Training Large Language Models To Reason In Parallel With Global Forking Tokens
Authors: Sheng Jia
Xiao Wang
Shiva Kasiviswanathan
Venue: ICLR 2026 Poster (2026)
Keywords: large language model
reasoning
chain of thoughts
Primary area: foundation or frontier models, including LLMs
TL;DR: We propose Set Supervised Fine-Tuning (SSFT), which treats parallel reasoning as a set prediction problem and incorporates a set-based global loss into SFT using bipartite matching between global forking tokens and diverse reasoning traces.

Abstract: Although LLMs have demonstrated improved performance by scaling parallel test-time compute, doing so relies on generating reasoning paths that are both diverse and accurate. For challenging problems, the forking tokens that trigger diverse yet correct reasoning modes are typically deep in the sampling tree. Consequently, common strategies to encourage diversity, such as temperature scaling, encounter a worsened trade-off between diversity and accuracy. Motivated by this challenge, we treat parallel reasoning as a set-of-next-token-prediction problem and incorporate a set-based global loss into Supervised Fine-Tuning (SFT) using bipartite matching between global forking tokens and unique reasoning traces. We observe that whereas naive fine-tuning with multiple reasoning traces collapses these unique reasoning modes, our proposed method, Set Supervised Fine-Tuning (SSFT), preserves these modes and produces emergent global forking tokens. Global Forking Policy Optimization (GFPO) leverages these maximally steerable tokens to incentivize complex reasoning, and the resulting models consistently outperform their SFT counterparts with GRPO on both math reasoning and execution-based code generation benchmarks.

Published as a conference paper at ICLR 2026
TRAINING LARGE LANGUAGE MODELS TO REASON IN PARALLEL WITH GLOBAL FORKING TOKENS
Sheng Jia!.3f Xiao Wang"; Shiva Prasad Kasiviswanathan"
University of Toronto, 2 Amazon, Vector Institute
(sheng. jia@mail. wtw.onoooooooooooooooooooo ca, waxao0amazon, com, kasiviswegmail.com
Q Code: https://githab... com Stee/nhhehggeeeeeeeeee Jaanang-//S00 SSFT
HE: httpoiiiatttooooo/seeeee
ABSTRACT
Although LLMs have demonstrated improved performance by scaling parallel test-time compute, doing so relies generating reasoning paths that are both diverse and accurate. For challenging problems, the forking tokens that trigger diverse yet correct reasoning modes are typically deep in the sampling tree. Con- sequently, common strategies to encourage diversity, such as temperature scaling, encounter worsened trade-off between diversity and accuracy. Motivated by this challenge, we treat parallel reasoning as a set-of-next-ton nex--tokent----eeeee pekeentpeeiition problem and incorporate a set-based global loss into Supervised Fine- Tuning (SFT) using bipartite matching between global forking tokens and unique reasoning traces. We observe that whereas naive fine-tuning with multiple reasoning traces col- lapses these unique reasoning modes, our proposed method, Set Supervised Fine- Tuning (SSFT), preserves these modes and produces emergent global forking to- kens. Global Forking Policy Optimization (GFPO) leverages these maximally steerable tokens to incentivize complex reasoning, and the resulting models con- sistently outperform their SFT counterparts with GRPO both math reasoning and incccuuion-bbeee code generation benchmarks.
INTRODUCTION
Large language models have recently improved reasoning by allocating more test-time compute to generate more tokens before producing the final answer (OpenAI (2025). However, extended se- quential scaling can lead to "overthinking" where performance decreases beyond certain sequence
length GCossl et al. 2025 Chen et al. 2024a). To mitigate this, another scaling dimension based on repeated parallel sampling and aggregation (Wang al., 2022 Brown et al. 2024) has shown success in further boosting reasoming performance. However, these methods rely on LLMs gener- ating diverse yet correct solutions; as tasks become harder, mechanism for increasing diversity is required. Recent work shows that only minority of tokens in Chain-of-
reasoning (Wei) jet al. 2022) can act as forking tokens that lead to distinet reasoning modes (Wang et al. 2025b), so the problem becomes harder and the generation becomes longer, it can become substantially harder to sample them. Also, common practices to encourage diversity, typically through tempera- ture scaling, inherently entail a diversity-accurae trade-off, the forking tokens that trigger diverse yet correct reasoning modes are typically located deeply within the sampling tree. Moreover, recent theoretical work also shows that increasing the temperature alone does not necessarily guarantee greater diversity unless the model is explicitly trained to ensure coverage. (Verine et al. 2025).
Building these observations, we aim to leverage diverse reasoning traces to train for coverage (uuo et al. 2025: Google| 2025b). We introduced global forking tokens prior to generate paral- lel reasoning traces and frame parallel reasoning as a set prediction problem. Specifically, given question, an LLM, conditioned on a reserved set of tokens in a chosen ordering, generates M rea- soning sequences in parallel, each aligned with one of M ground-truth reasoning traces. For each ordering, we compute the total autoregressive loss across the generated sequences. By enumerat- ing all possible orderings, we identify the minimum loss, which defines the set language modeling
Work done during internship at Amazon Science. "Equal advising.

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Published as a conference paper at ICLR 2026
loss, conditioned on the distinct forking tokens (Equation B} This formulation naturally incorpo- rates coverage into the training objective and is capable of learning global forking tokens that can serve prompts trigger reasoning modes that are both diverse and accurate. We operationalize this idea through our Set Supervised Fine-Tumm Fine-Tuning (SSFT) framework. Our main contributions are summarized below.
We introduce global forking tokens and incorporate set-based loss into SFT via bipartite matching between reserved control tokens and diverse traces (Section 2 Figure T The SSFT fine-tuned model globally steer reasoning modes from single global forking token, reducing dependence on sampling forking tokens mid-generation (Section 2.4 Figure
We show that SFT on diverse reasoning traces collapses control tokens into single reason- ing mode (Figure 5 whereas the global forking tokens learned by SSFT initiate distinct reasoning. This visualized quantitatively through discrepant reasoning effort and accu- racy over different control tokens (Figure 4 and improved Pass@k (Figure 33, and quali- tatively through distinct math reasoning strategies across forking tokens (Section
We demonstrate that, across multiple reasoning benchmarks, Global Forking Policy Op- timization can leverage the learned global forking tokens to incentivize diverse complex reasoning without collapsing them, and the resulting models consistently outperform their SFT counterparts with GRPO, improving Pass and Cons@k (Section 3
Table 2
2 LEARNING GLOBAL FORKING TOKENS VIA SET SUPERVISED FINETUNING
Background on Language Modeling with Reasoning. In language modeling, the goal is to train a model Te to approximate the joint distribution over a sequence of word tokens = ]T VT, where T is the sequence length, and each token is within a finite vocabulary set V. An autore- gressive model uses the chain rule to represent it as product of conditionals on the preceding tokens: Te (x) = II
To (Xe (x)).. This is known as next-token-prediitoon (NTP) (Radford al, 2019). For reasoning tasks, we break the sequence of word tokens into: (1) an input prompt x = {x (2) a special token (or a sequence of tokens) that initiates reasoming, and (3) a reasoning trace plus the final answer r = {r.) To simplify notation, we combine verifiable answer and a reasoning trace. A reasoning model autoregressively generates a reasoning path and the final answer: (r g) = I no (r, x. g. ret). To train a reasoning model, Supervised Fine-tuning (SFT) minimizes the negative log-likelihood of ground-truth reasoning trace, i.e., (0) =- Ex. (oggo(re/xx g,x,g r.
2.1 PARALLEL REASONING AS SET OF NEXT TOKEN PREDICTION
Problem Setup. In this paper, our goal is not only to instill new reasoning capabilities into a model, but also to ensure that prompting with a set of reserved special tokens, in parallel with a question, elicits distinct reasoning traces. We call these global forking tokens == ] instantiated as {<think i> J,N tags. We use g(i interchangeably with <think i>, depending on context for clarity. We consider a setting with multiple sources of reasoning traces, obtained at low cost without human annotation by distilling from diverse teachers, sampling repeatedly, and potentially filtering with a verifiable metric such as correctness. We adopt this low-cost regime to highlight the effectiveness of our algorithm, though the method extends naturally to settings with well-annotated, haman-labeled data. So our problem to (1) learn to do a set of next-token predictions on multiple distinct yet correct reasoning traces R := {rO))M 1-=1 in parallel for an input prompt x, and (2) ensure that distinct global forking tokens can uniquely initiate these distinct traces.
3
To do this, we make a simple change to the NTP loss, which now has two requirements: (1) Permutation--nvariance It should not depend on the order of elements in R and so we don't penalize a trace that incurs high NTP loss under one forking token if the model predicts it well when conditioned on another. (2) No shared global forking token: We want (g()) N to uniquely
initiate distinct reasoning traces, so this requirement prevents conditioning on the same g()) when generating distinct traces given a question.
To satisfy these requirements, we incorporate a subproblem in language modeling: finding the mini- mum cost bipartite matching configuration between the left vertices (g ( and the right vertices
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from) where the cost of each edge between left vertex i and right vertex is the NTP loss of pl)) conditioned on (()) and an input prompt x. A matching configuration is set edges connecting the left and right vertices where no two edges share common vertex. Without loss of generality, we assume N 2 M to simplify our notation. The total cost involves all vertices on the smaller side the bipartite graph, this allows to write the summation from I to min (N, MN, which equals M un- der this assumption. We denote a matching configuration as finite map {1,..., M] + {1, [1,.., N} such that (j)) = 3
r() is paired with gl Let Gp := { denote all the M x M!
(N)
configurations of bipartite graph. The total cost of each configuration represents the compatibil-
ity between (rO) JM and {g()) Ji=1 under this unique matching. Figure | matching configuration.
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1. Minimize Total Bipartite Matching Loss
Minimize Set NTP Loss
Figure 1: An illustration of one SSFT training step. Step 1: We first construct the cost ma- trix by evaluating all pairwise combinations: for each r(1) E {r(1), ) ,x((), (()),)()) r(4)} and each gg)]g
g(i) E {g(") g(2)_g g , (Equation
g we compute the NTP loss of r(() conditioned on g(() Then use Hungarian algorithm
find ā that minimizes the total bipartite matching cost. Here, this minimum is the sum of the losses highlighted in blue, which means
( {gg() p()), (g(5), ,r()), (g(²), r())), (g² r(())))))) Step 2: We optimize by backpropagating the set of NTP losses for ((),, each conditioned on g! g()))))
2.2 SSFT: MINIMIZING SET OF NTP LOSSES UNDER OPTIMAL. BIPARTITE MATCHING
Under this formulation, we propose Set Supervised Fine-Tuning (SSFT), which performs two opera- tions at each training step: (1) find the minimum-com cost bipartite matching that is optimal for uniquely initiating different reasoning traces (2) and then minimize the NTP losses under the matching con- figuration to instill diverse reasoning modes conditioned on the matched global forking tokens. We show our implementation in Algorithm
For the first step, we first compute all the entries in the cost matrix such as the one in Figure 1 and then apply the Hungarian algorithm (Kuhn 1955) efficiently find the optimal ē:
mmatching (g(ob)), x(1).
④ arg min Sp
where
T log
Cmatching
gg()) ((0) = -g
[ x g() (0))
(2)
T t=1
As noted in Equation 2 each matching cost in Equation is the negative log-likelihood of r()) conditioned on g g((()))) under the current model parameters. Here, sg(-) is stop- gradient, as the matching process is done by discrete optimization w.r.t. a, save VRAM by not storing
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intermediate activations. We explicitly indicate that length normalization is done to remove biases toward trace length, so that the matching is driven by semantic content, After solving for each (x. R), our second step optimizes model parameters 0 by backpropagating the matching loss in in mini-batch.
Equation The expectation is replaced by its sample mean over (x, (r()) )
Time complexity, In practice, we use only the first L < T, tokens in Equation 2 to compute the matching cost when early tokens already reveal sufficient differences between traces. This k t factor reduces the time complexity of matching to O ( Backpropagation is performed only on the M matched sequences and can use the full Tp length at the same compute by SFT.
M T
log To (1L))
(1 ( ( x22
Huunariin (0)
(3)
x, xRRwD
Remarks. The resulting model is not the same a simple routing of the models independently trained with the nonoverlapping subsets of these traces. Firstly, SSFT allows positive transfer in N
J== representation learning within [r()] even though they are matched to different (g(9) Ji=1- Sec-
ondly, it is not optimal to distill reasoning traces from the same fixed sources for every question if the goal is to maximize both diversity and correctness. Our algorithm supports a variable number of target reasoning traces across training steps, with sources that may also change for each x. Thirdly,
even the two sets of traces, [ra (r2))M Jj=1 for Xa and (r )زر for Xs, are from the same sources, their optimal configurations o. and on can still vary because a teacher model can reason differently under different questions. Lastly, reserve of traces (N > M), and empirically observe that all the forking tokens are being matched through- out the process. This is because the extra forking tokens can maximally differeifferentiaee similar traces.
global forking tokens than the maximum number
2.3 GFPO: OPTIMIZING THE SELECTION OF GLOBAL FORKING TOKEN
Although SSFT can leverage unbalanced bipartite graphs to learn correlations between questions and forking tokens given sufficient data, collecting additional traces often unnecessarily expensive once the model already learns distinct reasoning patterns. Instead, we apply small number of RL steps using only the policy gradients of global forking tokens an extremely efficient variant of Wang et al. (025b a). This Global Forking Policy Optimization (GFPO) properly shapes the output distribution of g conditioned on x. It can be implemented trivially by adding a few lines of Python slicing to any off-the-shell code (Sheng et al., 2024). since global forking tokens always begin the same indices of generations. Full generations used solely compute the advantage of gf), and they do not participate in backpropagation.
2.4 INFERENCE WITH LEARNED GLOBAL FORKING TOKENS
Inference protocols
Cons@k. Our inference protocol with parallel test-time compute is to prompt i-th response with <think (i&N)> > and then do majority voting on their answers.
Cons@k
rhark chase
Pass@1:
hankk
P To evaluate a single-path generation, we can either (1) apply GFPO and let the model sample the optimal g(() based on the question, or (2) prompt g(t) that is most likely to correlate with a more complex reasoning mode, as determined by a graph-based heuristic. Inspired by enumerating dissimilar bipartite matchings
Learned Matchings Gp
Hhink 30
?
reveal node-level varia- tion Blumenthal et al. 2022), choose the learned g(()) with largest coverage. Concretely, each time an optimal matching E (ox {o is computed, increment count c( empirically, only finite subset 6 := {o S Sp continues to accumulate mass late in train- ing, indicating the stable learned matchings. We then take the union of their edges, and select g((") that matched to the largest number of dis- an example of the matchings learned by aggregating all edges in Sp. More details
trink
20
+hink
Figure 2: Learned matchings by SSFT-32B in Exp obtained by connecting edges 6,p
tinct traces based on Equation (or Pass 1. F
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about 6 are in Appendix A.2
1" = arg max
(ia(()) = iH
(4)
P
3 EXPERIMENTS
We address the following research questions through experiments: (RQ1): In terms of Pass&1 I and Cons@k accuracy, how does model trained with SSFT perform reasoning benchmarks? (RQ2): Does finding the optimal bipartite matching matter in reasoning performance? (RQ3): Does training with diverse reasoning traces yield better accuracy and coverage under SSFT compared to standard SFT with temperature scaling? (RQ4): Does prompting with distinct {g(() Ji=1 genuinely make a model generate diverse reasoning traces? (RQ5): Is the performance gain from SSFT robust to different fine-tuning datasets and model scales?
3.1 EXPERIMENT SETUP
addi- tion to the R1 (Guo al. 2025) and Gemini Flash (Google 2024) traces provided by sl, also use Claude Opus 4.0/4.1 I (Anthropic 2025) and GPT-OSS- 10BB (Agarwal et al.) 2025) with high and medium reasoning effort to obtain pool of distilled targets for the 1,000 questions. For each question, we generate two traces per source to populate the pool. We then sample four traces from this pool. We call this 1kk-4mixed from DAPO-Math. 77 Yu et al. 2025).
Training Dataset. We use the 1,000 questions from slk dataset Muennighoff et al. (2025).
dataset. For additional GFPO, we use 1280 questions
Training Details. We fine-tune Bwen2.5-32B-Instruue (ang et al. 2025a) for six epochs with con- text length of 32,768. We reserve N = 6 global forking tokens and use M = 4 targets per question. To find the optimal bipartite matching for each input prompt, we consider only the first 1,000 tokens when computing the matching cost in Equation 2 This number is derived from |32768/ (MN) for computational efficiency, so matching costs fit one forward pass without extra VRAM. We call this model SSFT-32B. We also include SSFT but choose random bipartite matching at each step to fine-tune SSFT-32B (random σr) Models with an additional RI, step use the postfix GRPO/GFPO. Exact details on the pool of diverse distillation targets and selection procedure, as well as training hyperparameters, are provided in Appendix A.9 and Table
Baselines. Our baselines include two groups: (Single-Targel A) models trained with one trace per question and (Multi- Marget *) ) models trained with four traces per question. For (Single-Target), include s1.1-33 1-1BBBB (Muennighoff et al.) 2025), which uses 1k DeepSeck-RI traces. We also fine- tuned an SFT-OSS-distille ing effort, as these traces achieved the highest correctness on the k questions based on an evaluation by Claude 3.5 Sonnet comparing each attempt against the reference answer. For (Multi-Targed use sik-4mixed-reaasee question is duplicated for its distilled traces and <think i> is sampled without replacement for each duplicate, treating the four traces as four individual data points. We also include Multiverse- 32B (Yang et al., 22255b, which prompts Gemini Pr5-Pro (Google 2025a) to transform 1k sequential CoTs into parallel CoTs as their training data.
baseline that trains only on the 1 GPT-OSS traces with high reason-
we using standard SFT, where each
fine-tune SFT-miixed-disill--2BB
Evaluation Setup. Our evaluation tasks consist of AIME24/AIME25 (Ye et al., 2025)
MATH-500
(Hendrycks et al. 202T), GPQA-Diamond (Rein et al. 2024), and LiveCodeBench (L.CB), where LCB is considered distof-distribution relative to the fine-tuning dataset. We use LightEval (Habib) ctt al. 2023) as our evaluation framework with generation configurations: temperature=0.7 used in (Guha et al. 2025), top-p=0.95, max length=32768. For Pass@1 accuracy without any parallel test-time compute, we select learned (1) for SSFT-32B and g(4) for SSFT-32B (random σ)) based For each Pass@1 accuracy, we compute the average performance over 32 generations. For Consâ6, which applies each of the six global forking tokens once in our method and uses six generations for the baselines, we compute the average over II sets of generations to reduce variance in the results. We refer to this as Pass @ 1 of Native Cons @6 using a similar terminology the concurrent work (Wen et al. 2025 Appendix 1.10enreentts an example of the parallel generations.
on Equation 4 We let the model sample g()) for SSFT-32B-GFE
We choose the subscripts p and P emphasize that G is subset of Sp.
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Table 1: Performance SSFT compared to baselines on reasoning tasks, reported at Pass Cons@e, and Cons@32. SSFT selects <think 1> for Pass01 and replaces 6 generations with generations prompted by distinet <think for Cons@k. SSFT-GFPO samples <think for Pass@1. We observe consistent improvements over (i) SFT-OSS- which uses the Ik OSS-high traces; (ii) SFT-mmixeddditill-2BBB--
which uses four mixed traces but treats them as individual data; and (iii) SSFT-32B (random σ), which optimizes Equation with random o.
AIME 2024
AIME 2025
MATH-500
GPQA-D
LCB(v5)
Pass 1: Average performance of individual generations (* denotes out out- distribution tasks.)
Owen2.5-32B-Iman
15.80
10.40
80.40
47.00
23.35
s1.1-32B A
54.79
44.27
92.16
62.12
Multiverse-32B *
53.80
45.80
91.80
60.70
SFT-OSS--istill3228
57.82
48.75
89.54
60.06
34.13
SFT-mixcd-ddisill-32B---8
58.23
51.96
88.49
59.96
32.34
SSFT-32B (random σr) *
61.77
55.10
89.95
62.28
35.33
SSFT-32B *
64.06
58.13
90.02
60.39
38.92
SSFT-32B-GFPO *
64.22
58.80
89.90
62.48
42.10
@ass@1 / of Native Cons@6: Average performance of majority voting with 6 parallel generations
s1.1-32B A
70.30
53.33
95.60
61.45
72.12
65.45
95,47
61.52
SFT-mixed-disee distddd-itil-2228- 32B-t22B
*
73.94
70.00
95,88
58.75
SSFT-32B (random σ>>
73.03
67.58
95.67
61.87
SSFT-32B *
75.45
73.94
96.47
63.05
SSFT-32B-GFPO *
76.67
78.48
95.73
60.94
Cons@32: Majority voting performance with large number of parallel generations
s1.1-32B A
73.33
63.33
94.80
60.61
SFT-OSS--dsstil-322
76.66
73.33
96.00
61.60
-ii-mixcdddistil-32------ *
76.67
76.67
96.20
58.59
SSFT-32B (random σ)
80.00
80.00
95.60
62.63
SSFT-32B
83.33
86.67
96.80
61.62
SSFT-32B-GFPO *
83.33
83.33
96.80
62.12
indicates training with single-target data and * indicates training with multi-target data.
3.2
EVALUATING SSFT ON REASONING BENCHMARKS
For RQ1, we see in Table that SSFT delivers the best Pass@1 accuracy, 64.06 on AIME24 and 58.13 on AIME25, outperforming SFT-mixed-32B by 8.33% and 6.57%, respectively. We also observe consistent improvements all four tasks under parallel test-time compute at scales, Cons@6 and Cons@32, over SFT-mixed-320 which was trained on the same reasoning traces. Some notable results Cons@6 73.94%. Cons@32 86.67% AIME25. To answer RQ2, we observe consistent improvements over SSFT-32B (random σr), with especially strong gains at Cons@6 on AIME25, where effectiveness with few parallel generations is critical. As shown later in Figures and 5 optimal bipartite matching essential to prevent collapsing reasoning modes. Results with SSFT-GFPO confirm that GFPO successfully sharpens the output distribution of (i) conditioned on x. We observe positive gains in P Appendix reporrs the results for baselines with RL. For
generalization to LCB, SSFT also yields the largest gains. Coverage. For RQ3, compare method against SFT-mixed-32B under various in Pass@k accuracy with 32 generations to assess generation coverage. Figure 3 shows that SSFT achieves higher coverage across nearly all values of k. SFT-mixed-321 requires more allowed attempts and higher temperature to match the coverage of SSFT at the cost of lowering its Pass@1 1 and Cons
3.3 EVALUATING PARALLEL REASONING DIVERSITY AND LEARNED MATCHINGS
Addressing RQ4, we show that our global forking tokens genuinely initiate distinct reasoning traces and offer new mechanism for leveraging test-time compute.
Emerging Diverse Reasoning Modes. Using the Cons@6 results in Table 1 we form six sets of generations, each prompted by distinct (*) For each set, show the average accuracy and the

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Published as a conference paper at ICLR 2026
95
Consgr:: 75 73.369 36%
90-
6o A5%
Cons@6: 72.42%
80-
Comeie 73 03%
Consgr: 71.52%
Pazpgl (96)
Conoiges 67.50%
Contegi 09%
Consg6: 70.30%
75
SSFT-328, temp=0.7
Consert: 63 94%
SSFT: 228. temp=0,7
Consgré: 70
SSFT
3FT-mbodd-dicill-- tamp-0.7
60
SFT #FT-mindd-distlll- temp=077
80
SFT-mised dictllddddddl--555- tamp-0.
3FT-nixed-distll- 228.222. tamp-0.
55-
SFT-miodd-iiill--20 temp=10
SFT-miodd-tiiill- 32 temp-10
(logs
Bloga scale)
AIME24
(b) AIME25
Figure 3: Coverage of SSFT compared to SFT-mixed-distil-333 with temperature scaling, reported at Passūk. For convenience, we also report the Cons@6 accuracy next to each line. In AIME25, SFT-mixed-distie distilill-1132 needs to raise the inference temperature and use attempts match the coverage at the cost of lowering its Pass@1 and Cons@6 accuracy, further widening the gaps.
AIME24 Cons@6 75.45%: Avg stats (SSFT-optimal)
AIME25 Cons@6-73 73.94% Avg stats (SSFT-optimal)
35000
35000
30000
30000
Number of thinking tokens
25000
25000
Number thinking
20000
15000
15000
10000
10000
5000
0
think1
think2
think3
think4
thinkS
thinks
think1
think2
think3
think4
think5
think6
Figure 4: (SSFT, optimal matching). Distribution of thinking-tokee counts and average performance on AIME24 (left) and AIME25 (right) prompted by distinct <think 1>, <think 6>.
distribution of thinking-toker counts: Figure 4 for SSFT with optimal bipartite matching and Figure 5 for SSFT with random matching. Reasoning length: Length partially indicates the diversity in reasoning, and we see clear differences for SSFT with optimal matching. despite the absence of hand-crafted matching rule information about these traces. The consistency of these distributions across AIME24 and AIME25 indicates the differences is not from randomness, whereas randomly assigning a <t <think i>, in concurrent work (Wen al. 2025), does not yield clear consistent differences in reasoning length, as shown in Figure S Performance: After finetuning with random matching. prompting with distinct <think i> shows no meaningful impact (+e 61% AIME24 and 2a 55% on AIME25). With optimal matching, SSFT elicits distinct reasoning modes initiated by <thiin 1>,..., <think 4> that reach around 65% on AIME24 and 59% on AIME25, with different lengths. Although <think 5> and <think 6> are weaker due to shorter reasoning modes, the average of these and, especially, Cons@6 performance improve consistently with them.
Visualization of Learued Matchings. We observe in Figure 2 that some g() g from SSFT- 32B have unique configuration of matched edges with (1(3) j==1- This is positive indication that (g are likely to initiate distinct reasoning modes. We hypothesize that (g JN can still yield unique edge-matching configuration even subset of {r(")} are difficult to dis- tinguish. For interpretability, we fine-tune using the same four teacher models for each GPT- OSS high, medium, RI, and Gemini. We call this dataset s sik-4teachers-reasomii
dataset, and ask whether SSFT associate a unique g to the Gemini and RI traces which have easily identifi- able reasoning patterns, and then have the rest matched OSS-high/OSS medium traces in differ-

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Published as a conference paper at ICLR 2026
AIME24 73.006-73.03% Avg stats (SSFT-random)
AJME25 Cons@6-67.cm 67.58% Avg stats (SSFT-randoml
35000
35000
Aug A6C
Avg Asc
61.21%
Arg
Aeg
Aeg ACC
Ava Acc
Avg Arc
AVE AGE
Aug ASC
$4.03%
30000
30000
Number of thinking tokens
25000
Number thinking
20000
20000
15000
15000
10000
10000
5000
5000
think1
think2
think3
think44
thinkS
thinks
think1
think2
think3
think4
thinkS
think6
Figure 5: (SSFT, random matching). Distribution of thinking-token counts and average performance on AIME24 (left) and AIME25 (right) prompted by a distinct <think 1>,..., <think 6>.
ways (i.e. only matched to OSS-high, only matched to OSS-med, and matched both). We track source indices only for evaluation; the model still receives an unordered set traces with source information for each question. We study 3 bipartite matching settings for SSFT. Fig- ure [ shows the evolution of matchings learned under these 3 hyperparameters SSFT (a) random matching with four reserved g(), (b) optimal matching with four reserved g( and (c) optimal matching with six reserved g( Initially, all the configurations a E Sp accumulate mass as there is no correlations between g sand (r(1) " j-m1- Figure 6a shows SSFT under random match- ing does not shrink the size of configs computed as optimal, meaning that no correlations are learned between {g(*) }+= and {r(0)] By contrast, Figure 6b and Figure 6c show the emer- gence only strict subset of matching configurations in Sp. This indeed learned through SSFT. We first visualize the learned
This indicates some correlations
between (g(") 1 and (r()) 1==
matchings of the model with four g()) in Figure 8b We observe that g( and g(2) are uniquely matched to the RI and Gemini traces, showing that SSFT can indeed uniquely associate g to sufficiently diverse reasoning traces. Now to confirm our previous hypothesis, learned matchings (g(3), (OSS-high, OSS-med)), (g(4), OSS-med). Furthermore, by connecting all edges in G from SSFT with forking tokens (Figure 6cb, we also see unique learned match- OSS-high) in Figure & This confirms that the global forking tokens identify unique correlations even among highly similar traces.
the unique
ings (g(3) OSS-high), (g(4), OSS-med). (g(5),
ROBUSTNESS AND ABLATION STUDIES
Robustness to Fine-tuning with Code Gen- eration traces. We study whether SSFT improve Pass@1 more open-ended tasks by instilling specialized, complex rea- soning modes without mode collapse, unlike standard SFT where such modes can be corrupted by merging together. We select IK coding questions from Open- Thoughts Guha et al. 2025), and use RI generate four reasoning traces per question to construct codelk- dmixeds -eesoiindd--r-dsniinnnn
Table 2: Comparison of SSFT and baseline fine- tuned models trained code-generation data, evaluated on one coding task and three math rea- soning tasks that out-of-distribution relative to the fine-tuning set. At test time, <think i> is sampled by the models.
LCB(V)) AIME24' AIME25' MATH500'
somplex for ICR. denotes wit poat-triinnng)
Dwwen2.5-328-Ineee
23.35
15.80
10.40
We apply SSFT on this dataset, and then GFPO on 1k questions from Intellect-2-Bl (Team
24.17
SSFT-328-cude (random)
45.36
39.06
32.92
89.46
SSFT-32B-code
52.07
43.23
32,82
89.96
al. 2025). From the Pass@1 comparison Table 2 SSFT better elicits complex reasoning modes of higher accuracy than baselines that do not use a set loss. Full results are Appendix
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Gounds spfmaa
Stop Comulatvv S.
Stop Comurallv
qraniiy 20
dlpurtite Katching Confiy Indox
Nacching Config Indox (0. (0.23)
Bipaittt Natching Config Indox (0. (0.355)
(a) SSFT under random matching [ (11)))_))) No
(b) SSFT under optimal matching
(c) SSFT under optimal matching ((g (0) 1 Je - 6 [62, 110, 134, 180, 204, 230)
(gg Je. [22, 23}
correlations learned 6, 6p
Figure 6: Cumulative counts o E Op computed as optimal over training. Note that 6p and GG are defined in Sections 2.1 and 2.4 respectively. Front axis: matching configuration index k. Depth: training step t. Bar height is the cumulative counts. These are the evolution of matchings during training 3 Qwen-32B- Innttructt models under 3 bipartite matching settings. In this case study, always (GPT-OSS-higg high, GPT-OSS-med. RI, Gemini) for each question. Note that Random matching method does not minimize Eqn B under optimal matching, but we track it. We observe G Gp with random matching, meaning correlations learned. But by optimizing Hungarian loss, we see the emergence of 6, C 6p.
the [r()} J==1
Robustness to Fine-Tuning Data Quality. To test whether gains depend the traces generated by our procedure and on small high-quality dataset (Muennighoff et al.] (025). we fine-tune public dataset that already provides 2-4 reasoning traces per question: the 93k math set of Face (2025). Because this dataset has been successful for fine-tuning -.en2.5-MMth-7BB we adopt that model and compare SSFT against SFT trained on all available traces. Details are in Appendix A.9.4. Addressing RQ5, Table shows consistent improvements in both Pass@ I and Cons@32. results indicate SSFT effective for larger public dataset with less diverse traces.
shows
Table 3: Performance of SSFT versus SFT trained solely on publicly available distillation targets. The setup uses the 93k math questions from Face (2025) with 10en2.55-Mtt-77B as the base model. SFT-OpenR1-938 -93k-7BB uses the same distillation targets as SFT-mixxe-ddistill-555 in Table |
AIME 2024
AIME 2025
MATHI-500
GPQA-D
Model
Pass ICons @32
Pass @ ICons@32
Pass I ICons 32
Pass @Cons@32
10.42
20.00
9,48
23.33
81.87
87.4
30.29
30.30
SFT-OpenR1 -93k-7B
46.15
66.67
34.17
50.00
86.62
90.20
46.35
47.98
SSFT-OpenR1-933-77
51.25
73.33
35.52
56.66
89.74
93.60
46.86
48.90
Robustness to Model Family & Size. In Appendix A.3 SSFT also shows gains with Qwen3-4B- Base and --mma.1-88--nstmii though the improvements are larger for the bigger Qwen models.
Impact of Matching Length. We choose I. max seq len 1000 because it is roughly the largest MN
value that allows NM matching costs to computed a single forward pass without extra VRAM. shows Figure shows this adds very minimal computation because backpropagation and storing activations still dominate the training time. Note that in our setting, L = 1000 was sufficient for maximally dif- ferentiating complex reasoning modes and preserving high-accuracy traces, reflected by Pass and diversity in @o in Figure 7 Increasing L does not hurt Pass@1 or Cons@32. When ad- ditional compute time permits, L can be increased further and stopped once performance plateaus.
5 RELATED WORK
Test-time Scaling. There has been surge work line-tuning LLMs to reason longer, using re- inforcement learning for frontier models (OpenAl 2024 Shao al. 2024 XAI 2025 Yang al.

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Published as a conference paper at ICLR 2026
537 3571320 Vairki Time 2roug
Irpacc traniing Time As Mesromga, leccal
Impact SSFT-328 Passgg) w/ samples
100
Impact of on SSFT-32B Cons@32
12.3
1543
Conaa32
80.00
13.0
Pencea
NNZ25 Cone
AIM24 Coneg32
25:0
L-1000
Lo3de0
L-0
L-100
L-1000
L-2500 L-3505
L(Number of bookonss in matching coso
LOtumber of tokens is matching coul
L(Number of tokens in matching cost)
Figure 7: Left: Impact on training time. Middle: Impact on Pas@1. Right: Impact on Cons@32. Ablation on the number of tokens L. This a hyperparameter when computing the matching cost in Eq. 2 Setting L = 0 reduces to SSFT with random matching, For each L, we apply SSFT with matching (N = 6, M 4) to fine-tune Qwen2.5-32H Swren.5532B-Instruee on evaluate training time, @ass@ 1 on two challenging reasoning tasks, and C3n on the same tasks.
in and
2025a) and supervised fine-tuning for smaller ones (Muennighoff et al. 2025 Hu et al. 2025). These methods enable LLMs to improve reasoning by allocating more test-time compute to se- quential, iterative refinement such as self-reflection (Guo et al., 2025| Liu et al. 2025b). However, extended sequential reasoning
be more sensitive to the order of reasoning steps and may result in failures (Chen et al. 2024b). and performance can start to degrade beyond a certain length due to "overthinking" (Ghosal al., 2025). Our goal is to study the effective use of diverse reasoning traces to fine-tune small language models, essential for agentic AI (Belcak et al.) 2025).
Parallel Reasoning. Parallel scaling methods such self-consistency (Wang et al. 2022) and Best- of-N (Lightman et al. (023) improve LLM performance by generating multiple reasoning paths in parallel and aggregating them. These methods fundamentally require choosing temperature that can generate diverse reasoning paths, but a recent theoretical work shows that increasing temperature can sometimes fail increase diversity if language models are not trained towards coverage (Verine et al. 2025). Other search- based methods such as Monte Carlo tree search (MCTS) (Zhang et al., 2024 and Tree of Thoughts (ToT) (Yao et al. (023) apply parristic-giieed search with an external verifier to do more deliberate search to increase the coverage (Yao et al. 2023 Zhang et al., 2024).
However, their dependence on heuristics and domain-specific knowledge can limit their applicable tasks. Regarding training LLMs with parallel reasoning traces, Yang et al. (025b) proposes training with parallel CoTs decomposed from sequential CoTs, and our concurrent work Wen et al. (2025) proposes to train with multiple reasoning traces distilled from teacher models. These works show native parallel scaling can surpass sequential scaling within certain token limits. However, we aim to show that training on diverse distilled traces with set language modeling loss enables the model to learn global forking tokens that trigger distinct reasoning modes, improving Pass& I and Cons@k over baselines fine-tuned on the same dataset, whether on subsets or the full set.
Set-based Global Loss in Deep Learning. DETR introduces end-to-end object detection with a set global loss (Carion et al., 2220;; Minderer et al.] 2022). whose success in parallel bounding- box prediction inspires our approach. We are the first to extend this to language modeling: while DETR predicts a set of tokens parallel to match a list of bounding boxes, we predict a set of sequential reasoning paths initiated by global forking tokens and assess the matchings based on autoregressive losses. We adapt the set-based loss with autoregressive (AR) models, rather than diffusion-basee models (Nie et al. 2025), because AR models achieve superior reasoning performance.
CONCLUSION
We show that diverse reasoning traces can be leveraged to learn global forking tokens that glob- ally steer diverse and accurate reasoning modes. We propose Set Supervised Fine-Tuning (SSFT), which employs bipartite matching between such control tokens and traces to compute a set language modeling loss. We visualize that SSFT preserves distinct reasoning modes that SFT collapses, and produces emergent global forking tokens that Global Forking Policy Optimization leverages to in- centivize complex reasoning. The resulting models consistently outperform their SFT counterparts with GRPO, improving Pass@1 01 and Cons@k across math reasoning benchmarks and P on execution- based code generation benchmarks. For future work, we plan to scale up the underlying bipartite graph and explore more general settings beyond multi-teacher distillation.
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ACKNOWLEDGEMENTS
The authors thank Michael Zhang, Silviu Pitis, Andrew Li, and Elena Zhang at the University of Toronto for their valuable feedback. We also acknowledge Amazon Web Services for providing the computing resources that made this research possible.
REFERENCES
Sandhini Agarwal, Lama Ahmad, Jason Ai. Sam Altman, Andy Applebaum, Edwin Arbus, Rahul K Arora, Yu Bai, Bowen Baker. Haiming Bao, et al. gpt-oss-120b & gpt-oss-20b model card. arXiv preprint arxiv:2508. 10925, 2025.
Pranjal Aggarwal and Sean Welleck. LI: Controlling how long reasoning model thinks with reinforcement learning, arXiv preprint arXiv:2503.04667, 2025.
Anthropic. Claude 4.1 Opus. htttps:/ccaudde 2025. Generative Chatbot. Response to prompt [Date of Access).
Peter Belcak, Greg Heinrich, Shizhe Diao, Yonggan Fu, Xin Dong, Saurav Muralidharan, Yingyan Celine Lin, and Pavlo Molchanov. Small language models are the future of agentic ai. arXiv preprint arXiv:2506.02153, 2025.
David B. Blumenthal, Sébastien Bougleux, Anton Dignös, and Johann Gamper. Enumerating dis- similar minimum cost perfect and error-orreccing bipartite matchings for robust data matching. Inf. Sei., 596:202-221, 2022. TRLLhttps://di..orrs 10. 101 6 inn...in.... 2022. 03. 01
Bradley Brown, Jordan Juravsky, Ryan Ehrlich, Ronald Clark, Quoc V Le, Christopher Ré, and Azalia Mirhoseini. Large language monkeys: Scaling inference compute with repeated sampling. arXiv preprint arXiv:2407.2 1787, 2024.
Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-end object detection with transformers. In European conference on computer vision, pp. 213-229. Springer, 2020.
Xingyu Chen, Jiahao Xu, Tian Liang, Zhiwei He, Jianhui Pang, Dian Yu, Linfeng Song, Qiuzhi Liu, Mengfei Zhou, Zhuosheng Zhang, et al. Do not think that much for 2+ 3=? on the overthinking of ol-like llms. arXiv preprint arXiv:2412.21118
2024a.
Xinyun Chen, Ryan Andrew Chi, Xuezhi Wang, and Denny Zhou Premise order matters in reason- ing with large language models. In Forty-first International Conference Machine Learning, 2024b. httttttss///ppereeee
net fformm? 1d=4zAHgii00g
Tri Dao. Flashattention-2: Faster attention with better parallelism and work partitioning. arXiv preprint arXiv:2307.0880 2023.
Hugging Face. Open r1: A fully open reproduction of deepseek-rl, January 2025. URI https: hopen-pnn
Soumya Suvra Ghosal, Souradip Chakraborty, Avinash Reddy, Yifu Lu, Mengdi Wang, Dinesh Manocha, Furong Huang, Mohammad Ghavamzadeh, and Amrit Singh Bedi. Does think- ing more always help? understanding test-time scaling in reasoning models. arXiv preprint arXiv:2506.04211 2025.
Google. Gemini 2.0 flash thinking mode 2024.
December Decebrr URL https: htt.://ww..................aaaa.
thinking-mode
Google.
Gemini 2.5 (geminn-2.5-prooprevieee) httes///boe..ooole//ccho...o00
google-deed oet.e://eeeo..iiiaaaacraaoaaaaaa 2025a. Accessed: 2025-04-22.

---

Published as a conference paper at ICLR 2026
Google. Gemini 2.5: Our most intelligent ai model. https////bbooooooo//////coooo///
#gemini-2-S-thhnking March 2025b.
Etash Guha, Ryan Marten, Sedrick Kch, Negin Raoof, Georgios Smymis, Hritik Bansal, Marianna Nezhurina, Jean Mercat, Trung Vu, Zayne Sprague, et al. Openthoughts: Data recipes for reason- ing models. arXiv preprint arXiv:2506.04 04188, 2025.
Daya Guo, Dejian Yang, Haowei Zhang. Junxiao Song. Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. Deepseck-rl: Incentivizing reasoning capability in Ilms via reinforcement learning. arXiv preprint arXiv:2501. 12944889......... 2025.
Nathan Habib, Clémentine Fourrier, Hynek Kydlīček, Thomas Wolf, and Lewis Tunstall. Lighte- val: A lightweight framework for Ilm evaluation, 2023. URL https: // com/
buugiingface/1ighttev
Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. Measuring mathematical problem solving with the math dataset. arXiv preprint arXiv: 103.03874, 2021.
Pin-Lun Hsu, Yun Dai, Vignesh Kothapalli, Qingquan Song. Shao Tang, Siyu Zhu, Steven Shimizu, Shivam Sahni, Haowen Ning, and Yanning Chen. Liger kernel: Efficient triton kernels for Ilm training. arXiv preprint ar ArXiv:2410. 10989, 2024.
Xiao Hu, Xingyu Lu, Liyuan Mao, YiFan Zhang, Tianke Zhang, Bin Wen, Fan Yang, Tingting Gao, and Guorui Zhou. Why distillation can outperform zero-rl: The role of flexible reasoning. arXiv preprint arXiv:2505.2 2025.
H. W. Kuhn. The hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2(1-2):83 97, 1955. doi: /t00///0..//..000000...00000000 //onlinelibraa wiley. wwt.ccoo/aao http://o///ooo//...o./oo
URL https:
Hunter Lightman, Vineet Kosaraju, Yura Burda, Harri Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. Let's verify step by step, 2023. URL
https: /tt/aaiivv..../////// org/ abss 2305.20050
Zichen Liu, Changyu Chen, Wenjun Li, Penghui Qi, Tianyu Pang, Chao Du, Wee Sun Lee, and Min Lin. Understanding rl-zero-like training: A critical perspective. arXiv preprint arXiv:2503.20788 2025a.
Zichen Liu, Changyu Chen, Wenjun Li, Penghui Qi, Tianyu Pang, Chao Du, Wee Sun Lee, and Min Lin. Understanding rl-zero-like training: A critical perspective. arXiv preprint arXiv:2503.20783, 2025b.
Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint 11110:1711.01101, 2017.
Matthias Minderer, Alexey Gritsenko, Austin Stone, Maxim Neumann, Dirk Weissenborn, Alexey Dosovitskiy, Aravindh Mahendran, Anurag Amab, Mostafa Dehghani, Zhuoran Shen, et al. Sim- ple open-vocabulary object detection. In European conference on computer vision, pp. 728-755. Springer, 2022.
Niklas Muennighoff, Zitong Yang, Weijia Shi, Xiang Lisa Li, Li Fei-Fei, Hannanch Hajishirzi, Luke Zettlemoyer, Percy Liang, Emmanuel Candès, and Tatsunori Hashimoto. sl: Simple test-time scaling. arXiv preprint 2rXiv:2501.1993 2025.
Shen Nie, Fenggi Zhu, Zebin You, Xiaolu Zhang, Jingyang Ou, Jun Hu, Jun Zhou, Yankai Lin, Ji-Rong Wen, and Chongxuan Li. Large language diffusion models. arXiv preprint 00X0:92502.009992. 2025.
OpenAl. Learning to reason with 1ms. September 2024. URL BRR.ttp:://ppaa..cm learning-to- reason-wit th 1mm//
12

---

Published as a conference paper at ICLR 2026
OpenAL. Introducing openai o3 and 04-mini.
http:://openaa.ccommmmm
-00--03--n------m--m introducing--3- and- and-04-mina/ 2025.
Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever.
Language models unsupervised multitask learners. 2019. URL https://api.
semanticscholar: wwtp/orpsscooauoooooooooo 160025533
David Rein, Betty Li Hou, Asa Cooper Stickland, Jackson Petty, Richard Yuanzhe Pang, Julien Di- rani, Julian Michael, and Samuel R Bowman. Gpqa: A graduate-level google-proof q&a bench- mark. In First Conference on Language Modeling, 2024.
Rulin Shao, Shuyue Stella Li, Rui Xin, Scott Geng, Yiping Wang, Sewoong Oh, Simon Shaolei Du, Nathan Lambert, Sewon Min, Ranjay Krishna, et al. Spurious rewards: Rethinking training signals in rlvr. arXiv preprint arXiv:2506. 000006600947. 10947, 2025.
Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang. Y. K. Li, Y. Wu, and Daya Guo. Deepseekmath: Pushing the limits of mathe- matical reasoning in open language models, 2024. URL
03300
Guangming Sheng, Chi Zhang, Zilingfeng Ye, Xibin Wu, Wang Zhang, Ru Zhang, Yanghua Peng, Haibin Lin, and Chuan Wu. Hybridflow: A flexible and efficient rlhf framework. arXiv preprini arXiv: 2409.192000 2024.
Prime Intellect Team, Sami Jaghouar, Justus Mattern, Jack Min Ong, Jannik Straube, Manveer Basra, Aaron Pazdera, Kushal Thaman, Matthew Di Ferrante, Felix Gabriel, Fares Obeid, Ke- mal Erdem, Michael Keiblinger, and Johannes Hagemann. Intellect-2: reasoning model trained through globally decentralized reinforcement learning, 2025. URL https://arxiy: org/
abs/2505 07291
Alexandre Verine, Florian Le Bronnec, Kunhao Zheng, Alexandre Allauzen, Yann Chevaleyre, and benjamin negrevergne. Improving diversity in language models: When temperature fails, change the loss. In Forty-second International Conference on Machine Learning, 2025. URL https: 7/openrevi ew. nor/forum?i ds
Haozhe Wang, Qixin Xu, Che Liu, Junhong Wu, Fangzhen Lin, and Wenhu Chen. Emergent hi- erarchical reasoning in Ilms through reinforcement learning. arXiv preprint arXiv:2509.03644 03646, 2025a.
Shenzhi Wang, Le Yu, Chang Gao, Chujie Zheng, Shixuan Liu, Rui Lu, Kai Dang, Xionghui Chen, Jianxin Yang, Zhenru Zhang, et al. Beyond the 80/20 rule: High-entropy minority tokens drive effective reinforcement learning for I1m reasoning. arXiv preprint (xXiv:2506.0193) 2025b.
Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoe Le, Ed Chi, Sharan Narang, Aakanksha Chowdh- cry, and Denny Zhou, cofficonsis improves chain of thought reasoning in language models. arXiv preprint arXiv:2203. 11171, 2022.
Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi. Quoe V Le, Denny Zhou, et al. th-toutt prompting elicits reasoning in large language models. Advances in neural information processing systems, 35:24824-2488 2022.
Hao Wen, Yifan Su, Feifei Zhang, Yunxin Liu, Yunhao Liu, Ya-Qin Zhang, and Yuanchun Li. Para- thinker: Native parallel thinking as a new paradigm to scale Ilm test-time compute. arXiv preprint 04544455000 2025.
XAI. Grok 3 beta the age of reasoning agents. httts::// http::////.aai/ee////////////
February 2025.
An Yang, Anfeng Li, Baosong Yang. Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao. Chengen Huang, Chenxu Lv, et al. Qwen3 technical report. arXiv preprint 0988988005 2025a.
Xinyu Yang, Yuwei An, Hongyi Liu, Tianqi Chen, and Beidi Chen. Multiverse: Your language mod- els secretly decide how to parallelize and merge generation. arXiv preprint arXiv:2506.099 09991, 2025b.
13

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Shunyu Yao. Dian Yu, Jeftrey Zhao, Izhak Shafran, Tom Griffiths, Yuan Cao, and Karthik Narasimban. Tree of thoughts: Deliberate problem solving with large language models. Ad- vances in neural information processing systems, 36: 1809-11822, 2023.
Yixin Ye, Yang Xiao, Tiantian Mi, and Pengfei Liu, Aime-preview: A rigorous and immedi- ate evaluation framework for advanced mathematical reasoning. https://githun: com/ 2025. GitHub repository.
MAIR-NLP/AIME-PrevZ2
Qiying Yu, Zheng Zhang, Ruofei Zhu, et al. Dapo: An open- Ilm reinforcement learning
system scale, 2025. htt.l//tgg.//aaa.....//////2//.0 14476
Yang Yue, Zhiqi Chen, Rui Lu, Andrew Zhao, Zhaokai Wang, Shiji Song, and Gao Huang. Does inforcement learning really incentivize reasoning capacity in llms beyond the base model? arXiv preprint arXiv:2504 /3837, 2025.
Di Zhang, Xiaoshui Huang, Dongzhan Zhou, Yuqiang Li, and Wanli Ouyang. Accessing gpt-4 level mathemattcal olympiad solutions via monte carlo tree self-refine with llama-3 8b. arXiv preprint arXiv:2406.07339. 2024.
Xuechen Zhang, Zijian Huang, Chenshun Ni, Ziyang Xiong, Jiasi Chen, and Samet Oymak. Mak- ing small language models efficient reasoners: Intervention, supervision, reinforcement. arXiv preprint arXiv:2505.07961, 2025.
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A APPENDIX
A.1 LLM USAGE IN PAPER
We used ChatGPT (OpenAI, GPT-5) in September 2025 for occasional language polishing only. This is done when we really wanted to make sure there are no grammatical errors in few sen- tences. No text, code, experiment results, or figures were generated by the LLM. We made our own hypothesis, completed all technical content on our and made our conclusions. We also verified all outputs by ourselves. The occasional grammar check done by typing into LLM chatbox.
A.2 LEARNED MATCHINGS BETWEEN GLOBAL FORKING TOKENS AND TRACES
Initially, any of the bipartite matching configuration d Sp can be computed as optimal, the reserved global forking tokens {g( have specific correlations with these traces {r() j==1-
Jt=
This be observed in Figures 6 and g that c(orx), the count of configuration o being optimal during training, uniformly increases on all configuration indexes. However, as training goes with SSFT, notice only subset of Gp accumulates mass. This indicates there some unique correlations learned between
{g( Ji=1 and (r(5) j== This subset is denoted as Gp = and we call the unique edges S as learned matchings.
A.2.1 How TO CHOOSE THE GLOBAL FORKING TOKEN FOR PASS
To find 6p: we can simply track which configurations still accumulate mass in the last epoch. Then we can connect all the unique edges in Gp to visualize learned matchings. However, multiple global forking tokens may share the maximum number connected edges the learned matchings. To break the tie, we treat the counts as edge weights and select (x with the largest weighted degree. We provide this implementation in our code.
A.2.2 MORE VISUALIZATIONS ON LEARNED MATCHINGS
<think 1>
<think
OSS-high
<think 1>
OSS-high
<think
OSS-high
<think
medium
<think 2>
OSS-medium
<think 2
055-medium
<think 4
DeepSeek
<think 3>
R1
<think 3>
<think
Gemini
<think 4>
Gemini
<think 4>
Gemini
<think
(a) Visualization of learned matchings after SSFT with random matching. The fully connected graph shows no correlations were learned
(b) Visualization of learned matchings after SSFT with optimal matching between {g { and
(c) Visualization of learned matchings after SSFT with optimal matching between ( ) and
(r00)
(r(0))
Figure 8: These learned matching visualizations are obtained by connecting edges in the subset of configurations {arx 1P that still accumulate mass towards the end of training in Figure 6 These models are fine-tuned using the same GPT-OSS-high, GPT-OSS-medium RI, and Gemini traces for each question, so can better interpret the learned matchings
A.3 GLOBAL FORKING POLICY OPTIMIZATION FOR P INFERENCE BY SAMPLING (1)
For single-path generation, we prompt the ( most likely correlate with a complex reasoning mode using graph-based heuristic that selects the g()) matched the largest number of distinct reasoning traces. Since theoretically justifying any single heuristic is difficult, we apply small amount of RL to learn the optimal distribution global forking tokens conditioned each
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question. Using cold-start SFT before RL is common practice in LLM post-training (Guo et al. 2025 and recent work shows that RLVR alone does not incentivize new capabilities (uue et al,
2025
Liu et al. 2025a).
Setup. We use 1,280 questions from DAPO-Math-17k and GRPO for 10 steps using only the policy gradients of g( These are referred to GFPO. We also let a baseline op- timize full tokens even though that is much more computationally expensive. The base SFT models include SSFT-32B, SSFT-32B (random σ). STT--mieedddistil--22B---- trained using 511k-4mi xed reesonie Hyperparameters are in Table
Results. In Table 4 we show that SSFT instills diverse, high-accuracy reasoning modes without mode collapse, and that merely 10 steps of GFPO can shape the output distribution of g()) conditioned on x. By comparing against baselines with RL, we that the gains come from the synergy between GFPO and the global forking tokens learned by SSFT with a principled set loss, rather than from spurious rewards (Shao al. 2025). our setting, SFT-GRPO that optimizes all tokens still cannot match SSFT SSFT-GFPO, despite using substantially more compute.
Comparison of SSFT and baselines on three math reasoning tasks after further RL fine- tuning only on global forking tokens, denoted as GFPO. We also include an expensive GRPO base- line that optimizes the full sequences. SFT/SSFT uses s4k mrixed-reasoning data, and GRPO/GFPO uses only 1280 questions from DAPO-Math-I7 17K (Yu et al.) 2025). The optimal <think i> for each question is sampled by the models in this evaluation. By avoiding reasoning mode collapse ob- served in standard SFT-mixed-distinn and SSFT (random or) with random matching, SSFT preserves complex, aechaaccuaccy reasoning modes. Standard GRPO that applies gradients to full sequences does not improve performance and can even lead slight drop in this small-scale RL setting.
Table 4:
AIME 2024
AIME 2025
MATH-500
LCB(v5)*
Pass @ 1: Averuge performance of 32 individual generations (AIME24/25, MATH--500
WrT-mixed-distill 3-B Ba-B-ttas-Ga tags-GRPO
58.85
52.40
88.85
37.13
59.80
54.06
88.98
37.72
SSFT-32B (random ar)- GFPO
59.58
53.96
89.87
36.53
SSFT-32B- GFPO
64.22
58.80
89.90
42.10
A.4 ROBUSTNESS TO FINE-TUNING WITH CODE GENERATION TRACES
We investigate the advantage of SSFT over SFT when the fine-tuning dataset consists of traces from open-ended domains, such as code generation with unit cases. We study whether SSFT can improve P on competitive coding by instilling specialized coding reasoning modes without mode collapse.
Setup. We seleet IK coding questions from Then-hhoughhs (Guha et al. 2025). use DeepSeek-RI to generate four reasoning traces per question to construct code 1k www.mmiiiiiimmmmmmmmmmm reasoning. We apply SSFT on this dataset, and then GFPO on 1 questions from Intellece --- (Team et al., 2025).
At test time, each model samples <think i> for each question. For evaluation, LiveCodeBench(yS) and assess whether the models can generate valid code that passes unit tests when executed. Although the evaluation still relies the resulting code far more open-ended than in math tasks, where only a final numeric answer is extracted and scored. We still included the math tasks for OOD evaluation.
verifiable metric that avoids evaluation bias,
Results. Table 5 reports the full comparison between SSFT and baseline methods under fine- tuning with oode-generation traces. SSFT outperforms the baselines on LiveCodeBench and shows stronger generalization to the math domain, despite being fine-tuned only on code-generation traces. These results show the robustness of SSFT to using traces collected from more open-ended tasks. Although correctness and diversity are less clearly defined for code generation, SSFT instills diverse and specialized reasoning modes, and RL quickly leverage these find optimal forking path for each coding task. fact, the Pass gain on LCB after SSFT with code traces is slightly larger than the gain math tasks after SSFT with math traces (Table T). One explanation is that
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open-ended tasks involve much larger search space, and global forking tokens help the model avoid getting stuck in suboptimal reasoning modes.
Table 5: Comparison of SSFT and baselines on one coding benchmark, and three math reason- ing tasks that are out-of-distribution relative to the fine-tuning set, since the fine-tuning dataset codelk- 111-4mixedh deexxed-eeeee x contains only coding questions and traces. All methods use Qwen2.5-52B-Imstrucc as the base model. SFT-mixed-distie denotes standard SFT where differ- traces for the same prompt treated independent training examples. SSFT (random o) denotes SSFT with a random bipartite matching at each training step. All the three models have gone additional small-scale GFPO for optimal forking selection with 1000 coding questions from Intellect-2-Re At test time, <think i> is sampled by the models.
LCB(v5) AIME 2024"
AIME 2025'
MATH-500"
@ass@1 avg over n=16 for LCB, n=32 for math tasks. (* denotes out out-of -fi--iibittiiiiionnn relative to fine-tuning)
Swen2.5-32B-IInttee A
23.35
15.80
10.40
80.40
SFT-mixed-distiee -is-dddi-itttoooooooeoo *
47.13
34.69
24.17
89.39
SSFT-32B-code (random σ)) *
45.36
39.06
31.56
89,46
SSFT-32B-code *
52.07
43.23
32.82
89.96
Pass I of Cons@6; Average performance of majority voting with 6 parallel generations
SFT-mixcd-distill -------- *
41.52
30.30
91.70
SSFT-32B-code (random σ))
*
46.97
38.18
92.43
SSFT-32B-code *
51.82
37.88
93.25
Cons@32. 32: Majority voting performance with large number of parallel generations
SFT-mixed-disse -istsedddditti-l-3--3 -essilll-22---e -odec *
46.67
33.33
91.40
SSFT-32B-code (random σ)) *
46.67
43.33
92,00
SSFT-32B-code *
56.67
46.67
93,00
A indicates training with single-target data and indicates training with multi-target data.
A.5 ROBUSTNESS TO MODEL FAMILY AND SIZE
Setup. We use Qwen3-4B- -aae and Llama3. --s-rustt as additional base models, repeating the main experiment from Section Results are reported in Tables and 7 respectively.
smaller 4B model from the newer Qwen3 family Yang et al., 2025a), and Table shows that it is also robust on an 8B model from the older -- However, the performance gains are smaller than those observed with the larger Qwen2.5-32B.
Results. Table 6 shows that SSFT remains robust
family, achieving higher scores most reported benchmarks.
A.6 GENERALIZATION DOMAINS UNSEEN DURING SSFT.
SSFT using math traces - Coding. We follow Wang 2025b) and study out-of-distribution generalization to code generation using LiveCodeBeneht V5) as the evaluation task after fine-tuning with only math data slk- www-4mixddrreassniieee Tables and 4 we see that SSFT improves OOD generalization by about 5% over the baselines and about 19% over the base model in the 32B setting. For the model, Table 6 shows that is about 1% lower than one baseline, but still improves over the base model by about 7%.
1.28k questions from DAPO-Math-1 17k. In
SSFT using coding traces + Math. Since we fine-tuned models with only code generation traces, we also study the other direction. AIME24/25 and MATH-500 are used as OOD tasks. Table 5 shows that SSFT yields significant improvements the baselines across all math tasks, indicating that the advantage on LCB transfers to math reasoning, 5
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Table 6: Comparison of SSFT with baselines on three math reasoning tasks, where all methods use Qwen3-4B-Bast as the base model and are trained on the mixed-distill denotes standard SFT where different traces for the same prompt are treated inde- pendent training examples. SSFT (random σ¹) denotes SSFT with random bipartite matching. For Pass@l, SSFT uses graph-based heuristic: prompts the forking token, <think i>, with the largest number of connected edges in the learned matchings.
ieg.onnnnnn....... SFT-
AIME 2024
AIME 2025
MATH-500
LCB(v5)
Pass 1: Average performance of individual generations. LCB is OOD relative to the fine-tuning data.
Dwen3-4B-Bause
9.79
6.56
66.34
7.19
SFT-mixed-disse distill-Qwen3344BBB tagst
21.46
21.98
78.76
15.57
SSFT- weFT-QWen334BB (random o>
23.33
21.87
79.63
12.57
SSFT- SSFT-Qwen3-44+
23.44
23.13
81.85
14.37
Cons@d Averuge performance of majority voting with 6 parallel generations
Qwen3-4BB 40--Bace
13.03
10,91
78.63
-iimiiidddditill--eeett --t SSFT-Qwen3-4B (random o>
30.61
28.18
89.67
30.91
28.79
89.73
32.42
28.79
91.20
Cons @ 32: Majority voting performance with large number parallel generations
Qwen3 4
20.00
16.67
83.00
SFT-mixed-disse distill-Qweett Qwen3-4B--B--..e -eeen3--B-tags*
36.67
40.00
91.80
SSFT-Qwen3-4B (random o> S
43.33
36.67
91.60
40.00
43.33
91.80
A indicates training with single-target data and * indicates training with multi-target data.
Table 7: Comparison of SSFT with baselines on three math reasoning tasks, where all methods use Llama3.1 I-8B-Instruct as the base model and are trained on the s SFT-mixed-distill denotes standard SFT where different traces for the same prompt are treated as independent training examples. SSFT (random σr) denotes SSFT with random bipartite matching. For Pass@1, SSFT uses a graph- based heuristic: prompts the forking token, <think i>, with the largest number of connected edges in the learned matchings; result, <think 4> is selected for SSFT-Llama3. 1-8B and <think 1> for SSFT-L.aama31 1-88 (random σ).
ing.
AIME 2024
AIME 2025
MATH-500
Model
Pass @ 1
Cons@32
Pass@1
Cons@32
Pass @
Cons@32
3.65
13.33
0.94
6.67
47.53
65.40
DTT-mixcd-diistil--lamss 1-8B
4,90
13.33
4.90
10.00
61.95
80.00
SSFT-Llama3. 1-8B (random or)
6.77
13.33
4.79
10.00
62.22
79.60
SSFT-Llama3.1 1-8B
6.98
16.66
5.79
20.00
62.04
80.00
indicates training with single-target data and indicates training with multi-target data.
A.7 EXTENDED RELATED WORK.
Aggarwal & Welleck (025); Zhang et (2025) are seminal works that RL learn policy controlling how long a reasoning model thinks. During RLVR, Aggarwal & Welleck] (2025) appends an explicit "Think for n tokens" instruction the input prompt and adds length penalty based on the exact deviation. Zhang et al. (2025) instead appends a higher-level length description and adjusts the length penalty accordingly. We list the differences from our method below, though we believe our method is complementary to many of these RL works, since RLVR requires strong base model (Yue et al. 2025), which SSFT aims to provide.
Our focus on diversity and correctness vs. their focus on efficiency. They first enforce a relatively small token budget, < 4096, and then study accuracy. Our work instead aims address the plateaued or lower performance caused by overthinking. Even one particular reasoning mode does not improve with longer CoT, other reasoning modes can provide distinct and still potentially
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long COT. We use the distribution of thinking counts unbiased metric, in addition to accuracy differences, to assess diversity. We think length is a good diversity indicator for us because SSFT does not explicitly impose any length regularization. Controlling reasoning length through global forking tokens is an emergent behavior, in contrast to these RI. works that use explicit length penalties in their objectives.
Experiment settings. Our SFT-OpenR1-93k-77 is strong enough baseline scoring 46.15% on AIME24, compared to 26% by LI- LB-Exact-7B (Aggarwal & Welleck (025). and 30% by TLDR-7B-44 (Aggarwal & Welleck 2025). Part of the difference likely comes from setup asymmetries: our baseline does not constrain the token limit and uses teacher distillations, whereas these works rely on -easSeek-RI-istillle
a base model that has gone through SFT using private data. So we would need to run non-trivial number of experiments to first verify whether their length-controlled optimization remains competitive at much larger token budgets (at Ieast 15,000), and whether their results hold with a base SFT model fine-tuned with public data. Even then, the research question would be whether their RL methods can make SSFT unnecessary, so we would need to compare SSFT plus their method against SFT plus their method. We plan to investigate the synergy between an SSFT base model and different RL methods in future work.
A.8 ALGORITHM: SSFT IMPLEMENTATION
Algorithm D presents the core SSFT implementation with optimal bipartite matching. In practice, the nested-loop computation used to populate C is fully vectorized and can be executed in single forward pass. This does not blow up VRAM because (i) we do not store activations for these cost evaluations (no backprop through matching costs), and (ii) We only need to the first TE < Tr NTP losses to compute the matching cost, as the NTP loss over the first few thousand tokens can already differentiate many reasoning traces in terms of their modes. Nevertheless, our code also supports matching over the full Tr tokens by chunking the computation into a few batches, so this step does not become a VRAM bottleneck. Fine-tuning on 1k questions with traces each, SSFT (optimal matching) took 6.5 h for 6 epochs, compared to 6.1 h for standard SFT, adding only a small overhead.
To support training with variable-size parallel generations, we propose simple and scalable imple- mentation that incurs no additional VRAM overhead. Rather than concatenating diverse reasoning traces along the sequence dimension, which would require complex sequence parallelism for mem- ory efficiency, our algorithm expands variable-size parallel generations along the batch dimension under distributed training (Appendix A.8)
The primary VRAM bottleneck in SSFT remains the backpropagation Step g regardless of whether use optimal or random matching, because the effective batch size scales with M. To mitigate this, we split the backward pass into several gradient-accumulltion steps. Although our experiments use the same number of reasoning traces per question, we also support variable number of targets using our gueue-based batching in Algorithm 2 The complication arises when using distributed training with a variable-sized batch, as different processes require the same per-device batch size to perform collective operations. This is mitigated by padding with PAD" sequences to align batch sizes. Our implementation minimizes the number of "PAD" sequences by storing a variable number of targets in queue and dequeuing multiple items to form per-device global batch, so smaller batches can be stitched together instead of always being padded.
A.9 TRAINING DETAILS
A.9.1 TRAINING DATASETS
32B experiments with questions from sl(Miin): We explain the process of generating our training
dataset for experiments in Table I Figure 3 Figure E Figure 5 Figure 6 First, we use the 1000 questions from sl MMennighoff et al. 2025) and populate a pool of reasoning traces by distilling from PPT-OSS--20B high reasoning, GPT-OSS-- 1DSS-120-mmaa 2200-medium reasoning, DeepSeek R1, Gemini Flash2.0 Thinking, and Claude4/4.1. We use temperature 1.0, maximum length of 32768, and sets high reasoning effort unless specified. We generate two traces per teacher model. We use Claude3.5
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Algorithm Set Supervised Fine-tuning (SSFT)
Require:
Tg: base model
N: Number of global forking tokens {g (g)
Ji=1
D: Dataset with (at most) M reasoning traces per question
B: Global batch size
Tl: The first TL number of tokens to match in matching cost (Equation 5
Ensure: Output
for each training step do for k: 1 B do
2:
3: Sample input prompt and the corresponding reasoning traces (xxs (r( D
}z=1)
4:
Initialize cost matrix C NMM for i 1,..,, do
5:
6:
for j = 1, ..., M do Compute the matching cost between g(() and r by Equation 5
7:
Tc
Lematching (8 -sg
log no Fe Xk: 8 (0)
(i)
(5)
8:
Store the matching cost in C.
C(i,j)
mattchnng (0))) rkk
(1)
(6)
9:
end for
10:
end for
Compute optimal matching o% between (g()) (N) and (r(() M
1==1- Hungarian algorithm
Kuhn 1955
can be applied to C to efficiently compute Equation 7 Equution
M
o = arg min o
C(a()).))
12:
end for
13:
Compute the empirical set language modeling loss (Equation 8):
B
M
T. log no (c (e2lxx.g(oud
Liuungrrian (0) =
k,ttt)
.))
(8)
B
14:
Update model parameters using gradients V Hungariinn (0) end for
15:
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Algorithm 2 Queue-based Distributed SSFT with variable number of traces for each question
Require:
Ng. base model
Number of global forking tokens {g (
D: Dataset with (at most) M reasoning traces per question, variable m number of traces per
question
Global batch size
TL: The first TL number of tokens to match in matching cost (Equation 5
b: Original per-device global batch size (b 2 M), This is "micro batch s original grad accumulation steps"
Ensure: Output To
1: for each epoch do
2: Initialize Queue Q for storing sequence of (Xk, (rx 1m where m is variable number that differs between input questions and different processes (GPUs)
3: Initialize Queue q for storing sequence of sizes of sets
4: for every (xk: {rk m 1-=1) do
5:
Q + Q.enqueue((xx: (rk mm )
6:
f g.enqueue(m) all-q = All-gather(g)
7:
8:
Initialize list temp.
9:
while All processes have at least b sequences based ali-q do while temp batch does not have at least b sequences do
10:
11:
temp batch temp buteh.oppend( Q (deequeue() q.deque()
12:
13:
end while
14:
compute the the maximum per_device global batch size bmaz currently in all pro- cesses using all Jist (inferred. no collective operation)
15:
Pad temp_eetaae batch to size bmaz by appending "pad sequences" as needed. Update all entries in all q list based on inferred usage
17:
Perform one SSFT training step, SSFT (mo, lemp. batch, bmax)
end while
19:
end for
20:
end for
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Sonnet to extract the answer from the distilled solutions and compare with the ground- -oound-truth answer. The correctness of these distilled traces are shown in g
dataset, sample 4 traces per question from this pool, the dataset consists of 1000 questions, each paired with reasoning traces. This dataset was used to fine-tune SSFT-32B, SSFT-32B (random σ), and SFT-mixedd-iistll-3333
For slk-4mixed-reason
For fine-tuning SFT-OSSS-distll-32B model, we only use the 1000 traces from "Runl" GPT-OSS- 120B with high reasoning effort.
For obtaining the visualizations in Figure Figure 8 we fine-tune models using the same teacher models for all 1000 questions. we always choose the 4 traces from "Runl" of GPT-OSS-1209 1200S-hh -hhh,,hh, GPT-OSS- CELOSS--12B-meee 20B-medium, DeepSeek R1, and Gemini Flash2.0 Thinking. As mentioned Section
3.3 and Figure we fine-tune under 3 bipartite matching hyperparameters conduct this case study.
GPT-OSS- 120B-high
GPT-OSS- 120B -eB--medumm
DeepSeekR1
Gemini
Claude Opus4/4.1
Flash2.0
Thinking
Runl 796/1000 Run2 785/1000
769/1000
620/1000
538/1000
656/1000
753/1000
641/1000
545/1000
647/1000
Table 8: The number of correct reasoning traces distilled for the 1,000 questions in s by different teacher models. This evaluation is done by Sonnet comparing the predictions and the ground-truth answers. We see that GPT-OSS has the highest accuracy for sl dataset.
A.9.2 SSFT TRAINING DETAILS
For consistency, we use Qwen2.5 1-w.5----B---n-rue (Yang et al. 2025a) as the base model for all of our 32B experiments. We use standard fine-tuning hyperparameters: batch size of 32, which is derived from 4 gradient accumulation steps and distributed training with 8 GPUS (4 x 8 32). This results in 756 gradient steps. The maximum sequence length is set to 32,768. We train with 6 with a learning rate of l - 5 warmed up linearly for 5% and then decayed to 0 using cosine schedule. We choose AdamW optimizer (Loshchilov & Hutter) 2017) with 31 = 0.9, 32 = 0.95, and weight decay 1e 4. We only backpropagates the completion loss. which the loss on reasoning traces and the answers. Fine-tuning SSFT-32B plus loggings took 6.5 hours on 8 NVIDIA B200 GPUs using Py PyTocch FSDP, Liger Kernel (Hsu et al., 20244 for fused cross entropy loss, and FlashAAtentiinn22 2 (Dao 2023) for fused attention computation. Fine- tuning SSFT-32B (random σr) took 6.3 hours, and Fine-tuning hours. Even our baseline WUT--SS-distill-- reproduce s1.1 took 1.66 hours, which is longer than the time reported by Muennighoff et al. (2025). This is due to using GPUs instead of 16 GPUs, hardware and package differences. When training with s 1lk-4mixed-reasonin 4x reasoning traces, but we did not linearly increase the number of epochs, as these traces can be similar, and the number of distinct questions is still 1,000. Overall, made sure all of our models are fine-tuned with consistent hyperapameters.
train for epochs with global
STT-mixed-distil--3
took 6.1
with only one trace per question, and our attempt to
added one extra epoch from 5 epochs to 6 epochs, since
have
A.9.3 VISUALIZZTIONN OF SSFT TRAINING DYNAMICS
Figure 9 shows the standard training dynamics of SSFT with optimal bipartite matching. The result- ing model is SSFT-32B The loss plotted here is Equation
Figure shows the evolution of bipartite matching during SSFT. Figures IO shwlshow that the gap between optimal bipartite matching cost and non-optimal bipartite matching cost under other keeps widening during training. This means that these reasoning traces are indeed starting to match unique global forking tokens. Even though SSFT effectively optimizes non-stationary objective which depends on model parameters O, the wideming gap shows the inner discrete optimization is converging as training goes. Figure [0c also confirms that the model learned some unique correla- tions between {g()) _ and (10)) as only subset of matchings are still computed optimal.
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Training Loss
Learning Rate Schedule
Gradient Norm
x10-3
1.05
Gradient Norm 2
Leaminn Rate
Less
0.6
0.30
-0.05
Training Steps
Training Steps
Training Steps
Figure 9: Training dynamics of SSFT-322 -2B sik-4mixed-reasoning
Compared to Figure we see more a accumulating mass towards the end. This due to having more mixed diverse reasoning traces, so the model learned more intricate associations between these global forking tokens and truly diverse reasoning traces.
Gap between rest and 1st opt total cost
Maturing Contgg 10 (0..35M)
The gap between the optimal bipartite matching cost Equation and second smallest bipartite matching cost. The increase shows the global forking tokens preferring a specific matching.
(b) The gap between the optimal matching cost Equation and average of bipartite matching cost under other a. The increase shows the global forking tokens start preferring specific matching.
(c) Cumulative counts of o computed as optimal over training. We observe that only subset 6, accumulates mass towards the end, indicating some unique correlations learned between (g ( ) and
{r"")
Figure 10: Dynamics of bipartite matching during the fine-tuning of SSFT-32B s 41k--mieed- reasoning.
A.9.4 ABLATION STUDY TRAINING DETAILS (REMOVING HIGH QUALITY SMALL DATASET)
For this ablation study, choose Open- R MM-aht220k default split, which has 93,000 math ques- tions and 2 ~ traces. Since Qwen2.5-Matee Mathhh a widely fine-tuned model using this dataset, we also choose it our base model. We train for 3 epochs using A100 GPUs, which took around days. Our hyperparameters are mostly consistent with the recommended hyperparameters by Face (025). We fine-tune both SSFT-OpenR 193k-7B and SFT-OpenRI -93k--BB with maximum length of 32768, learning rate of 4.0e 05 warmed up linearly for 3% and decayed to 0 following cosine schedule, gradient accumulation steps. For our SSFT method, we reserve N 4 global forking tokens, and use the first 1000 tokens for matching. Again, only the completion loss is used for optimizing the model parameters.
A.9.5 OTHER TRAINING SETUP AND HYPERPARAMETERS
Table 9 shows the hyperparameters used for all of our key experiments.
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Table 9: Trainning details of SSFT and GFPO. -FT- optimizes over full sequences, whereas GFPO only optimizes <think i>, tokenized in 4 tokens. 8xB200 is used for full GRPO.
SSFT
GFPO (GRPO Forking Tokens)
Parameter
Value
Parameter
Value
Global Batch Size
32
Global Batch Size
128
GPUs
8xB200
GPUs
8xA 100
Max Sequence Length
32768
Max Generation Length
31000
Global Forking Tokens
Prompt Length
1024
M distillation targets
Number rollouts
L Matching Length
1000
PPO minibatch size
128
Learning Rate
le-5
Learning Rate
1c-6
Adam $1
0.9
Adam $1
Adam Sz
0.95
Adam 8g
0.95
Warmup ratio
5%
Rollout Temperature
0.7
Epochs
Epochs
KI. coeffcient
0.001
Training data for math models
sik weeixeeddeeeoonee
1280 questions from DAPO-Math-1l
sl questions with 4 mixed traces each.
Training data for coding models
code 1ooeckk-miin xexxeexeee rend-reasoniin ing Ik questions from Open-Thoughts 4 RI traces each
1280 questions from Intelloct-2-RI
Base models being used
SSFT-32B
TT--mixdd-disill-2BBBB
Owen2.55332-Innen Qwea3-4B (wen3-4BBBase Base
SSFT-32B (random σ²)
SSFT-32B-code (before RI.)
Llama3.1-80 -nsttrrruuctttt
SFT-mixad-distill-22----oo (hefore RL)
SSFT-32B (random --ccd (before RL)
Owwnn.55Matth-7-Insneee Separate training details and hyperparameters reported in Appendix A.9.4
A.10 EXAMPLES OF SSFT-32B REASONING TRACES INITIATED BY DIFFERENT GLOBAL FORKING TOKENS
We randomly picked set set six parallel generations from those where SSFT-32B correctly answered a problem from AIME 2025 under all six global forking tokens. This is question 11.
Problem 11 (AIME25)
A piecewise linear periodic function is defined by i E [-1, 2- i E (1,3),
x,
f(x)
and
f(x + 4) - fxx for all real
The graph of f() has sawtooth pattern. The parabola x 34y² intersects the graph of f(x) at finitely many points. The sum of the y-coordinates of these intersection points can be expressed in the form
+ bvC
d
where a, b, c, are positive integers, ged(a, b, d) and is squarefree. Find a a+b+otd.
Here are the 6 parallel generations by SSFT-32B. We distinct reasoning traces produced by prompting with unique g( [g(1) J The differences highlighted in red comments.
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<thi nk Periodization + Quadratic Families + Vieta/Telescoping
< <lim_start: >assistant
< <lim_start]
<thinkl>
"write 1 4k with & E [-1,3)" "Case (t [-1,1), y 1) ... Case
1 1 VI + 544k
(t E [13),
t)a 34/²
4h: and
"let
68
- and -1 273 + 544k
Thus - +1+ 5v
"telescopes
68
68
to rational constant 9/34" (/Comment: periodize = 1. + 4k; solve two quadratic
families, use Vieta pair-sums, edge-period gives the irrational y) </thinkl>
< <lim_start >answer
Answer: 259
<think 2> Nearest-Muliple-of44 Reduction
< inn_start >assistant
<lim_start
<think2>
Start from y = f(34y²)
write 34y² 4k + r with r E [-1,3) (choose
... Branch A (r [-1, y - 34y² y 4k 0 Branch B (r [1,3)) y = 2 r + 34y² + y - (2 + 4k) = ... Vieta: pair-sums + and
34y
cancel over full periods
only the edge/partial period remains (k- = 8, descending
-1 + 273 + 544-8
5v15
branch)
68
68
([Comment:) reduce E [-1,3) via nearest multiple of 4; full periods cancel by +
pair-sums; only the boundary k = 8 root contributes)
</think2>
< 1im start >answer
Answer: .... 259
<think 3> Paper-Style Sections; Explicit Pair-Sums and Totals
<| <lim_stare start >assistant
[im starstart|
<think3>
1+ 44225
"Now we need sum of all y-coordinates"
vl,n
UC
y
68
1=0
5V185 17
"hheefoore S2
"Total sum S ...."
([Comment:) formal
68
write-up; separates sums by family, explicitly displays the boundary /125 then totals)
68
</think3>
<lim_start >answer
Answer: 259
<think 4> Set/System Framing Before Algebra
< in >assistant
lim start
<think4>
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"the set ((x)) Intersecting with the parabola I = 34y² requires solving
y f(z), 34y (then proceeds the same two-case algebra)
-1 227 544.8
-1 185
u4
([Comment:) definition-first framing;
68
68
after setting up the system, it reaches the same boundary u4 value the partial period)
</think4>
<lim_starm starttrrt. >answer
Answer: 259
<think 5> Outline & Symmetry: Full Periods Cancel, Last Period Contributes
< <lim_start start >assistant
<|imstarrt
<think5>
"the sum is zero That's a key insight." ... "for each full period (k: 0 to k. 7). the sum is zero now consider the partial period = 8" Compute Upos (k=8) =
4662
(term-by-term cancellation examples for h: = 0, (/Comment:)
68
symmetry explains full full-period cancellation; the lone boundary contribution at supplies the irrational term)
</think5>
<|im__t start >answer
Answer: 259
<think Streamnof-Consscouuseess Scratchwork
< <lim_starts >assistant
<| <lim_start start
<think6>
"Okay, so need to solve this problem me try to break this down step by step."
1 + /662
"Wait, actually, no. Wait," VAS8 =
([Comment:) exploratory
68
self-checks; reaches the same boundary y on the partial period)
</think6>
>answer
Answer: 259
A.11 ALTERNATIVE IMPLEMENTATION OF TRAINING WITH MULTI-TARGET SEQUENCES
Figure shows an alternative implementation for training with multiple target parallel sequences. Instead of flattening multi-target data along the batch dimension, reasoning traces along the sequence dimension, modify the causal attention matrix, and position ids for positional embeddings. While easy to implement, this version extends the sequence length and cannot perform gradient accumulation along the batch dimension to maintain the same VRAM fine-tuning already the micro-batch size
also concatenate parallel
training with single reasoning trace. Since to due to the increased number of reasoning tokens already present in single reasoning path, this implementation also cannot further shrink the micro-batch size accommodate the extended context length. In addition, the code that uses the initial flash-attention implementation, which requires causal attention matrix, not compatible with this setup. terms of choosing the number of sequences in global batch size, this implementation is also less flexible.
26

---

Published as a conference paper at ICLR 2026
Prompt: that there infinitely many prime numbers.
LL
Xyrcest
<thinkt> </thinkl>-ttinkkt
R, < <tthink2>-thinkdt>
<fthinkls X
Asalstini grouf fratele mony prime, 11 wovegg....
thez draiik.....
soudies but enery openin is aufilte.
$3 ( 22 3 04 Ts 06)
Figure 11: An illustration of training with multiple parallel targets along the sequence dimension.
27


Reviews:
[{"review_id": "GFXsBiMlWo", "reviewer": "Reviewer_MjCW", "summary": "This paper introduces **Set Supervised Fine-Tuning (SSFT)**, a novel framework designed to make Large Language Models (LLMs) generate multiple, diverse, and correct reasoning paths simultaneously. SSFT reframes parallel reasoning as a \"set-of-next-token-prediction\" task. It employs a set of learnable **\"global forking tokens\"** and a **bipartite matching** loss function to associate these tokens with different reasoning solutions during training. This approach enables the model to produce varied reasoning patterns when prompted with different forking tokens, effectively overcoming the typical trade-off between diversity and accuracy. Experiments show that SSFT-trained models significantly outperform standard Supervised Fine-Tuning (SFT) on challenging reasoning benchmarks.", "questions": "- What is the expected performance of SSFT on much smaller (<7B) or larger (>70B) models?\n- Regarding the number of global forking tokens, N, I would like to explore the potential impact of its fixed setting. The inherent diversity of reasoning paths differs for any given problem. Is it possible that a uniform N value could create redundancy for simpler problems while failing to fully capture the diversity of more complex ones? I am curious about the authors' perspective on the feasibility and potential of dynamically adjusting N for each problem. For instance, could performance be improved by a mechanism that predicts or assigns an optimal N' value tailored to each specific question?\n- This paper brilliantly demonstrates how SSFT addresses mode collapse in SFT. An interesting parallel occurs in the reinforcement learning (RL) domain, where \"entropy collapse\" is also a core challenge—the model overfits to a few high-reward trajectories, thereby losing its exploratory capacity. I would like to ask the authors if they believe SSFT's core mechanism (i.e., using forking tokens and set matching to \"protect\" diversity) could offer new insights for solving entropy collapse in RL. Specifically, how might the ideas from SSFT be combined with or complement existing RL techniques aimed at maintaining policy entropy, such as entropy regularization?\n- I've noticed that the experiments in this paper are primarily focused on mathematical reasoning datasets. A notable characteristic of these datasets is that a single problem often has multiple, distinct, and structurally varied correct solution paths. SSFT has achieved great results by using N explicit forking tokens to capture this diversity. However, my question is whether this advantage can naturally generalize to other, non-mathematical domains. In many general-purpose scenarios (such as open-ended question answering, creative writing, or code generation), \"diversity\" may manifest in a more implicit and subtle manner, rather than as the several clearly distinct \"solutions\" seen in math problems. In these contexts, is forcing a match between N discrete forking tokens and this more continuous or ambiguous diversity still the optimal strategy? Could the performance of SSFT in these general-purpose domains be limited by this \"explicit forking\" design?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.  **High Innovation**: The SSFT framework and the concept of \"global forking tokens\" are highly original. Formalizing parallel reasoning as a set prediction problem with bipartite matching is a significant and clever departure from standard fine-tuning paradigms, effectively mitigating \"mode collapse.\"\n2.  **Effective and Elegant Method**: The use of the Hungarian algorithm for optimal matching allows the model to end-to-end learn associations between forking tokens and different reasoning styles without explicit labels. This design is both elegant and highly effective.\n3.  **Strong Empirical Results**: The SSFT model demonstrates superior performance on difficult reasoning benchmarks like AIME and MATH-500. The ablation studies are thorough, clearly proving the necessity of the optimal matching component and showing a better diversity-accuracy balance compared to baselines.\n4.  **High-Quality Presentation**: The paper is well-written and logically structured. The visualizations provide clear, intuitive evidence for the method's effectiveness and interpretability.", "weaknesses": "1.  **Implementation Complexity**: SSFT's training process, which requires solving a bipartite matching problem at each step, is more complex than standard SFT. This could pose a barrier for application on larger-scale models or datasets.\n2.  **Data Dependency**: The method's success hinges on access to a high-quality, diverse set of reasoning paths, which can be costly to obtain. The performance might degrade if such data is unavailable. But considering that this work explicitly aims to preserve the model’s diversity under such settings, this weakness is not particularly limiting.\n3.  **Heuristic for Single-Path Inference**: The strategy for choosing a forking token in single-pass (Pass@1) inference is based on a heuristic (the token with the most connections). This choice lacks strong theoretical justification and could be explored further.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YaSHKr1Juj", "reviewer": "Reviewer_6BW4", "summary": "This paper proposes Set Supervised Fine-Tuning (SSFT), which introduces global forking tokens prior to generating parallel reasoning traces and frames parallel reasoning as a set prediction problem. This method expands the diversity of generated traces and keeps the accuracy of the generation at the same time.", "questions": "1. Could you please include an analysis of out-of-distribution performance after training?\n2. Could you please include 1 or 2 more different sizes of models in the main experiment part?\n3. Could you please include 1 or 2 other model families to show the generalization of this method, such as Qwen3, Llama, or Phi?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The motivation of this work is reasonable, which points out the shortcomings of the current parallel scaling methods and the trade-off between diversity and accuracy.\n2. Based on the motivation in (1), the logic of the design of the framework and matching algorithm makes sense.\n3. The experiment part shows the effectiveness of this method compared to other baselines in both mathematical and STEM reasoning tasks. \n4. This work also introduces a way to deploy the method efficiently, which is a good engineering contribution, although this might not be the main focus of this paper.", "weaknesses": "1. Some obvious and important typos, such as Fine-tuning not Fine-turninig.\n2. The main result section might need more diversity on model families and model sizes. \n3. Maybe an analysis of the generalization ability of this method is needed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZNaMhgwp0Z", "reviewer": "Reviewer_Ut4D", "summary": "The paper treats parallel reasoning as a set-of-next-token-prediction problems. The paper proposes Set Supervised Fine-Turning (SSFT). They incorporate a set-based global loss and use the loss for Supervised Fine-Turning (SFT). Experiments on multiple reasoning benchmarks show our SSFT method consistently outperforms SFT.", "questions": "1. \"To find the optimal bipartite matching for each input prompt, we consider only the first 1,000 tokens when computing the matching cost in Equation 2 for computational efficiency.\" Why did you only choose the first 1,000 tokens? Is it enough for computing the matching cost? How accurate is it? \n2. Have you ever compared the methods to the RL method? [1,2] can also generate different length of the outputs guided by instructions/tokens.\n3. When generate Training Dataset, what to do if a teacher model can't generate correct traces? \n [1] Zhang, Xuechen, et al. \"Making small language models efficient reasoners: Intervention, supervision, reinforcement.\" arXiv preprint arXiv:2505.07961 (2025).\n\n[2] Aggarwal, Pranjal, and Sean Welleck. \"L1: Controlling how long a reasoning model thinks with reinforcement learning.\" arXiv preprint arXiv:2503.04697 (2025).", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. I think the paper treats parallel reasoning as a set-of-next-token-prediction problem is an interesting and useful idea.\n2. The paper focuses on an important question.\n3. The motivation and intuition under the paper is pretty clear and easy to follow.", "weaknesses": "1. The experiments are only conducted with a single base model. But I think it will be helpful is models with different size and from different model families can also be used.\n2. I think the comparison to baseline is a bit unfair. For (MultiTarget), they use one <think> token for all four traces and treating them as four individual data points. This can make all traces mixed together and confuse the model. I think a fairer comparison is to use different tokens for different traces. As you do for SSFT. (Single-Target △) models trained with one trace per question and (Multi-Target ⋆) models trained with four traces per question. Does it mean that the total number of training tokens is different?\n3. It's unclear to me how robust the algo is, especially the matching part. I think there will be probamatic if the generated trace of the base model is very similar to one if the teacher model. For example, if the base model is deepseek-distilled, and deepseek is one of the teacher model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Gb7B79vRjY", "reviewer": "Reviewer_8WvB", "summary": "This paper tackles \"mode collapse\" in LLMs trained on diverse reasoning traces. It introduces Set Supervised Fine-Tuning (SSFT), which frames parallel reasoning as a set prediction problem. SSFT uses a set of global forking tokens $\\{g^{(i)}\\}$ and multiple reasoning traces $\\{r^{(j)}\\}$. At each step, it finds an optimal bipartite matching $\\hat{\\sigma}$ between tokens and traces by minimizing the NTP loss. The model only optimizes this set-based loss, $\\mathcal{L}_{Hungarian}(\\theta)$. This forces tokens to learn distinct reasoning modes, improving both Pass@1 (single) and Cons@k (voting) accuracy on reasoning benchmarks.", "questions": "1. The caption of the table should be placed above.\n2. See Weakness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper introduces a novel problem formulation, treating parallel reasoning as a \"set-of-next-token-prediction\" task. It applies a set-based loss via bipartite matching (inspired by DETR in vision) to LLM finetuning.\n2. The empirical evaluation is rigorous. By comparing SSFT against baselines trained on the exact same data (SFT-mixed) and an ablation (SSFT-random), the paper proves the method's core components are responsible for its success. Visualizations (Fig. 4 vs 5) powerfully confirm that SSFT prevents the \"mode collapse\" that plagues naive finetuning.\n3. The paper is clearly written, and its core, complex mechanism is made simple and intuitive by the visualization in Figure 1, which illustrates the entire training step.\n4. This work provides the first effective method for multi-teacher distillation, allowing a single model to learn diverse reasoning strategies. It significantly improves the sample efficiency of parallel decoding (Cons@k) and introduces \"global forking tokens\" as a new, powerful form of steerable generation.", "weaknesses": "1. Computational Scalability: The SSFT algorithm requires $\\mathcal{O}(N \\times M)$ forward passes (where $N$=tokens, $M$=traces) at each training step to build the cost matrix. This is a potential bottleneck that may not scale efficiently if researchers want to use many more tokens or traces.\n2. Reliance on Matching Heuristic: The optimal matching is computed using only the first $T_L=1000$ tokens of each trace for efficiency. The paper lacks an ablation study on this critical $T_L$ hyperparameter, making it unclear if the method is robust for reasoning paths that only diverge after this prefix.\n3. Limited Task Scope: The experiments are confined to mathematical and structured-knowledge reasoning. It is not demonstrated how SSFT would perform on more open-ended, creative, or argumentative tasks where \"correctness\" and \"diversity\" are less clearly defined.\n4. Weak Justification for Pass@1 Token: The heuristic for choosing the best token for Pass@1 inference (the one with the \"largest coverage\") is not well-justified. The paper does not provide strong evidence for why this \"flexibility\" heuristic should correlate with the highest single-attempt accuracy.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)