 Sentence probabilityWord probabilityMasked sentence zhè gè bān jí zhēn gān jì ng， yí gè hēi là jī dōu méi yǒu。 这个班级真干净，一个嘿辣鸡都没有。 What a clean classroom, without any Hey Spicy Chicken. … … Input textToxic words matching gān jì ng 干净gāng jīng 杠精 hēi là jī 嘿辣鸡hēi lā jī 黑垃圾 là jī 辣鸡lā jī 垃圾辣 垃拉 𝑤 𝑙 Homo -graph Toxic lexicon Output textToxic words filteringContinue Break outOR Filtering iteration Matched substrings Matched toxic wordsFigure 1: The main workflow of C2TU method. C2TU consists of two key stages: matching and filtering. In the matching stage, we identify candidate toxic terms in the input text by leveraging a homo-graph and a toxic lexicon. In the filtering stage, we use a language model (BERT model or LLM) to compute the probability gap for each (w, l)pair and iteratively replace the pair with the most significant gap. Once the iteration terminates, the unveiled sentence is returned. 2.2 Chinese Spelling Correction Our task shares similarities with the CSC task, which aims to identify and correct Chinese spelling errors based on contextual, phonetic and graphemic information. Some existing methods are based on small language models (e.g., BERT) [ 27,4,16]. For example, SCOPE [ 16] utilize phonetic knowledge of Chinese characters in conjunction with BERT for spelling correction. With the advent of LLMs, there are also LLM-based models [ 10,29,17]. A representative model Simple-CSC [ 29] is a training-free and prompt-free method, treating LLMs purely as language models to perform spelling correction at the token level. However, the CSC task is not well aligned with our task, as the substitution of toxic words is often intentional, and toxic texts themselves are abnormal texts that deviate from regular language usage. Therefore, we pay attention to Chinese cloaked toxicity unveiling in this paper. 3 Methodology 3.1 Graph and Lexicon Construction 3.1.1 Chinese Homo-graph Given a Chinese toxic speech dataset, we first construct a homophone graph G= (N,E)to capture the likely cloak between tokens, where each node in Nrepresents a Chinese character, and each edge inEindicates the phonetically similar relation. Specifically, we extract all Chinese characters from the dataset as nodes, then use the open-source library pypinyin2to | https://arxiv.org/abs/2505.22184v1 |
obtain their pinyin regardless of tones. Then characters with identical pinyin are connected. Each node also has a self-loop, as each character is considered to have a homophonic relation with itself. We further consider polyphonic characters and phonetically similar pronunciations based on regional dialects. Polyphonic Characters: Some Chinese characters have multiple pronunciations. If a character has at least one pronunciation matching another character’s, an edge is linked between them. 2https://pypi.org/project/pypinyin/ 3 Dialectal Relationships: There also exist dialectal phonetic confusions in Chinese, where users may replace characters based on their dialect pronunciations. Common confusions include: •Retroflex and non-retroflex sounds, such as “ 山” (sh ¯an, means “mountain”) and “ 三” (s¯an, means “three”). •Front and back nasal sounds, such as “ 应” (y¯ıng, means “should”) and “ 因” (y¯ın, means “reason”). •Initial consonants “n” and “l”, such as “ 男” (nán, means “male”) and “ 蓝” (lán, means “blue”). Considering these confusions, we introduce five types of additional relations: “n”↔“l”, “zh” ↔ “z”, “ch” ↔“c”, “sh” ↔“s” and “*ng” ↔“*n” . For example, “ 男” (nán) and “ 蓝” (lán) share an “n”↔“l”confusion, so an edge is added between node “ 男” and node “ 蓝”, although their pinyin representations are not entirely identical. 3.1.2 Toxic Lexicon We also utilize a toxic lexicon to introduce external knowledge about toxicity. The lexicon is a set of toxic words denoted as L={l1,···, lm}. While we notice that some public datasets (e.g., ToxiCN [19]) have their own toxic lexicon released, they contain many toxic words with homophone cloak. We take “ 垃圾 ” (l¯a j¯ı, means “rubbish”) and “ 辣鸡” (là j ¯ı, means “spicy chicken”) as an example. Here, the proto toxic word is “ 垃圾 ”, while “ 辣鸡” is a cloaked substitution commonly used in social media, even more frequently than the protoword. Therefore, to keep a simple and clean toxic lexicon, we only retain protowords. 3.2 Toxic Word Matching Given a text sequence X, we enumerate substring winXand match each wagainst toxic words l∈ L using the homo-graph G. Formally, a substring wof length Nis considered as a candidate toxic word if and only if ∃l∈ L,len(w) = len( l)and∀k∈ {1,2,···, N},G.HasEdge( wk, lk) = 1 , where wkisk-th Chinese character in w, and so does lk. Here, len(·)is the length of the input string. The function G.HasEdge( ·,·) = 1 indicates that two nodes are linked in G. The above matching method could have a small false negative rate , i.e., any homophonic substitution will be unveiled given a comprehensive homo-graph Gand lexicon L. However, it may also yield a high false positive rate by incorrectly matching non-toxic word as toxic word. For example, “ 干 净” (g¯an jìng, means “clean”) will be matched to “ 杠精” (gàng j ¯ıng, means “troll”), even though “干净” is typically a non-toxic word. False positives are especially severe for single-character toxic words, as all homophones of the single toxic character (e.g., “ 操” (cào, means “f*ck”)) would be matched. The pseudocode of the matching algorithm is summarized in Alg. 1, which outputs M candidate | https://arxiv.org/abs/2505.22184v1 |
toxic words pairs, denoted as Wp={(w(i), l(i))}fori= 1,···, M. Here, l(i)is a probable toxic unveiling of w(i). 3.3 Filtering Candidate Toxic Words To further filter out the incorrect matches (see Figure 2) in Wp, we leverage the full semantics of the text sequence X. Formally, for each (w, l)pair, we define XpreandXtailas the prefix sequence and suffix sequence of winX, respectively. Given XpreandXtail, we next calculate the probabilities P(w|Xpre, Xtail)andP(l|Xpre, Xtail), shorted as PwandPl, respectively. For all pairs (w, l)that satisfy Pw< Pl, we select the one that yields the largest difference, denoted as ( ˜w,˜l). Formally, the probability difference is calculated by: ProbDiff (Pw, Pl) = log Pw−logPl= logPw Pl. (1) After that, in the given text sequence X, we replace ˜wwith˜l, remove ( ˜w,˜l)fromWpand then repeat the previous process until Wpis empty or all the remaining pairs in WphavePw≥Pl. Note that in each iteration, we need to recalculate the probabilities for each pair in Wpbecause once ˜wis replaced, XpreandXtailfor remaining pairs change. 4 Sentences after substitution mǔ rén jiǎng de bú shì hàn yǔ shì mù yǔ 。 母人讲的不是汉语是 穆语。 Femoids don’t speak Chinese, but do speak Mu language. Mù rén jiàn de bú shì hàn yǔ shì mù yǔ 。 穆人贱的不是汉语是 穆语。 The Mu people are trashy, not in Chinese, but in Mu language. mù rén jiǎng de bú shì hàn yǔ shì mǔ yǔ 。 穆人讲的不是汉语是 母语。 The Mu people don’t speak Chinese, but do speak mother tongues.Probability 8.71𝑒˗1 3.54𝑒˗8 7.45𝑒˗3母人 femoid 贱 trashy 母 motherMatching 穆人 Mu people 讲 speak 穆 MuSubstitutionmù rén jiǎng de bú shì hàn yǔ shì mù yǔ 。 穆人讲的不是汉语是 穆语。 The Mu people don’t speak Chinese, but do speak Mu language.Input:Figure 2: Three candidate toxic words pairs are matched in the input text. Each pair is individually substituted to generate three new sentences. It is evident that the pair [“ 讲”, “贱”] is incorrect, as the resulting sentence is highly unnatural in meaning, leading to a significantly smaller probability. In contrast, the substitutions of [“ 穆人”, “母人”] and [“穆”, “母”] are correct, and their corresponding sentences yield higher probabilities. 3.4 Computing Probability Difference In this section, we introduce two methods for computing P(x|Xpre, Xtail), where xcan be either wor l, with both BERT and LLMs. 3.4.1 BERT-based Method During the pre-training phase, BERT utilizes an unsupervised Masked Language Modeling (MLM) task, whose goal is to predict tokens that are randomly masked. Hence, BERT is well-suited for computing P(x|Xpre, Xtail). Given a text sequence X= [Xpre, x, X tail], for the i-th token xiinx, we first mask it with the [ mask ] token and derive a text sequence Xmask i= [Xpre,···, xi−1,[mask ], xi+1,···, Xtail]. Then we can output the probability for the masked token PBERT(xi|Xmask i). For a toxic word xthat includes N tokens, we sequentially predict each of these tokens and combine them by: PBERT(x|Xpre, Xtail) =NY i=1PBERT(xi|Xmask i). (2) Due to the varying word lengths, we adjust Equation 2 to calculate the geometric mean of all N probabilities. Finally, the log probability to predict xis formally given by: logPx= | https://arxiv.org/abs/2505.22184v1 |
logNq PBERT(x|Xpre, Xtail) =1 NNX i=1logPBERT(xi|Xmask i)(3) 3.4.2 LLMs-based Method We next explore to use LLMs to filter candidate toxic words. Due to the auto-regressive nature, mainstream LLMs compute the occurrence probability of xbased on the pre-context Xpreby: PLLM(x|Xpre) =NY i=1PLLM(xi|Xpre, x1,···, xi−1). (4) 5 In this way, only pre-context is used while the post-context is totally ignored. To address the issue, instead of directly computing the probability for a word, we calculate the probability difference, and transform the word probability difference into the sentence probability difference based on Bayes’ theorem [3]. Formally, we prove: Theorem 1 Given a text sequence X= [Xpre, w, X tail]and a toxic word lwithlen(w)=len(l)=N, letX′denote the new sequence with wreplaced by l. Then, the following equation holds: ProbDiff( Pw, Pl) = log PLLM(X)−logPLLM(X′), (5) where PLLM(X)andPLLM(X′)denote the probability of XandX′output by LLM, respectively. Both the probabilities can be calculated in the auto-regressive manner as Equation 4. In this way, we can capture the semantics of the entire sentence, rather than only the pre-context information. The proof of Theorem 1 is provided in Appendix C. 3.5 Time Complexity Analysis C2TU consists of two main stages: toxic word matching and filtering. Suppose we have an input text sequence Xof length |X|and a toxic lexicon with mtoxic words. In the matching stage, we enumerate substrings in Xand match them with toxic words in the lexicon, whose largest length is Nmax. We enumerate substrings of length ≤Nmax for matching. Considering thatNmax≪ |X|, the number of enumerated substrings is linear to |X|. For each substring, we only need to match it with the toxic word of the same length, whose average number is assumed to be m Nmax. Therefore, the overall matching time complexity is O(m Nmax|X|). In the filtering stage, we iteratively handle Mcandidate (w, l)pairs. In each iteration, for each pair(w, l), we calculate the occurrence probability difference of the two words, leading to a time complexity of O(M|X|2)no matter whether BERT or LLMs are used. The overall time complexity is thus O(M2|X|2). Different from other LLM-based models that generate tokens with a time complexity of O(|X|3), our method C2TU-LLM only uses LLMs to output the probability difference, which is highly more efficient. 4 Experiment 4.1 Experimental Details Datasets We conduct experiments on two public datasets ToxicloakCN [24] and COLDataset [7]. Further, for fairness, we fine-tune the BERT-based CSC model on a subset of CHSD [21] named as CHSD-subset . Details on these datasets are given in Appendix E. Baselines To our best knowledge, we are the first to unveil Chinese cloaked toxicity. Considering the relevance between our task and CSC task, we thus compare our method with three categories of baselines. (1) Naive : Directly replace all candidate toxic words without any filtering. (2) BERT-based method : SCOPE [ 16] is a BERT-based CSC model that introduces an auxiliary task of Chinese pronunciation prediction to improve CSC task. We also finetune the pre-trained SCOPE model onCHSD-subset , obtaining the ft-SCOPE model. (3) LLM-based method : i) The prompt-based LLMs Baichuan2-7B-Base [ 25] and Deepseek-V3 [ 18]. ii) Simple-CSC | https://arxiv.org/abs/2505.22184v1 |
[ 29] is a training-free and prompt-free LLM-based CSC model that considers character pronunciation and shape similarities. Setup We construct homo-graph by pypinyin, and utilize lexicons after manual correction and deduplication for toxicity matching. Our filtering model consists of two versions: BERT-based and LLM-based. For the former, we utilize bert-chinese-base [ 8]. For the latter, we use Baichuan2-7B- Base [ 25] as our backbone model. None of the models’ parameters have been modified, nor have any prompts been used, as our method is entirely training-free and prompt-free. For SCOPE, we follow its original pre-training stage and finetune it on CHSD-subset . For Simple-CSC, we employ it on Baichuan2-13B-Base as its original paper. All models can be downloaded on Huggingface3. We also 3https://huggingface.co/ 6 Table 1: The main results of all baseline models and our method w.r.t. the F1 score/% and the accuracy/% metrics, where Det. and Cor. mean detection and correction. We highlight the best score in bold and underline the runner-up one. Metric/% MethodToxicloakCN COLDataset Sentence Character Sentence Character Det. Cor. Det. Cor. Det. Cor. Det. Cor. F1 scoreNaive 3.96 2.77 20.29 17.31 72.32 64.00 90.42 87.41 SCOPE 13.23 6.33 7.81 2.22 14.80 9.91 18.41 7.01 ft-SCOPE 35.79 26.99 17.48 11.73 62.66 59.15 41.21 35.83 Simple-CSC 47.81 41.28 50.18 43.77 59.60 57.36 71.60 69.28 Baichuan2-7B-Base 2.62 2.18 3.46 0.77 1.26 1.03 5.31 0.90 Deepseek-V3 28.66 20.21 17.15 8.56 40.89 31.99 40.15 26.84 C2TU-BERT 62.65 58.53 69.77 65.81 87.06 82.56 91.00 90.00 C2TU-LLM 74.67 70.01 78.56 75.04 90.52 88.54 93.73 93.30 AccuracyNaive 7.28 6.41 81.08 80.69 71.87 64.00 98.73 98.38 SCOPE 35.35 31.33 79.42 78.79 15.47 11.22 78.35 76.78 ft-SCOPE 52.88 47.36 80.72 80.01 63.52 60.14 82.32 81.33 Simple-CSC 66.38 63.58 98.13 97.97 53.85 51.99 97.15 96.97 Baichuan2-7B-Base 40.03 39.86 92.55 92.44 4.55 4.43 90.99 90.78 Deepseek-V3 46.36 41.04 83.46 82.57 40.59 32.26 85.67 83.82 C2TU-BERT 76.32 74.38 98.66 98.53 84.65 80.51 98.94 98.84 C2TU-LLM 83.60 81.36 99.02 98.89 88.37 86.55 99.25 99.20 notice that there exist some other CSC baselines. However, some of them [ 27,10] underperform the two used CSC baselines, while others [ 4,17] lack complete pretraining datasets and source codes. Thus, we exclude them from comparison. The prompt template used for prompt-based LLMs is presented in Appendix G. Our experiments are conducted on a server equipped with an Nvidia A800 80G GPU. 4.2 Main Results We evaluate the model performance by F1 score and accuracy from two aspects (sentence-level and character-level), with each one containing both detection and correction tasks. The sentence-level task requires the complete detection/correction of all Chinese cloaked characters in a sentence and keeps other non-cloaked characters not mis-detected/mis-corrected; otherwise, the task fails. For the character-level task, detection and correction are applied exclusively to individual characters. The effectiveness of the model is reflected by its ability to identify and correct more cloaked characters while minimizing modifications to non-toxic ones—the higher the true positive rate and the lower the false positive rate, the better the performance. Details on the evaluation metrics can be found in Appendix D. From Table 1, we observe: (1) The Naive model | https://arxiv.org/abs/2505.22184v1 |
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