Upload folder using huggingface_hub

iclr26/3MifB0f7qR/appendix_chunks.jsonl

@@ -0,0 +1,100 @@
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0087", "section": "A THE USE OF LARGE LANGUAGE MODELS (LLMS)", "page_start": 14, "page_end": 14, "type": "Text", "text": "The Large Language Models (LLMs) were exclusively utilized to polish the writing and detect potential typos, with no involvement in other aspects.", "source": "marker_v2", "marker_block_id": "/page/13/Text/2"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0088", "section": "B.1 RLVR ALGORITHMS", "page_start": 14, "page_end": 14, "type": "Text", "text": "We have employed the GRPO algorithm enhanced with a subset of DAPO techniques. Primarily, we have integrated DAPO's Dynamic Sampling Trick and eliminated the KL divergence term, resulting in an optimization objective that is:", "source": "marker_v2", "marker_block_id": "/page/13/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0089", "section": "B.1 RLVR ALGORITHMS", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\begin{split} \\mathcal{J}(\\theta) &= & \\mathbb{E}_{q \\sim \\mathcal{D}, \\{o_i\\}_{i=1}^G \\sim \\pi_{\\theta_{\\text{old}}}(\\cdot|q)} \\\\ & \\left[ \\frac{1}{G} \\sum_{i=1}^G \\frac{1}{|o_i|} \\sum_{t=1}^{|o_i|} \\min\\left(r_{i,t}(\\theta) \\hat{A}_{i,t}, \\, \\text{clip}\\Big(r_{i,t}(\\theta), 1-\\varepsilon, 1+\\varepsilon\\Big) \\hat{A}_{i,t}\\Big) \\right] \\\\ & \\text{s.t.} \\quad 0 < \\#\\Big\\{o_i \\mid [o_i \\text{ is correct}]\\}\\Big\\} < G, \\end{split} \\tag{1}", "source": "marker_v2", "marker_block_id": "/page/13/Equation/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0090", "section": "B.1 RLVR ALGORITHMS", "page_start": 14, "page_end": 14, "type": "Text", "text": "where", "source": "marker_v2", "marker_block_id": "/page/13/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0091", "section": "B.1 RLVR ALGORITHMS", "page_start": 14, "page_end": 14, "type": "Equation", "text": "r_{i,t}(\\theta) = \\frac{\\pi_{\\theta}(o_{i,t} \\mid q, o_{i, < t})}{\\pi_{\\theta_{\\text{old}}}(o_{i,t} \\mid q, o_{i, < t})}, \\quad \\hat{A}_{i,t} = \\frac{R_i - \\text{mean}(\\{R_i\\}_{i=1}^G)}{\\text{std}(\\{R_i\\}_{i=1}^G)}. (2)", "source": "marker_v2", "marker_block_id": "/page/13/Equation/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0092", "section": "B.1 RLVR ALGORITHMS", "page_start": 14, "page_end": 14, "type": "Text", "text": "Our reward function R mirrors that of DeepScaleR (Luo et al., 2025b), employing an Outcome Reward Model. It returns 1 if and only if both the answer and format are correct; otherwise, it returns 0. In summary, our reward function yields:", "source": "marker_v2", "marker_block_id": "/page/13/Text/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0093", "section": "B.1 RLVR ALGORITHMS", "page_start": 14, "page_end": 14, "type": "Equation", "text": "R = \\begin{cases} 1, & \\text{if the answer (e.g. passes basic LaTeX/Sympy checks)} \\\\ & \\text{and format (e.g. exists and ) are both correct,} \\\\ 0, & \\text{otherwise.} \\end{cases} (3)", "source": "marker_v2", "marker_block_id": "/page/13/Equation/10"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0094", "section": "B.2 LOW-VARIANCE PASS@K ESTIMATION", "page_start": 14, "page_end": 14, "type": "Text", "text": "Pass@k is a measure of a model's problem - solving ability, indicating the probability that the model can generate at least one correct solution in k attempts. Specifically, for each problem x_i in the evaluation dataset \\mathcal{D} , we generate n samples (where n \\geq k ) and count the correct ones as c_i . The direct calculation formula is:", "source": "marker_v2", "marker_block_id": "/page/13/Text/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0095", "section": "B.2 LOW-VARIANCE PASS@K ESTIMATION", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\operatorname{pass}@k := \\mathbb{E}_{x_i \\sim \\mathcal{D}} \\left[ 1 - \\left( 1 - \\frac{c_i}{n} \\right)^k \\right] \\tag{4}", "source": "marker_v2", "marker_block_id": "/page/13/Equation/13"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0096", "section": "B.2 LOW-VARIANCE PASS@K ESTIMATION", "page_start": 14, "page_end": 14, "type": "Text", "text": "However, this formula has excessive variance and insufficient accuracy. To solve this problem, we adopt the unbiased estimation method proposed by Chen et al. (Chen et al., 2021), using the unbiased estimator of pass@k over the dataset:", "source": "marker_v2", "marker_block_id": "/page/13/Text/14"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0097", "section": "B.2 LOW-VARIANCE PASS@K ESTIMATION", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\operatorname{pass}@k := \\mathbb{E}_{x_i \\sim \\mathcal{D}} \\left[ 1 - \\frac{\\binom{n - c_i}{k}}{\\binom{n}{k}} \\right] \\tag{5}", "source": "marker_v2", "marker_block_id": "/page/13/Equation/15"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0098", "section": "B.2 LOW-VARIANCE PASS@K ESTIMATION", "page_start": 14, "page_end": 14, "type": "Text", "text": "In our experiments, to ensure sufficient accuracy, we set n such that 2k \\le n , which helps further reduce the variance of the estimate.", "source": "marker_v2", "marker_block_id": "/page/13/Text/16"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0099", "section": "B.3 MORE RELATED WORKS", "page_start": 14, "page_end": 14, "type": "Text", "text": "Recent studies show that RL algorithms, such as PPO (Schulman et al., 2017) and GRPO (Guo et al., 2025), can greatly enhance model reasoning capabilities. Building on this, several works have refined this paradigm from different perspectives. One method can be adjusting the reward function. Some studies (Zhu et al., 2025; Shao et al., 2025) directly modify the reward function to improve training efficiency. Other methods introduced intermediate process rewards (Wang et al., 2024; Malik et al., 2025), while Wen et al. (Wen et al., 2025) set up a separate correctness judgment for CoT to obtain rewards.", "source": "marker_v2", "marker_block_id": "/page/13/Text/18"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0100", "section": "B.3 MORE RELATED WORKS", "page_start": 15, "page_end": 15, "type": "Text", "text": "Another novel perspective aims to improve sample efficiency by measuring certainty. For example, TreeRL (Hou et al., 2025) and VinePPO (Kazemnejad et al., 2025) enhanced sample effects by introducing entropy or confidence. MRT (Qu et al., 2025), on the other hand, reused partial trajectories during testing to boost sample efficiency. R3 (Xi et al., 2024) improves RL sample efficiency by decomposing human solution steps and providing preceding steps to guide the model in completing subsequent ones. Further, some research adopted a multi-stage training or reasoning mode, exploring from different angles such as training length (Luo et al., 2025b), question difficulty (Parashar et al., 2025), and fixed-length summaries during reasoning (Yan et al., 2025).", "source": "marker_v2", "marker_block_id": "/page/14/Text/1"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0101", "section": "B.3 MORE RELATED WORKS", "page_start": 15, "page_end": 15, "type": "Text", "text": "In addition to designing better algorithms, another line of research (Shao et al., 2024; Yue et al., 2025; Zhao et al., 2025) has investigated how reinforcement learning affects the frontier of model capabilities, observing a decay in pass@k when k becomes large. In response to this phenomenon, some works (Yu et al., 2025; Liu et al., 2025; An et al., 2025) maintained entropy stability by adjusting training entropy through methods such as increasing the clipping upper bound, enlarging the temperature coefficient, extending the training length, and periodically updating the KL reference model. StepHint (Zhang et al., 2025) also preserved entropy stability by leveraging intermediate thinking content of iterative length as a prompting signal.", "source": "marker_v2", "marker_block_id": "/page/14/Text/2"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0102", "section": "B.3 MORE RELATED WORKS", "page_start": 15, "page_end": 15, "type": "Text", "text": "In contrast to the aforementioned research, our work adopts an orthogonal approach by using part of the ground-truth solution as a hint, without requiring any modifications to the existing reinforcement learning infrastructure. We provide both theoretical justification and empirical evidence that this strategy maintains pass@k without compromising the exploratory capacity of the underlying reinforcement learning algorithm.", "source": "marker_v2", "marker_block_id": "/page/14/Text/3"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0103", "section": "B.4 BENCHMARKS", "page_start": 15, "page_end": 15, "type": "Text", "text": "We evaluate the models' breadth across various tasks in multiple domains, including mathematics, coding, reasoning, and logical inference. For mathematics, we follow DeepScaleR (Luo et al., 2025b) and Nemotron (Moshkov et al., 2025), and conduct assessments on more challenging mathematical datasets such as AIME2024 (MAA, 2024), AIME2025 (MAA, 2025), Olympiad Bench (He et al., 2024), HMMT FEB 25 (hmm, 2025), and BRUMO25 (bru, 2025). Specifically, HMMT25 Feb and BRUMO25 are both sourced from MathArena (Balunović et al., 2025). In the realm of coding, we utilize commonly employed datasets, including Code Contests (Li et al., 2022), Codeforces<sup>1</sup>, and LCB V5 202410-202502 (Jain et al., 2024). For logical reasoning tasks, we assess our models' capabilities using GPQA Diamond (Rein et al., 2023) <sup>2</sup> and Zebraliogic (Lin et al., 2025). The benchmarks related to coding and logical reasoning are all referenced from AReaL (Fu et al., 2025).", "source": "marker_v2", "marker_block_id": "/page/14/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0104", "section": "B.5 TRAINING DATASET", "page_start": 15, "page_end": 15, "type": "Text", "text": "The dataset employed in our study is OpenR1-Math-220K (Open-R1 Team, 2025). Prior to commencing the training of the Partial Solution, we conducted a preliminary screening of the dataset. Specifically, we utilized the DeepSeek-R1-Distill-1.5B (Guo et al., 2025) model to perform eight inference operations on each of the 220k data entries in the OpenR1 dataset. Subsequently, we compared the annotated answers in the OpenR1 dataset with the results generated from each inference to tally the number of correct instances for each data entry. Ultimately, we selected the data entries with 0 or 1 correct instance as the training samples for our study. The final dataset size is 26K.", "source": "marker_v2", "marker_block_id": "/page/14/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0105", "section": "B.5 TRAINING DATASET", "page_start": 15, "page_end": 15, "type": "Text", "text": "For controlled comparisons, we further split this 26K subset by re-sampling Nemotron-1.5B eight times per item and counting correct completions. We define Easy Data as questions with correct counts in [7,8] and train a model on this split, denoted <code>Easy-Nemotron-1.5B</code>. Similarly, we define Hard Data as questions with correct counts in [0,1] and train <code>Hard-Nemotron-1.5B</code> on this split.", "source": "marker_v2", "marker_block_id": "/page/14/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0106", "section": "B.5 TRAINING DATASET", "page_start": 15, "page_end": 15, "type": "Text", "text": "Additionally, for the augmented data, we perform eight inference passes using the model currently under training. We then select samples for which the number of correct predictions falls within the range of [0,4]. This criterion is motivated by the finding that samples exhibiting higher variance are more beneficial for training (Gao et al., 2025; Wang et al., 2025b). The range [0,4] is chosen because it includes the point of maximum sample variance, which is achieved with four correct predictions out of eight trials. For convenience, we refer to augmented data with partial ratio p as Partial-p data.", "source": "marker_v2", "marker_block_id": "/page/14/Text/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0107", "section": "B.5 TRAINING DATASET", "page_start": 15, "page_end": 15, "type": "Footnote", "text": "1", "source": "marker_v2", "marker_block_id": "/page/14/Footnote/10"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0108", "section": "B.5 TRAINING DATASET", "page_start": 15, "page_end": 15, "type": "Footnote", "text": "& lt;sup>2</sup>In the GPQA Diamond dataset, multiple-choice questions are presented in the form of options rather than directly providing the answer, requiring the model to output only A, B, C, or D.", "source": "marker_v2", "marker_block_id": "/page/14/Footnote/11"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0109", "section": "B.6 THE RATIONALE FOR THE CHOICE OF p", "page_start": 16, "page_end": 16, "type": "TableGroup", "text": "Table 5: Number of problems vs pass rate under different hint levels on OpenMath-Nemotron-1.5B before training. We evaluated OpenMath-Nemotron-1.5B on the OpenR1 dataset after the first round of filtering, with each problem assessed 8 times. The table illustrates the distribution of correct answers (n) where n ∈ {0, 1, . . . , 8}. Hint Levels 0 / 8 1 / 8 2 / 8 3 / 8 4 / 8 5 / 8 6 / 8 7 / 8 8 / 8 Partial-50 143 224 304 472 710 1013 1779 3655 17741 Partial-25 3155 1997 1814 1785 1902 2175 2614 3440 7159 Partial-10 3589 2090 1865 1842 1905 2176 2653 3415 6506 Partial-0 3812 2218 1854 1842 2007 2136 2517 3264 6391", "source": "marker_v2", "marker_block_id": "/page/15/TableGroup/271"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0110", "section": "B.6 THE RATIONALE FOR THE CHOICE OF p", "page_start": 16, "page_end": 16, "type": "Text", "text": "In this study, we evaluated the performance of OpenMath-Nemotron-1.5B on the OpenR1 dataset under various hint levels. The evaluation was performed after the first round of filtering, and each problem was assessed 8 times to capture the predictive distribution. The resulting table (Table 5) shows the distribution of correct answers across different hint levels, where the values represent the number of times the model answered correctly (n ∈ {0, 1, . . . , 8}).", "source": "marker_v2", "marker_block_id": "/page/15/Text/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0111", "section": "B.6 THE RATIONALE FOR THE CHOICE OF p", "page_start": 16, "page_end": 16, "type": "Text", "text": "The selection of the hint parameter p was primarily based on these evaluation results. As shown in the table, the performance with a Partial-50 hint significantly reduces task difficulty, as evidenced by the high pass rates across most levels. In contrast, Partial-25 (25% hint) exhibits a performance pattern similar to that of the no-hint scenario (Partial-0), with only marginal differences in task difficulty.", "source": "marker_v2", "marker_block_id": "/page/15/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0112", "section": "B.6 THE RATIONALE FOR THE CHOICE OF p", "page_start": 16, "page_end": 16, "type": "Text", "text": "This minimal difference in difficulty between Partial-0 and Partial-25 suggests that training with Partial-25 does not provide substantial gains compared to Partial-0. Consequently, we adopted a stepwise design in which the hint level is first set to p = 50%, followed by p = 25%, to evaluate the model's performance under varying conditions.", "source": "marker_v2", "marker_block_id": "/page/15/Text/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0113", "section": "B.7 EVALUATION SETUP", "page_start": 16, "page_end": 16, "type": "Text", "text": "We configured the models to have a maximum generation length of 32,768 tokens. In line with DeepSeek-R1 (Guo et al., 2025) , we utilized pass@k evaluation (Chen et al., 2021) , with the formula detailed in B.2. We reported pass@1 using a non-zero temperature. Specifically, we used a sampling temperature of 0.7 and a top-p value of 0.95 to generate k responses per question, typically set at 32, with deviations explicitly noted. Particular attention should be paid to the fact that, although we incorporated partial Solution during training, it was not included in the evaluation phase.", "source": "marker_v2", "marker_block_id": "/page/15/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0114", "section": "DeepScaleR Others' Inference", "page_start": 16, "page_end": 16, "type": "Text", "text": "<|User|>{input}", "source": "marker_v2", "marker_block_id": "/page/15/Text/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0115", "section": "DeepScaleR Others' Inference", "page_start": 16, "page_end": 16, "type": "Text", "text": "Please reason step by step, and put your final answer within \\boxed{}.<|Assistant|><think>", "source": "marker_v2", "marker_block_id": "/page/15/Text/13"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0116", "section": "Nemotrion Coding's Inference", "page_start": 16, "page_end": 16, "type": "Text", "text": "<|im_start|>user", "source": "marker_v2", "marker_block_id": "/page/15/Text/15"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0117", "section": "Nemotrion Coding's Inference", "page_start": 16, "page_end": 16, "type": "Text", "text": "{input}", "source": "marker_v2", "marker_block_id": "/page/15/Text/16"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0118", "section": "Nemotrion Coding's Inference", "page_start": 16, "page_end": 16, "type": "Text", "text": "<|im_end|> <|im_start|>assistant", "source": "marker_v2", "marker_block_id": "/page/15/Text/17"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0119", "section": "Nemotrion Coding's Inference", "page_start": 16, "page_end": 16, "type": "Text", "text": "<think>", "source": "marker_v2", "marker_block_id": "/page/15/Text/18"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0120", "section": "Nemotrion Others' Inference", "page_start": 16, "page_end": 16, "type": "Text", "text": "<|im_start|>system", "source": "marker_v2", "marker_block_id": "/page/15/Text/20"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0121", "section": "Nemotrion Others' Inference", "page_start": 16, "page_end": 16, "type": "Text", "text": "Please reason step by step, and put your final answer within \\boxed{}.<|im_end|> <|im_start|>user", "source": "marker_v2", "marker_block_id": "/page/15/Text/21"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0122", "section": "Nemotrion Others' Inference", "page_start": 17, "page_end": 17, "type": "Text", "text": "{input}<lim_endl> <lim_startl>assistant", "source": "marker_v2", "marker_block_id": "/page/16/Text/1"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0123", "section": "Training prompt with partial solutions (math RL)", "page_start": 17, "page_end": 17, "type": "Text", "text": "{Problem}", "source": "marker_v2", "marker_block_id": "/page/16/Text/3"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0124", "section": "Training prompt with partial solutions (math RL)", "page_start": 17, "page_end": 17, "type": "Text", "text": "## Hint: {Partial Solution}", "source": "marker_v2", "marker_block_id": "/page/16/Text/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0125", "section": "Training prompt with partial solutions (math RL)", "page_start": 17, "page_end": 17, "type": "Text", "text": "Please reason step by step, and put your final answer within \\boxed{}.", "source": "marker_v2", "marker_block_id": "/page/16/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0126", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "Theorem 4.4 (Lower Bound on RL Learnability under Solution Inaccessibility). Given a probability threshold \\delta_p > 0 , if for every question q \\in \\mathcal{Q} , the model capacity set C(q, \\delta_p) does not intersect with the solution set \\mathcal{S}(q) , i.e.,", "source": "marker_v2", "marker_block_id": "/page/16/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0127", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Equation", "text": "C(q, \\delta_n) \\cap \\mathcal{S}(q) = \\emptyset, \\quad \\forall q \\in \\mathcal{Q},", "source": "marker_v2", "marker_block_id": "/page/16/Equation/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0128", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "then under Assumption 4.3, when training RL for T steps with B samples per step such that TB = \\Theta(1/\\delta_p) , there is a constant probability that the RL algorithm will not update the model.", "source": "marker_v2", "marker_block_id": "/page/16/Text/10"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0129", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "Proof. Let p_{\\text{sol}} = \\sum_{\\tau^* \\in \\mathcal{S}(q)} P_{\\mu}(\\tau^*|q) denote the cumulative generation probability of any solution trajectory. By C(q, \\delta_p) \\cap \\mathcal{S}(q) = \\emptyset and Def 4.2:", "source": "marker_v2", "marker_block_id": "/page/16/Text/11"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0130", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Equation", "text": "p_{\\text{sol}} = \\sum_{\\tau^* \\in \\mathcal{S}(q)} P_{\\mu}(\\tau^*|q) < \\delta_p", "source": "marker_v2", "marker_block_id": "/page/16/Equation/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0131", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "For N=TB independent samples across T steps with batch size B, the probability of complete failure (no solution sampled) is:", "source": "marker_v2", "marker_block_id": "/page/16/Text/13"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0132", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Equation", "text": "\\mathbb{P}(\\text{failure}) = (1 - p_{\\text{sol}})^N > (1 - \\delta_p)^N", "source": "marker_v2", "marker_block_id": "/page/16/Equation/14"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0133", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "Given TB = \\Theta(1/\\delta_p) , we have:", "source": "marker_v2", "marker_block_id": "/page/16/Text/15"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0134", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Equation", "text": "(1 - \\delta_p)^N > (1 - \\delta_p)^{\\Theta(1/\\delta_p)} = \\Theta(1).", "source": "marker_v2", "marker_block_id": "/page/16/Equation/16"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0135", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "The last inequality follows from the fact that (1-x)^{1/x} > \\exp(-1/(1-x)) for x \\in (0,1) . By Assumption 4.3, if no solution is found, the model weights remain unchanged.", "source": "marker_v2", "marker_block_id": "/page/16/Text/17"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0136", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "Lemma C.1 (Upper Bound on Sampling Budget for Solution Given Hint). Given a question q \\in \\mathcal{Q} , if there exists a hint h_q for the question q (Def. 4.5), then if we perform TB = \\Theta(1/\\delta_p') = \\Theta(\\delta_p^\\epsilon/\\sqrt{\\delta_p}) i.i.d sampling over the initial model conditioned on (q, h_q) , we can find a valid solution with a constant probability.", "source": "marker_v2", "marker_block_id": "/page/16/Text/18"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0137", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "Proof. By Definition 4.5, we know:", "source": "marker_v2", "marker_block_id": "/page/16/Text/19"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0138", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Equation", "text": "1. P_{\\mu}(h_a|q) \\geq \\delta'_n", "source": "marker_v2", "marker_block_id": "/page/16/Equation/20"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0139", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Equation", "text": "2. \\exists s_q \\in \\mathcal{S}(q) : P_{\\mu}(s_q|(q,h_q)) \\geq \\delta_p'", "source": "marker_v2", "marker_block_id": "/page/16/Equation/21"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0140", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "With N = TB \\ge 10/\\delta'_p independent samples conditioned on (q, h_q) , the probability of not finding the solution s_q is:", "source": "marker_v2", "marker_block_id": "/page/16/Text/22"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0141", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Equation", "text": "\\mathbb{P}(\\text{no solution}) = (1 - P_{\\mu}(s_q|(q, h_q)))^N \\le (1 - \\delta_p')^{10/\\delta_p'} \\le \\exp(-10) < 0.01.", "source": "marker_v2", "marker_block_id": "/page/16/Equation/23"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0142", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "Therefore, \\mathbb{P}(\\text{finding solution}) > 0.99 .", "source": "marker_v2", "marker_block_id": "/page/16/Text/24"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0143", "section": "C.1 Proofs", "page_start": 17, "page_end": 17, "type": "Text", "text": "Theorem 4.6 (Informal Upper Bound on RL Learnability with Hint). If we have a hint h_q for every question q \\in \\mathcal{Q} (Def. 4.5), then there exists an RL algorithm that can output a policy \\pi_\\theta such that \\mathbb{E}_{q \\sim \\mathrm{Uniform}(\\mathcal{Q})}[\\mathbb{P}_{\\tau \\sim \\pi_\\theta(\\cdot|q)}(\\tau \\in \\mathcal{S}(q))] \\geq 0.99 with O(1/\\delta_p') sampling budget with high probability.", "source": "marker_v2", "marker_block_id": "/page/16/Text/25"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0144", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "This theorem is a direct corollary of the Theorem 5 regarding the bandit setup in (Mei et al., 2022). Because the setup here is relatively simple, we also present a detailed proof for this special case here. We first formalize our setup as follows:", "source": "marker_v2", "marker_block_id": "/page/17/Text/1"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0145", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "Assumption C.2 (Tabular RL with Hint). We consider the tabular RL setting with softmax policy parameterization. There exists a finite set of possible questions \\mathcal{Q} and a finite set of possible solutions \\mathcal{S} . For each question q \\in \\mathcal{Q} , there exists a hint h_q , which is a subset of solutions h_q \\subseteq \\mathcal{S} .", "source": "marker_v2", "marker_block_id": "/page/17/Text/2"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0146", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "The policy is parameterized by a |S| \\times |Q| matrix \\theta in the following way:", "source": "marker_v2", "marker_block_id": "/page/17/Text/3"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0147", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Equation", "text": "\\mu_{\\theta}(s|q) = \\frac{\\exp(\\theta_{s,q})}{\\sum_{s' \\in S} \\exp(\\theta_{s',q})}", "source": "marker_v2", "marker_block_id": "/page/17/Equation/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0148", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "Here the setup is different than the autoregressive setting in our experiments and simplify the model to a tabular setup for the simplicity of analysis. We now restate the assumption on the existence of hint in this setup.", "source": "marker_v2", "marker_block_id": "/page/17/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0149", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "Assumption C.3 (Hint Existence, Formal Version of Definition 4.5). For each question q \\in \\mathcal{Q} , there exists a hint h_q \\subseteq \\mathcal{S} such that \\sum_{s \\in h_q} P_{\\mu}(s|q) \\ge \\delta'_p . Further, there exists a solution s_q \\in \\mathcal{S} such that P_{\\mu}(s_q|q) \\ge \\delta'_p \\sum_{s \\in h_q} P_{\\mu}(s|q) .", "source": "marker_v2", "marker_block_id": "/page/17/Text/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0150", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "RL Algorithm: We will first sample \\Theta(1/\\delta'_p) action based on the policy \\mu_\\theta conditioned on the question q and the hint h_q . Then we will do a one-step policy gradient update on our policy. Noted that here we can reach high reward within one step because the reward function is deterministic.", "source": "marker_v2", "marker_block_id": "/page/17/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0151", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "Theorem C.4 (Formal Version of Theorem 4.6). Under Assumption C.2 and Assumption C.3, running 1 steps of policy gradient update with sampling budget \\Theta(1/\\delta'_{v}) , the learned policy achieves:", "source": "marker_v2", "marker_block_id": "/page/17/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0152", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Equation", "text": "\\mathbb{E}_{q \\sim \\text{Uniform}(\\mathcal{Q})}[\\mathbb{P}_{\\tau \\sim \\mu_{\\theta}(\\cdot|q)}(\\tau \\in \\mathcal{S}(q))] \\geq 0.99", "source": "marker_v2", "marker_block_id": "/page/17/Equation/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0153", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "with probability 0.99.", "source": "marker_v2", "marker_block_id": "/page/17/Text/10"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0154", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "Proof. First, by Assumption C.3, for any question q, we have:", "source": "marker_v2", "marker_block_id": "/page/17/Text/11"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0155", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Equation", "text": "\\sum_{s \\in h_q} P_{\\mu}(s|q) \\geq \\delta_p \\quad \\text{and} \\quad \\exists s_q : P_{\\mu}(s_q|q) \\geq \\delta_p' \\sum_{s \\in h_q} P_{\\mu}(s|q)", "source": "marker_v2", "marker_block_id": "/page/17/Equation/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0156", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "With sampling budget N = \\Theta(|\\mathcal{Q}|/\\delta_p') , by Lemma C.1 and the union bound, we will find a solution s_q for every question q with probability at least 0.99. Suppose the found set of solutions for question q is S_q and all sampled solutions are s^{(1)}, \\ldots, s^{(N)} . Then because", "source": "marker_v2", "marker_block_id": "/page/17/Text/13"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0157", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Equation", "text": "\\nabla_{\\theta} \\log \\mu_{\\theta}(s|q) = e_s - \\sum_{s' \\in \\mathcal{S}} \\mu_{\\theta}(s'|q)e_{s'}", "source": "marker_v2", "marker_block_id": "/page/17/Equation/14"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0158", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "We have the policy gradient being", "source": "marker_v2", "marker_block_id": "/page/17/Text/15"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0159", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Equation", "text": "\\begin{aligned} \\text{PG}_{:,q} &= \\frac{1}{N} \\sum_{i=1}^{N} \\mathbf{1}[s^{(i)} \\in S_q] \\nabla_{\\theta} \\log \\mu_{\\theta}(s^{(i)}|q) \\\\ &= \\frac{1}{N} \\sum_{i=1}^{N} \\mathbf{1}[s^{(i)} \\in S_q] (e_{s^{(i)}} - \\sum_{s' \\in \\mathcal{S}} \\mu_{\\theta}(s'|q) e_{s'}) \\\\ &= \\frac{1}{N} \\sum_{i=1}^{N} \\mathbf{1}[s^{(i)} \\in S_q] e_{s^{(i)}} - \\left(\\frac{1}{N} \\sum_{i=1}^{N} \\mathbf{1}[s^{(i)} \\in S_q]\\right) \\left(\\sum_{s' \\in \\mathcal{S}} \\mu_{\\theta}(s'|q) e_{s'}\\right). \\end{aligned}", "source": "marker_v2", "marker_block_id": "/page/17/Equation/16"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0160", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "We can make two simple observations:", "source": "marker_v2", "marker_block_id": "/page/17/Text/17"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0161", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "ListGroup", "text": "1. For every s \\notin S_a , PG_{s,a} < 0 . 2. There exists a s^* \\in S_q such that PG_{s^*,q} > 0 .", "source": "marker_v2", "marker_block_id": "/page/17/ListGroup/169"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0162", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "Therefore, consider the updated parameters", "source": "marker_v2", "marker_block_id": "/page/17/Text/20"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0163", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Equation", "text": "\\theta'_{s,q} = \\theta_{s,q} + \\eta PG_{s,q}", "source": "marker_v2", "marker_block_id": "/page/17/Equation/21"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0164", "section": "C.1 Proofs", "page_start": 18, "page_end": 18, "type": "Text", "text": "If \\eta is large enough, we know that \\sum_{s \\in S_-} P_{\\mu_{\\theta'}}(s|q) \\ge 0.99 . This completes the proof.", "source": "marker_v2", "marker_block_id": "/page/17/Text/22"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0165", "section": "D.1 ABLATION STUDY WITHOUT HINT", "page_start": 19, "page_end": 19, "type": "TableGroup", "text": "Table 6: Ablation without hint on Nemotron-1.5B: Pass@1 (avg@32) on challenging maths benchmarks. \"QUESTA-Nemotron-1.5B w/o hint\" trains RL on the same data but removes hints from the prompt, while \"w/ hint\" uses partial-solution hints during training. With hints, the model improves all benchmarks and achieves a +2.82 average gain over w/o hint (63.26 vs. 60.44), on top of the improvements over the base model. The one using hint requires nearly half the number of steps compared to the one not using hint to achieve the same performance. Model AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 Avg Nemotron-1.5B 61.77 49.50 31.56 64.62 58.23 53.14 QUESTA-Nemotron-1.5B w/o hint (2K step) 69.48 59.79 38.85 68.05 66.04 60.44 QUESTA-Nemotron-1.5B w/ hint (1.1K step) 69.27 60.00 37.92 69.72 68.33 61.05 QUESTA-Nemotron-1.5B w/ hint (2K step) 72.50 62.29 41.67 70.36 69.48 63.26", "source": "marker_v2", "marker_block_id": "/page/18/TableGroup/93"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0166", "section": "D.1 ABLATION STUDY WITHOUT HINT", "page_start": 19, "page_end": 19, "type": "FigureGroup", "text": "Figure 7: Pass@k comparison on Nemotron-1.5B for RL with vs. without hints. Training with hints consistently dominates across k and avoids the performance drop at larger k seen in standard RL. Hints are used only during training; evaluation uses no hints.", "source": "marker_v2", "marker_block_id": "/page/18/FigureGroup/94"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0167", "section": "D.1 ABLATION STUDY WITHOUT HINT", "page_start": 19, "page_end": 19, "type": "Text", "text": "We ablate the role of hints by training RL on Nemotron-1.5B with and without partial-solution hints. Here, QUESTA-Nemotron-1.5B w/o hint denotes RL on the same data and schedule but with the hint removed from the prompt; QUESTA-Nemotron-1.5B w/h hint uses identical settings except that the partial solution is provided as a hint during training. As summarized in Table 6, removing the hint still improves over the base model (average Pass@1: 53.14 \\rightarrow 60.44 ), but adding the hint yields a", "source": "marker_v2", "marker_block_id": "/page/18/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0168", "section": "D.1 ABLATION STUDY WITHOUT HINT", "page_start": 20, "page_end": 20, "type": "Text", "text": "further +2.82 average gain ( 60.44 \\rightarrow 63.26 ), with consistent improvements across all five benchmarks. The one using hint requires nearly half the number of steps compared to the one not using hint to achieve the same performance. Note that hints are used only during training; all evaluations are conducted without hints.", "source": "marker_v2", "marker_block_id": "/page/19/Text/1"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0169", "section": "D.1 ABLATION STUDY WITHOUT HINT", "page_start": 20, "page_end": 20, "type": "Text", "text": "Figure 7 compares Pass@k curves. The w/hint model lifts the entire curve across k and avoids the degradation at larger k commonly observed in standard RL, while the w/o hint variant brings smaller gains that taper off as k increases. A possible reason for this phenomenon is that, without hints, extremely difficult problems remain unlearned. Consequently, during reinforcement learning training, the model prioritizes improving performance on problems that have become relatively easier as training progresses. This leads the model to become overly confident, thereby reducing its Pass@k metric. In contrast, when hints are provided, the model still prioritizes learning more difficult problems—this is because such problems can provide effective learning signals.", "source": "marker_v2", "marker_block_id": "/page/19/Text/2"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0170", "section": "D.2 ABLATION STUDY WITH DIFFERENT MODELS", "page_start": 20, "page_end": 20, "type": "TableGroup", "text": "Table 7: Performance comparison on DeepScaleR-1.5B: Pass@1 (avg@32) across maths benchmarks. QUESTA consistently improves all tasks and raises the average by +6.50 points. Model AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 Avg DeepScaleR-1.5B 40.42 31.35 19.27 52.97 37.40 36.28 QUESTA-DeepScaleR-1.5B 49.16 35.94 21.77 58.69 48.33 42.78 Table 8: Performance comparison (Pass@1, averaged over 32 samples) showing the impact of QUESTA across benchmarks in other domains, including general knowledge, logic, and coding tasks. We observe minor cross-domain generalization on all these benchmarks, despite QUESTA being applied exclusively in the maths domain.", "source": "marker_v2", "marker_block_id": "/page/19/TableGroup/103"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0171", "section": "D.2 ABLATION STUDY WITH DIFFERENT MODELS", "page_start": 20, "page_end": 20, "type": "Table", "text": "Model | GPQA Diamond Zebralogic Code Contest All Codeforces LCB V5 202410-202502 Avg DeepScaleR-1.5B 38.5 14.26 9.07 8.79 19.57 18.04 QUESTA-DeepScaleR-1.5B 39.2 14.98 10.1 8.9 20.9 18.82", "source": "marker_v2", "marker_block_id": "/page/19/Table/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0172", "section": "D.2 ABLATION STUDY WITH DIFFERENT MODELS", "page_start": 20, "page_end": 20, "type": "FigureGroup", "text": "Figure 8: Pass@k on DeepScaleR-1.5B: QUESTA raises the entire curve across k and avoids the large-k drop often seen in standard RL. The increasing gap with k indicates improved sample diversity rather than overconfident collapse. No hints are used at evaluation.", "source": "marker_v2", "marker_block_id": "/page/19/FigureGroup/104"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0173", "section": "D.2 ABLATION STUDY WITH DIFFERENT MODELS", "page_start": 20, "page_end": 20, "type": "Text", "text": "We next examine model-family transfer by applying QUESTA to DeepScaleR-1.5B. We train for 750 steps on DeepScaleR-1.5B Stage 2 Luo et al. (2025b) on the QUESTA first stage.", "source": "marker_v2", "marker_block_id": "/page/19/Text/10"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0174", "section": "D.2 ABLATION STUDY WITH DIFFERENT MODELS", "page_start": 21, "page_end": 21, "type": "FigureGroup", "text": "Figure 9: We conducted training of the DeepScaleR model employing the QUESTA with a dataset comprising 26,000 questions. The average pass rate was calculated from a sample of 8 instances. The initial graph represents the scenario without the incorporation of partial solutions, while the subsequent graph depicts the situation where partial solutions were included. The application of QUESTA significantly diminishes the incidence of unsolved or partially addressed problems within the training dataset. Concurrently, it has come to our attention that our previous method of data curation was not entirely accurate; in fact, the amount of data providing meaningful training signals is less abundant than anticipated, suggesting the potential for further refinement of the dataset.", "source": "marker_v2", "marker_block_id": "/page/20/FigureGroup/95"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0175", "section": "D.2 ABLATION STUDY WITH DIFFERENT MODELS", "page_start": 21, "page_end": 21, "type": "Text", "text": "As shown in Table 7, QUESTA-DeepScaleR-1.5B improves every maths benchmark over the base model, achieved an average improvement of 6%, indicating that the benefits of QUESTA are not tied to a single architecture.", "source": "marker_v2", "marker_block_id": "/page/20/Text/3"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0176", "section": "D.2 ABLATION STUDY WITH DIFFERENT MODELS", "page_start": 21, "page_end": 21, "type": "Text", "text": "Pass@k behavior mirrors these gains. In Figure 8, QUESTA-DeepScaleR lifts the entire Pass@k curve across k and avoids the degradation at larger k reported in standard RL settings. The widening gap at larger k suggests improved candidate diversity rather than overfitting to a single trajectory, consistent with our general pass@k analysis in Appendix B.2. Complementing this, Figure 9 shows that on the 26K training set (evaluated without hints), mass shifts away from the 0/8-1/8 bins toward higher pass rates, reducing unsolved or partially solved cases. Hints are used only during training and are removed at evaluation time.", "source": "marker_v2", "marker_block_id": "/page/20/Text/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0177", "section": "D.2 ABLATION STUDY WITH DIFFERENT MODELS", "page_start": 21, "page_end": 21, "type": "Text", "text": "Beyond maths, Table 8 reports out-of-distribution (OOD) results on general knowledge, logic, and coding. QUESTA-DeepScaleR-1.5B achieves small but consistent gains (Avg: 18.04 \\rightarrow 18.82; +0.78), suggesting that the improved reasoning patterns transfer modestly beyond the training domain.", "source": "marker_v2", "marker_block_id": "/page/20/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0178", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 21, "page_end": 21, "type": "FigureGroup", "text": "Figure 10: Training dynamics of QUESTA-Nemotron-1.5B w/o hint.", "source": "marker_v2", "marker_block_id": "/page/20/FigureGroup/96"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0179", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 22, "page_end": 22, "type": "FigureGroup", "text": "Figure 11: Training dynamics of QUESTA-Nemotron-1.5B-50.", "source": "marker_v2", "marker_block_id": "/page/21/FigureGroup/102"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0180", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 22, "page_end": 22, "type": "FigureGroup", "text": "Figure 12: Training dynamics of QUESTA-Nemotron-1.5B-50-25-0.", "source": "marker_v2", "marker_block_id": "/page/21/FigureGroup/103"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0181", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 22, "page_end": 22, "type": "FigureGroup", "text": "Figure 13: Training dynamics of Easy-Nemotron-1.5B.", "source": "marker_v2", "marker_block_id": "/page/21/FigureGroup/104"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0182", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 22, "page_end": 22, "type": "FigureGroup", "text": "Figure 14: Training dynamics of Hard-Nemotron-1.5B.", "source": "marker_v2", "marker_block_id": "/page/21/FigureGroup/105"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0183", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 23, "page_end": 23, "type": "FigureGroup", "text": "Figure 15: The training dynamics of QUESTA-DeepScaleR-1.5B. The first and second charts show the changes in the average reward and average inference length of the rollout samples that include all incorrect/correct ones, respectively. The third chart shows the average entropy excluding all incorrect/correct rollout samples.", "source": "marker_v2", "marker_block_id": "/page/22/FigureGroup/64"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0184", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 23, "page_end": 23, "type": "FigureGroup", "text": "Figure 16: Training dynamics of QUESTA-Nemotron-1.5B on OpenMathReasoning (Moshkov et al., 2025). The first and second charts show the changes in the average reward and average inference length of the rollout samples that include all incorrect/correct ones, respectively. The third chart shows the average entropy excluding all incorrect/correct rollout samples. Dynamics closely mirror those on OpenR1-Math-220K, with no entropy collapse.", "source": "marker_v2", "marker_block_id": "/page/22/FigureGroup/65"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0185", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 24, "page_end": 24, "type": "FigureGroup", "text": "Figure 17: We compare pass@k curves of RLVR-trained models with and without QUESTA.", "source": "marker_v2", "marker_block_id": "/page/23/FigureGroup/58"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0186", "section": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL", "page_start": 25, "page_end": 25, "type": "FigureGroup", "text": "Figure 18: Comparison of RL training dynamics: Training with only hard problems (green) makes progress very slowly due to sparse rewards, while our method with partial solutions (orange) accelerates learning and consistently achieves higher accuracy across training steps.", "source": "marker_v2", "marker_block_id": "/page/24/FigureGroup/58"}

iclr26/3MifB0f7qR/appendix_text_v3.txt

@@ -0,0 +1,299 @@
+[p. 14 | section: A THE USE OF LARGE LANGUAGE MODELS (LLMS) | type: Text]
+The Large Language Models (LLMs) were exclusively utilized to polish the writing and detect potential typos, with no involvement in other aspects.
+
+[p. 14 | section: B.1 RLVR ALGORITHMS | type: Text]
+We have employed the GRPO algorithm enhanced with a subset of DAPO techniques. Primarily, we have integrated DAPO's Dynamic Sampling Trick and eliminated the KL divergence term, resulting in an optimization objective that is:
+
+[p. 14 | section: B.1 RLVR ALGORITHMS | type: Equation]
+\begin{split} \mathcal{J}(\theta) &= & \mathbb{E}_{q \sim \mathcal{D}, \{o_i\}_{i=1}^G \sim \pi_{\theta_{\text{old}}}(\cdot|q)} \\ & \left[ \frac{1}{G} \sum_{i=1}^G \frac{1}{|o_i|} \sum_{t=1}^{|o_i|} \min\left(r_{i,t}(\theta) \hat{A}_{i,t}, \, \text{clip}\Big(r_{i,t}(\theta), 1-\varepsilon, 1+\varepsilon\Big) \hat{A}_{i,t}\Big) \right] \\ & \text{s.t.} \quad 0 < \#\Big\{o_i \mid [o_i \text{ is correct}]\}\Big\} < G, \end{split} \tag{1}
+
+[p. 14 | section: B.1 RLVR ALGORITHMS | type: Text]
+where
+
+[p. 14 | section: B.1 RLVR ALGORITHMS | type: Equation]
+r_{i,t}(\theta) = \frac{\pi_{\theta}(o_{i,t} \mid q, o_{i, < t})}{\pi_{\theta_{\text{old}}}(o_{i,t} \mid q, o_{i, < t})}, \quad \hat{A}_{i,t} = \frac{R_i - \text{mean}(\{R_i\}_{i=1}^G)}{\text{std}(\{R_i\}_{i=1}^G)}. (2)
+
+[p. 14 | section: B.1 RLVR ALGORITHMS | type: Text]
+Our reward function R mirrors that of DeepScaleR (Luo et al., 2025b), employing an Outcome Reward Model. It returns 1 if and only if both the answer and format are correct; otherwise, it returns 0. In summary, our reward function yields:
+
+[p. 14 | section: B.1 RLVR ALGORITHMS | type: Equation]
+R = \begin{cases} 1, & \text{if the answer (e.g. passes basic LaTeX/Sympy checks)} \\ & \text{and format (e.g. exists and ) are both correct,} \\ 0, & \text{otherwise.} \end{cases} (3)
+
+[p. 14 | section: B.2 LOW-VARIANCE PASS@K ESTIMATION | type: Text]
+Pass@k is a measure of a model's problem - solving ability, indicating the probability that the model can generate at least one correct solution in k attempts. Specifically, for each problem x_i in the evaluation dataset \mathcal{D} , we generate n samples (where n \geq k ) and count the correct ones as c_i . The direct calculation formula is:
+
+[p. 14 | section: B.2 LOW-VARIANCE PASS@K ESTIMATION | type: Equation]
+\operatorname{pass}@k := \mathbb{E}_{x_i \sim \mathcal{D}} \left[ 1 - \left( 1 - \frac{c_i}{n} \right)^k \right] \tag{4}
+
+[p. 14 | section: B.2 LOW-VARIANCE PASS@K ESTIMATION | type: Text]
+However, this formula has excessive variance and insufficient accuracy. To solve this problem, we adopt the unbiased estimation method proposed by Chen et al. (Chen et al., 2021), using the unbiased estimator of pass@k over the dataset:
+
+[p. 14 | section: B.2 LOW-VARIANCE PASS@K ESTIMATION | type: Equation]
+\operatorname{pass}@k := \mathbb{E}_{x_i \sim \mathcal{D}} \left[ 1 - \frac{\binom{n - c_i}{k}}{\binom{n}{k}} \right] \tag{5}
+
+[p. 14 | section: B.2 LOW-VARIANCE PASS@K ESTIMATION | type: Text]
+In our experiments, to ensure sufficient accuracy, we set n such that 2k \le n , which helps further reduce the variance of the estimate.
+
+[p. 14 | section: B.3 MORE RELATED WORKS | type: Text]
+Recent studies show that RL algorithms, such as PPO (Schulman et al., 2017) and GRPO (Guo et al., 2025), can greatly enhance model reasoning capabilities. Building on this, several works have refined this paradigm from different perspectives. One method can be adjusting the reward function. Some studies (Zhu et al., 2025; Shao et al., 2025) directly modify the reward function to improve training efficiency. Other methods introduced intermediate process rewards (Wang et al., 2024; Malik et al., 2025), while Wen et al. (Wen et al., 2025) set up a separate correctness judgment for CoT to obtain rewards.
+
+[p. 15 | section: B.3 MORE RELATED WORKS | type: Text]
+Another novel perspective aims to improve sample efficiency by measuring certainty. For example, TreeRL (Hou et al., 2025) and VinePPO (Kazemnejad et al., 2025) enhanced sample effects by introducing entropy or confidence. MRT (Qu et al., 2025), on the other hand, reused partial trajectories during testing to boost sample efficiency. R3 (Xi et al., 2024) improves RL sample efficiency by decomposing human solution steps and providing preceding steps to guide the model in completing subsequent ones. Further, some research adopted a multi-stage training or reasoning mode, exploring from different angles such as training length (Luo et al., 2025b), question difficulty (Parashar et al., 2025), and fixed-length summaries during reasoning (Yan et al., 2025).
+
+[p. 15 | section: B.3 MORE RELATED WORKS | type: Text]
+In addition to designing better algorithms, another line of research (Shao et al., 2024; Yue et al., 2025; Zhao et al., 2025) has investigated how reinforcement learning affects the frontier of model capabilities, observing a decay in pass@k when k becomes large. In response to this phenomenon, some works (Yu et al., 2025; Liu et al., 2025; An et al., 2025) maintained entropy stability by adjusting training entropy through methods such as increasing the clipping upper bound, enlarging the temperature coefficient, extending the training length, and periodically updating the KL reference model. StepHint (Zhang et al., 2025) also preserved entropy stability by leveraging intermediate thinking content of iterative length as a prompting signal.
+
+[p. 15 | section: B.3 MORE RELATED WORKS | type: Text]
+In contrast to the aforementioned research, our work adopts an orthogonal approach by using part of the ground-truth solution as a hint, without requiring any modifications to the existing reinforcement learning infrastructure. We provide both theoretical justification and empirical evidence that this strategy maintains pass@k without compromising the exploratory capacity of the underlying reinforcement learning algorithm.
+
+[p. 15 | section: B.4 BENCHMARKS | type: Text]
+We evaluate the models' breadth across various tasks in multiple domains, including mathematics, coding, reasoning, and logical inference. For mathematics, we follow DeepScaleR (Luo et al., 2025b) and Nemotron (Moshkov et al., 2025), and conduct assessments on more challenging mathematical datasets such as AIME2024 (MAA, 2024), AIME2025 (MAA, 2025), Olympiad Bench (He et al., 2024), HMMT FEB 25 (hmm, 2025), and BRUMO25 (bru, 2025). Specifically, HMMT25 Feb and BRUMO25 are both sourced from MathArena (Balunović et al., 2025). In the realm of coding, we utilize commonly employed datasets, including Code Contests (Li et al., 2022), Codeforces<sup>1</sup>, and LCB V5 202410-202502 (Jain et al., 2024). For logical reasoning tasks, we assess our models' capabilities using GPQA Diamond (Rein et al., 2023) <sup>2</sup> and Zebraliogic (Lin et al., 2025). The benchmarks related to coding and logical reasoning are all referenced from AReaL (Fu et al., 2025).
+
+[p. 15 | section: B.5 TRAINING DATASET | type: Text]
+The dataset employed in our study is OpenR1-Math-220K (Open-R1 Team, 2025). Prior to commencing the training of the Partial Solution, we conducted a preliminary screening of the dataset. Specifically, we utilized the DeepSeek-R1-Distill-1.5B (Guo et al., 2025) model to perform eight inference operations on each of the 220k data entries in the OpenR1 dataset. Subsequently, we compared the annotated answers in the OpenR1 dataset with the results generated from each inference to tally the number of correct instances for each data entry. Ultimately, we selected the data entries with 0 or 1 correct instance as the training samples for our study. The final dataset size is 26K.
+
+[p. 15 | section: B.5 TRAINING DATASET | type: Text]
+For controlled comparisons, we further split this 26K subset by re-sampling Nemotron-1.5B eight times per item and counting correct completions. We define Easy Data as questions with correct counts in [7,8] and train a model on this split, denoted <code>Easy-Nemotron-1.5B</code>. Similarly, we define Hard Data as questions with correct counts in [0,1] and train <code>Hard-Nemotron-1.5B</code> on this split.
+
+[p. 15 | section: B.5 TRAINING DATASET | type: Text]
+Additionally, for the augmented data, we perform eight inference passes using the model currently under training. We then select samples for which the number of correct predictions falls within the range of [0,4]. This criterion is motivated by the finding that samples exhibiting higher variance are more beneficial for training (Gao et al., 2025; Wang et al., 2025b). The range [0,4] is chosen because it includes the point of maximum sample variance, which is achieved with four correct predictions out of eight trials. For convenience, we refer to augmented data with partial ratio p as Partial-p data.
+
+[p. 15 | section: B.5 TRAINING DATASET | type: Footnote]
+1
+
+[p. 15 | section: B.5 TRAINING DATASET | type: Footnote]
+& lt;sup>2</sup>In the GPQA Diamond dataset, multiple-choice questions are presented in the form of options rather than directly providing the answer, requiring the model to output only A, B, C, or D.
+
+[p. 16 | section: B.6 THE RATIONALE FOR THE CHOICE OF p | type: TableGroup]
+Table 5: Number of problems vs pass rate under different hint levels on OpenMath-Nemotron-1.5B before training. We evaluated OpenMath-Nemotron-1.5B on the OpenR1 dataset after the first round of filtering, with each problem assessed 8 times. The table illustrates the distribution of correct answers (n) where n ∈ {0, 1, . . . , 8}. Hint Levels 0 / 8 1 / 8 2 / 8 3 / 8 4 / 8 5 / 8 6 / 8 7 / 8 8 / 8 Partial-50 143 224 304 472 710 1013 1779 3655 17741 Partial-25 3155 1997 1814 1785 1902 2175 2614 3440 7159 Partial-10 3589 2090 1865 1842 1905 2176 2653 3415 6506 Partial-0 3812 2218 1854 1842 2007 2136 2517 3264 6391
+
+[p. 16 | section: B.6 THE RATIONALE FOR THE CHOICE OF p | type: Text]
+In this study, we evaluated the performance of OpenMath-Nemotron-1.5B on the OpenR1 dataset under various hint levels. The evaluation was performed after the first round of filtering, and each problem was assessed 8 times to capture the predictive distribution. The resulting table (Table 5) shows the distribution of correct answers across different hint levels, where the values represent the number of times the model answered correctly (n ∈ {0, 1, . . . , 8}).
+
+[p. 16 | section: B.6 THE RATIONALE FOR THE CHOICE OF p | type: Text]
+The selection of the hint parameter p was primarily based on these evaluation results. As shown in the table, the performance with a Partial-50 hint significantly reduces task difficulty, as evidenced by the high pass rates across most levels. In contrast, Partial-25 (25% hint) exhibits a performance pattern similar to that of the no-hint scenario (Partial-0), with only marginal differences in task difficulty.
+
+[p. 16 | section: B.6 THE RATIONALE FOR THE CHOICE OF p | type: Text]
+This minimal difference in difficulty between Partial-0 and Partial-25 suggests that training with Partial-25 does not provide substantial gains compared to Partial-0. Consequently, we adopted a stepwise design in which the hint level is first set to p = 50%, followed by p = 25%, to evaluate the model's performance under varying conditions.
+
+[p. 16 | section: B.7 EVALUATION SETUP | type: Text]
+We configured the models to have a maximum generation length of 32,768 tokens. In line with DeepSeek-R1 (Guo et al., 2025) , we utilized pass@k evaluation (Chen et al., 2021) , with the formula detailed in B.2. We reported pass@1 using a non-zero temperature. Specifically, we used a sampling temperature of 0.7 and a top-p value of 0.95 to generate k responses per question, typically set at 32, with deviations explicitly noted. Particular attention should be paid to the fact that, although we incorporated partial Solution during training, it was not included in the evaluation phase.
+
+[p. 16 | section: DeepScaleR Others' Inference | type: Text]
+<|User|>{input}
+
+[p. 16 | section: DeepScaleR Others' Inference | type: Text]
+Please reason step by step, and put your final answer within \boxed{}.<|Assistant|><think>
+
+[p. 16 | section: Nemotrion Coding's Inference | type: Text]
+<|im_start|>user
+
+[p. 16 | section: Nemotrion Coding's Inference | type: Text]
+{input}
+
+[p. 16 | section: Nemotrion Coding's Inference | type: Text]
+<|im_end|> <|im_start|>assistant
+
+[p. 16 | section: Nemotrion Coding's Inference | type: Text]
+<think>
+
+[p. 16 | section: Nemotrion Others' Inference | type: Text]
+<|im_start|>system
+
+[p. 16 | section: Nemotrion Others' Inference | type: Text]
+Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>user
+
+[p. 17 | section: Nemotrion Others' Inference | type: Text]
+{input}<lim_endl> <lim_startl>assistant
+
+[p. 17 | section: Training prompt with partial solutions (math RL) | type: Text]
+{Problem}
+
+[p. 17 | section: Training prompt with partial solutions (math RL) | type: Text]
+## Hint: {Partial Solution}
+
+[p. 17 | section: Training prompt with partial solutions (math RL) | type: Text]
+Please reason step by step, and put your final answer within \boxed{}.
+
+[p. 17 | section: C.1 Proofs | type: Text]
+Theorem 4.4 (Lower Bound on RL Learnability under Solution Inaccessibility). Given a probability threshold \delta_p > 0 , if for every question q \in \mathcal{Q} , the model capacity set C(q, \delta_p) does not intersect with the solution set \mathcal{S}(q) , i.e.,
+
+[p. 17 | section: C.1 Proofs | type: Equation]
+C(q, \delta_n) \cap \mathcal{S}(q) = \emptyset, \quad \forall q \in \mathcal{Q},
+
+[p. 17 | section: C.1 Proofs | type: Text]
+then under Assumption 4.3, when training RL for T steps with B samples per step such that TB = \Theta(1/\delta_p) , there is a constant probability that the RL algorithm will not update the model.
+
+[p. 17 | section: C.1 Proofs | type: Text]
+Proof. Let p_{\text{sol}} = \sum_{\tau^* \in \mathcal{S}(q)} P_{\mu}(\tau^*|q) denote the cumulative generation probability of any solution trajectory. By C(q, \delta_p) \cap \mathcal{S}(q) = \emptyset and Def 4.2:
+
+[p. 17 | section: C.1 Proofs | type: Equation]
+p_{\text{sol}} = \sum_{\tau^* \in \mathcal{S}(q)} P_{\mu}(\tau^*|q) < \delta_p
+
+[p. 17 | section: C.1 Proofs | type: Text]
+For N=TB independent samples across T steps with batch size B, the probability of complete failure (no solution sampled) is:
+
+[p. 17 | section: C.1 Proofs | type: Equation]
+\mathbb{P}(\text{failure}) = (1 - p_{\text{sol}})^N > (1 - \delta_p)^N
+
+[p. 17 | section: C.1 Proofs | type: Text]
+Given TB = \Theta(1/\delta_p) , we have:
+
+[p. 17 | section: C.1 Proofs | type: Equation]
+(1 - \delta_p)^N > (1 - \delta_p)^{\Theta(1/\delta_p)} = \Theta(1).
+
+[p. 17 | section: C.1 Proofs | type: Text]
+The last inequality follows from the fact that (1-x)^{1/x} > \exp(-1/(1-x)) for x \in (0,1) . By Assumption 4.3, if no solution is found, the model weights remain unchanged.
+
+[p. 17 | section: C.1 Proofs | type: Text]
+Lemma C.1 (Upper Bound on Sampling Budget for Solution Given Hint). Given a question q \in \mathcal{Q} , if there exists a hint h_q for the question q (Def. 4.5), then if we perform TB = \Theta(1/\delta_p') = \Theta(\delta_p^\epsilon/\sqrt{\delta_p}) i.i.d sampling over the initial model conditioned on (q, h_q) , we can find a valid solution with a constant probability.
+
+[p. 17 | section: C.1 Proofs | type: Text]
+Proof. By Definition 4.5, we know:
+
+[p. 17 | section: C.1 Proofs | type: Equation]
+1. P_{\mu}(h_a|q) \geq \delta'_n
+
+[p. 17 | section: C.1 Proofs | type: Equation]
+2. \exists s_q \in \mathcal{S}(q) : P_{\mu}(s_q|(q,h_q)) \geq \delta_p'
+
+[p. 17 | section: C.1 Proofs | type: Text]
+With N = TB \ge 10/\delta'_p independent samples conditioned on (q, h_q) , the probability of not finding the solution s_q is:
+
+[p. 17 | section: C.1 Proofs | type: Equation]
+\mathbb{P}(\text{no solution}) = (1 - P_{\mu}(s_q|(q, h_q)))^N \le (1 - \delta_p')^{10/\delta_p'} \le \exp(-10) < 0.01.
+
+[p. 17 | section: C.1 Proofs | type: Text]
+Therefore, \mathbb{P}(\text{finding solution}) > 0.99 .
+
+[p. 17 | section: C.1 Proofs | type: Text]
+Theorem 4.6 (Informal Upper Bound on RL Learnability with Hint). If we have a hint h_q for every question q \in \mathcal{Q} (Def. 4.5), then there exists an RL algorithm that can output a policy \pi_\theta such that \mathbb{E}_{q \sim \mathrm{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \pi_\theta(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99 with O(1/\delta_p') sampling budget with high probability.
+
+[p. 18 | section: C.1 Proofs | type: Text]
+This theorem is a direct corollary of the Theorem 5 regarding the bandit setup in (Mei et al., 2022). Because the setup here is relatively simple, we also present a detailed proof for this special case here. We first formalize our setup as follows:
+
+[p. 18 | section: C.1 Proofs | type: Text]
+Assumption C.2 (Tabular RL with Hint). We consider the tabular RL setting with softmax policy parameterization. There exists a finite set of possible questions \mathcal{Q} and a finite set of possible solutions \mathcal{S} . For each question q \in \mathcal{Q} , there exists a hint h_q , which is a subset of solutions h_q \subseteq \mathcal{S} .
+
+[p. 18 | section: C.1 Proofs | type: Text]
+The policy is parameterized by a |S| \times |Q| matrix \theta in the following way:
+
+[p. 18 | section: C.1 Proofs | type: Equation]
+\mu_{\theta}(s|q) = \frac{\exp(\theta_{s,q})}{\sum_{s' \in S} \exp(\theta_{s',q})}
+
+[p. 18 | section: C.1 Proofs | type: Text]
+Here the setup is different than the autoregressive setting in our experiments and simplify the model to a tabular setup for the simplicity of analysis. We now restate the assumption on the existence of hint in this setup.
+
+[p. 18 | section: C.1 Proofs | type: Text]
+Assumption C.3 (Hint Existence, Formal Version of Definition 4.5). For each question q \in \mathcal{Q} , there exists a hint h_q \subseteq \mathcal{S} such that \sum_{s \in h_q} P_{\mu}(s|q) \ge \delta'_p . Further, there exists a solution s_q \in \mathcal{S} such that P_{\mu}(s_q|q) \ge \delta'_p \sum_{s \in h_q} P_{\mu}(s|q) .
+
+[p. 18 | section: C.1 Proofs | type: Text]
+RL Algorithm: We will first sample \Theta(1/\delta'_p) action based on the policy \mu_\theta conditioned on the question q and the hint h_q . Then we will do a one-step policy gradient update on our policy. Noted that here we can reach high reward within one step because the reward function is deterministic.
+
+[p. 18 | section: C.1 Proofs | type: Text]
+Theorem C.4 (Formal Version of Theorem 4.6). Under Assumption C.2 and Assumption C.3, running 1 steps of policy gradient update with sampling budget \Theta(1/\delta'_{v}) , the learned policy achieves:
+
+[p. 18 | section: C.1 Proofs | type: Equation]
+\mathbb{E}_{q \sim \text{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \mu_{\theta}(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99
+
+[p. 18 | section: C.1 Proofs | type: Text]
+with probability 0.99.
+
+[p. 18 | section: C.1 Proofs | type: Text]
+Proof. First, by Assumption C.3, for any question q, we have:
+
+[p. 18 | section: C.1 Proofs | type: Equation]
+\sum_{s \in h_q} P_{\mu}(s|q) \geq \delta_p \quad \text{and} \quad \exists s_q : P_{\mu}(s_q|q) \geq \delta_p' \sum_{s \in h_q} P_{\mu}(s|q)
+
+[p. 18 | section: C.1 Proofs | type: Text]
+With sampling budget N = \Theta(|\mathcal{Q}|/\delta_p') , by Lemma C.1 and the union bound, we will find a solution s_q for every question q with probability at least 0.99. Suppose the found set of solutions for question q is S_q and all sampled solutions are s^{(1)}, \ldots, s^{(N)} . Then because
+
+[p. 18 | section: C.1 Proofs | type: Equation]
+\nabla_{\theta} \log \mu_{\theta}(s|q) = e_s - \sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q)e_{s'}
+
+[p. 18 | section: C.1 Proofs | type: Text]
+We have the policy gradient being
+
+[p. 18 | section: C.1 Proofs | type: Equation]
+\begin{aligned} \text{PG}_{:,q} &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] \nabla_{\theta} \log \mu_{\theta}(s^{(i)}|q) \\ &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] (e_{s^{(i)}} - \sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q) e_{s'}) \\ &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] e_{s^{(i)}} - \left(\frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q]\right) \left(\sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q) e_{s'}\right). \end{aligned}
+
+[p. 18 | section: C.1 Proofs | type: Text]
+We can make two simple observations:
+
+[p. 18 | section: C.1 Proofs | type: ListGroup]
+1. For every s \notin S_a , PG_{s,a} < 0 . 2. There exists a s^* \in S_q such that PG_{s^*,q} > 0 .
+
+[p. 18 | section: C.1 Proofs | type: Text]
+Therefore, consider the updated parameters
+
+[p. 18 | section: C.1 Proofs | type: Equation]
+\theta'_{s,q} = \theta_{s,q} + \eta PG_{s,q}
+
+[p. 18 | section: C.1 Proofs | type: Text]
+If \eta is large enough, we know that \sum_{s \in S_-} P_{\mu_{\theta'}}(s|q) \ge 0.99 . This completes the proof.
+
+[p. 19 | section: D.1 ABLATION STUDY WITHOUT HINT | type: TableGroup]
+Table 6: Ablation without hint on Nemotron-1.5B: Pass@1 (avg@32) on challenging maths benchmarks. "QUESTA-Nemotron-1.5B w/o hint" trains RL on the same data but removes hints from the prompt, while "w/ hint" uses partial-solution hints during training. With hints, the model improves all benchmarks and achieves a +2.82 average gain over w/o hint (63.26 vs. 60.44), on top of the improvements over the base model. The one using hint requires nearly half the number of steps compared to the one not using hint to achieve the same performance. Model AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 Avg Nemotron-1.5B 61.77 49.50 31.56 64.62 58.23 53.14 QUESTA-Nemotron-1.5B w/o hint (2K step) 69.48 59.79 38.85 68.05 66.04 60.44 QUESTA-Nemotron-1.5B w/ hint (1.1K step) 69.27 60.00 37.92 69.72 68.33 61.05 QUESTA-Nemotron-1.5B w/ hint (2K step) 72.50 62.29 41.67 70.36 69.48 63.26
+
+[p. 19 | section: D.1 ABLATION STUDY WITHOUT HINT | type: FigureGroup]
+Figure 7: Pass@k comparison on Nemotron-1.5B for RL with vs. without hints. Training with hints consistently dominates across k and avoids the performance drop at larger k seen in standard RL. Hints are used only during training; evaluation uses no hints.
+
+[p. 19 | section: D.1 ABLATION STUDY WITHOUT HINT | type: Text]
+We ablate the role of hints by training RL on Nemotron-1.5B with and without partial-solution hints. Here, QUESTA-Nemotron-1.5B w/o hint denotes RL on the same data and schedule but with the hint removed from the prompt; QUESTA-Nemotron-1.5B w/h hint uses identical settings except that the partial solution is provided as a hint during training. As summarized in Table 6, removing the hint still improves over the base model (average Pass@1: 53.14 \rightarrow 60.44 ), but adding the hint yields a
+
+[p. 20 | section: D.1 ABLATION STUDY WITHOUT HINT | type: Text]
+further +2.82 average gain ( 60.44 \rightarrow 63.26 ), with consistent improvements across all five benchmarks. The one using hint requires nearly half the number of steps compared to the one not using hint to achieve the same performance. Note that hints are used only during training; all evaluations are conducted without hints.
+
+[p. 20 | section: D.1 ABLATION STUDY WITHOUT HINT | type: Text]
+Figure 7 compares Pass@k curves. The w/hint model lifts the entire curve across k and avoids the degradation at larger k commonly observed in standard RL, while the w/o hint variant brings smaller gains that taper off as k increases. A possible reason for this phenomenon is that, without hints, extremely difficult problems remain unlearned. Consequently, during reinforcement learning training, the model prioritizes improving performance on problems that have become relatively easier as training progresses. This leads the model to become overly confident, thereby reducing its Pass@k metric. In contrast, when hints are provided, the model still prioritizes learning more difficult problems—this is because such problems can provide effective learning signals.
+
+[p. 20 | section: D.2 ABLATION STUDY WITH DIFFERENT MODELS | type: TableGroup]
+Table 7: Performance comparison on DeepScaleR-1.5B: Pass@1 (avg@32) across maths benchmarks. QUESTA consistently improves all tasks and raises the average by +6.50 points. Model AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 Avg DeepScaleR-1.5B 40.42 31.35 19.27 52.97 37.40 36.28 QUESTA-DeepScaleR-1.5B 49.16 35.94 21.77 58.69 48.33 42.78 Table 8: Performance comparison (Pass@1, averaged over 32 samples) showing the impact of QUESTA across benchmarks in other domains, including general knowledge, logic, and coding tasks. We observe minor cross-domain generalization on all these benchmarks, despite QUESTA being applied exclusively in the maths domain.
+
+[p. 20 | section: D.2 ABLATION STUDY WITH DIFFERENT MODELS | type: Table]
+Model | GPQA Diamond Zebralogic Code Contest All Codeforces LCB V5 202410-202502 Avg DeepScaleR-1.5B 38.5 14.26 9.07 8.79 19.57 18.04 QUESTA-DeepScaleR-1.5B 39.2 14.98 10.1 8.9 20.9 18.82
+
+[p. 20 | section: D.2 ABLATION STUDY WITH DIFFERENT MODELS | type: FigureGroup]
+Figure 8: Pass@k on DeepScaleR-1.5B: QUESTA raises the entire curve across k and avoids the large-k drop often seen in standard RL. The increasing gap with k indicates improved sample diversity rather than overconfident collapse. No hints are used at evaluation.
+
+[p. 20 | section: D.2 ABLATION STUDY WITH DIFFERENT MODELS | type: Text]
+We next examine model-family transfer by applying QUESTA to DeepScaleR-1.5B. We train for 750 steps on DeepScaleR-1.5B Stage 2 Luo et al. (2025b) on the QUESTA first stage.
+
+[p. 21 | section: D.2 ABLATION STUDY WITH DIFFERENT MODELS | type: FigureGroup]
+Figure 9: We conducted training of the DeepScaleR model employing the QUESTA with a dataset comprising 26,000 questions. The average pass rate was calculated from a sample of 8 instances. The initial graph represents the scenario without the incorporation of partial solutions, while the subsequent graph depicts the situation where partial solutions were included. The application of QUESTA significantly diminishes the incidence of unsolved or partially addressed problems within the training dataset. Concurrently, it has come to our attention that our previous method of data curation was not entirely accurate; in fact, the amount of data providing meaningful training signals is less abundant than anticipated, suggesting the potential for further refinement of the dataset.
+
+[p. 21 | section: D.2 ABLATION STUDY WITH DIFFERENT MODELS | type: Text]
+As shown in Table 7, QUESTA-DeepScaleR-1.5B improves every maths benchmark over the base model, achieved an average improvement of 6%, indicating that the benefits of QUESTA are not tied to a single architecture.
+
+[p. 21 | section: D.2 ABLATION STUDY WITH DIFFERENT MODELS | type: Text]
+Pass@k behavior mirrors these gains. In Figure 8, QUESTA-DeepScaleR lifts the entire Pass@k curve across k and avoids the degradation at larger k reported in standard RL settings. The widening gap at larger k suggests improved candidate diversity rather than overfitting to a single trajectory, consistent with our general pass@k analysis in Appendix B.2. Complementing this, Figure 9 shows that on the 26K training set (evaluated without hints), mass shifts away from the 0/8-1/8 bins toward higher pass rates, reducing unsolved or partially solved cases. Hints are used only during training and are removed at evaluation time.
+
+[p. 21 | section: D.2 ABLATION STUDY WITH DIFFERENT MODELS | type: Text]
+Beyond maths, Table 8 reports out-of-distribution (OOD) results on general knowledge, logic, and coding. QUESTA-DeepScaleR-1.5B achieves small but consistent gains (Avg: 18.04 \rightarrow 18.82; +0.78), suggesting that the improved reasoning patterns transfer modestly beyond the training domain.
+
+[p. 21 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 10: Training dynamics of QUESTA-Nemotron-1.5B w/o hint.
+
+[p. 22 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 11: Training dynamics of QUESTA-Nemotron-1.5B-50.
+
+[p. 22 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 12: Training dynamics of QUESTA-Nemotron-1.5B-50-25-0.
+
+[p. 22 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 13: Training dynamics of Easy-Nemotron-1.5B.
+
+[p. 22 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 14: Training dynamics of Hard-Nemotron-1.5B.
+
+[p. 23 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 15: The training dynamics of QUESTA-DeepScaleR-1.5B. The first and second charts show the changes in the average reward and average inference length of the rollout samples that include all incorrect/correct ones, respectively. The third chart shows the average entropy excluding all incorrect/correct rollout samples.
+
+[p. 23 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 16: Training dynamics of QUESTA-Nemotron-1.5B on OpenMathReasoning (Moshkov et al., 2025). The first and second charts show the changes in the average reward and average inference length of the rollout samples that include all incorrect/correct ones, respectively. The third chart shows the average entropy excluding all incorrect/correct rollout samples. Dynamics closely mirror those on OpenR1-Math-220K, with no entropy collapse.
+
+[p. 24 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 17: We compare pass@k curves of RLVR-trained models with and without QUESTA.
+
+[p. 25 | section: D.3 SUPPLEMENTAL EXPERIMENT DETAIL | type: FigureGroup]
+Figure 18: Comparison of RL training dynamics: Training with only hard problems (green) makes progress very slowly due to sparse rewards, while our method with partial solutions (orange) accelerates learning and consistently achieves higher accuracy across training steps.

iclr26/3MifB0f7qR/assets.json

@@ -0,0 +1,200 @@
+[
+  {
+    "filename": "_page_0_Figure_6.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_0_Figure_6.jpeg",
+    "bytes": 56713,
+    "width": 796,
+    "height": 487,
+    "aspect_ratio": 1.6344969199178645,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_0_Figure_6.jpeg"
+  },
+  {
+    "filename": "_page_18_Figure_5.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_18_Figure_5.jpeg",
+    "bytes": 138816,
+    "width": 1088,
+    "height": 977,
+    "aspect_ratio": 1.1136131013306039,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_18_Figure_5.jpeg"
+  },
+  {
+    "filename": "_page_19_Figure_8.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_19_Figure_8.jpeg",
+    "bytes": 78616,
+    "width": 1108,
+    "height": 549,
+    "aspect_ratio": 2.0182149362477233,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_19_Figure_8.jpeg"
+  },
+  {
+    "filename": "_page_1_Figure_1.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_1_Figure_1.jpeg",
+    "bytes": 77325,
+    "width": 1011,
+    "height": 534,
+    "aspect_ratio": 1.8932584269662922,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_1_Figure_1.jpeg"
+  },
+  {
+    "filename": "_page_20_Figure_1.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_20_Figure_1.jpeg",
+    "bytes": 50172,
+    "width": 1128,
+    "height": 427,
+    "aspect_ratio": 2.641686182669789,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_20_Figure_1.jpeg"
+  },
+  {
+    "filename": "_page_20_Figure_7.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_20_Figure_7.jpeg",
+    "bytes": 47836,
+    "width": 1112,
+    "height": 363,
+    "aspect_ratio": 3.0633608815426996,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_20_Figure_7.jpeg"
+  },
+  {
+    "filename": "_page_21_Figure_1.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_21_Figure_1.jpeg",
+    "bytes": 44424,
+    "width": 1118,
+    "height": 366,
+    "aspect_ratio": 3.0546448087431695,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_21_Figure_1.jpeg"
+  },
+  {
+    "filename": "_page_21_Figure_3.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_21_Figure_3.jpeg",
+    "bytes": 47576,
+    "width": 1108,
+    "height": 367,
+    "aspect_ratio": 3.019073569482289,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_21_Figure_3.jpeg"
+  },
+  {
+    "filename": "_page_21_Figure_5.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_21_Figure_5.jpeg",
+    "bytes": 43154,
+    "width": 1099,
+    "height": 370,
+    "aspect_ratio": 2.9702702702702704,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_21_Figure_5.jpeg"
+  },
+  {
+    "filename": "_page_21_Figure_7.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_21_Figure_7.jpeg",
+    "bytes": 45577,
+    "width": 1107,
+    "height": 360,
+    "aspect_ratio": 3.075,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_21_Figure_7.jpeg"
+  },
+  {
+    "filename": "_page_22_Figure_1.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_22_Figure_1.jpeg",
+    "bytes": 43212,
+    "width": 1124,
+    "height": 358,
+    "aspect_ratio": 3.1396648044692737,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_22_Figure_1.jpeg"
+  },
+  {
+    "filename": "_page_22_Figure_3.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_22_Figure_3.jpeg",
+    "bytes": 46469,
+    "width": 1126,
+    "height": 370,
+    "aspect_ratio": 3.0432432432432432,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_22_Figure_3.jpeg"
+  },
+  {
+    "filename": "_page_23_Figure_1.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_23_Figure_1.jpeg",
+    "bytes": 173521,
+    "width": 1106,
+    "height": 1157,
+    "aspect_ratio": 0.9559204840103717,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_23_Figure_1.jpeg"
+  },
+  {
+    "filename": "_page_24_Figure_1.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_24_Figure_1.jpeg",
+    "bytes": 111542,
+    "width": 1100,
+    "height": 1078,
+    "aspect_ratio": 1.0204081632653061,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_24_Figure_1.jpeg"
+  },
+  {
+    "filename": "_page_2_Figure_1.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_2_Figure_1.jpeg",
+    "bytes": 44786,
+    "width": 914,
+    "height": 440,
+    "aspect_ratio": 2.077272727272727,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_2_Figure_1.jpeg"
+  },
+  {
+    "filename": "_page_3_Picture_9.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_3_Picture_9.jpeg",
+    "bytes": 2037,
+    "width": 133,
+    "height": 61,
+    "aspect_ratio": 2.180327868852459,
+    "keep": false,
+    "reject_reason": "too_small",
+    "model_path": null
+  },
+  {
+    "filename": "_page_7_Figure_1.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_7_Figure_1.jpeg",
+    "bytes": 47360,
+    "width": 1116,
+    "height": 356,
+    "aspect_ratio": 3.134831460674157,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_7_Figure_1.jpeg"
+  },
+  {
+    "filename": "_page_7_Figure_3.jpeg",
+    "path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/marker_raw/3MifB0f7qR/marker_markdown/3MifB0f7qR/_page_7_Figure_3.jpeg",
+    "bytes": 43787,
+    "width": 1006,
+    "height": 387,
+    "aspect_ratio": 2.599483204134367,
+    "keep": true,
+    "reject_reason": null,
+    "model_path": "assets/_page_7_Figure_3.jpeg"
+  }
+]

iclr26/3MifB0f7qR/dataset_meta.json

@@ -0,0 +1,64 @@
+{
+  "paper_id": "3MifB0f7qR",
+  "pipeline": "Paper2Markdown-V3",
+  "ok": true,
+  "page_count": 25,
+  "chunk_count": 187,
+  "main_body_chunk_count": 77,
+  "appendix_chunk_count": 100,
+  "reference_chunk_count": 10,
+  "model_text_chars": 36398,
+  "raw_markdown_chars": 76702,
+  "sanitized_chars": 72894,
+  "page_provenance": {
+    "min_page": 1,
+    "max_page": 25,
+    "invalid_count": 0
+  },
+  "marker_block_type_counts": {
+    "Caption": 1,
+    "Equation": 24,
+    "FigureGroup": 17,
+    "Footnote": 6,
+    "ListGroup": 7,
+    "PageFooter": 25,
+    "PageHeader": 25,
+    "Picture": 1,
+    "SectionHeader": 38,
+    "Table": 2,
+    "TableGroup": 6,
+    "Text": 134
+  },
+  "asset_count_raw": 18,
+  "asset_count_model_kept": 17,
+  "asset_count_rejected": 1,
+  "asset_reject_reasons": {
+    "kept": 17,
+    "too_small": 1
+  },
+  "artifact_leak_audit": {
+    "ok": true,
+    "hits": {
+      "Anonymous Authors": [],
+      "ACKNOWLEDGMENT": [],
+      "OpenReview": [],
+      "\"accept_label\"": [],
+      "\"decision\"": [],
+      "\"decision_tier\"": [],
+      "\"source_status\"": [],
+      "Meta-review": [],
+      "Official Review": [],
+      "official_reviews": [],
+      "meta_reviews": [],
+      "suggested_verdict_score": []
+    },
+    "artifact_count": 2
+  },
+  "default_model_input": "model_text_v3.txt",
+  "appendix_input": "appendix_text_v3.txt",
+  "reference_input": "reference_text_v3.txt",
+  "source": "iclr2026_test_public",
+  "run_name": "iclr26_20260429_1952_all",
+  "partial_snapshot": true,
+  "snapshot_created_at": "2026-04-30T02:34:47.606558+00:00"
+}

iclr26/3MifB0f7qR/main_body_chunks.jsonl

@@ -0,0 +1,77 @@
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0000", "section": "ABSTRACT", "page_start": 1, "page_end": 1, "type": "Text", "text": "Reinforcement learning (RL) has emerged as a central paradigm for training large language models (LLMs) in reasoning tasks. Yet recent studies (Yue et al., 2025; Liu et al., 2025) question RL's ability to incentivize reasoning capacity beyond the base model. This raises a key challenge: how can RL be adapted to solve harder reasoning problems more effectively? To address this challenge, we propose a simple yet effective strategy via Question Augmentation: introduce partial solutions during training to reduce problem difficulty and provide more informative learning signals. Our method, QuestA, when applied during RL training on math reasoning tasks, not only improves pass@1 but also pass@k—particularly on problems where standard RL struggles to make progress. This enables continual improvement over strong open-source models such as DEEPSCALER and OPENMATH NEMOTRON, further enhancing their reasoning capabilities. We achieve new state-of-the-art results on math benchmarks using 1.5B-parameter models: 72.50% (+10.73%) on AIME24, 62.29% (+12.79%) on AIME25, and 41.67% (+10.11%) on HMMT25. Code, data and model are available at", "source": "marker_v2", "marker_block_id": "/page/0/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0001", "section": "ABSTRACT", "page_start": 1, "page_end": 1, "type": "FigureGroup", "text": "Figure 1: QUESTA is a data augmentation method that injects partial solutions to effectively scaffold RL training on hard reasoning problems. We construct 26K high-quality augmented prompts from challenging instances in OpenR1 (Open-R1 Team, 2025), and fine-tune models using 32K-context-length RL. When applied to Nemotron-1.5B, QUESTA delivers substantial performance gains—achieving new state-of-the-art results across all math benchmarks for 1.5B-parameter models.", "source": "marker_v2", "marker_block_id": "/page/0/FigureGroup/112"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0002", "section": "1 Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "Frontier large language models (LLMs), including OpenAI-O1, O3 (Jaech et al., 2024), DeepSeek-R1 (Guo et al., 2025), Qwen3 (Yang et al., 2025), and Gemini 2.5 (Gemini Team, Google DeepMind,", "source": "marker_v2", "marker_block_id": "/page/0/Text/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0003", "section": "1 Introduction", "page_start": 1, "page_end": 1, "type": "Footnote", "text": "& lt;sup>‡</sup>This work is independent of and outside of the work at Amazon.", "source": "marker_v2", "marker_block_id": "/page/0/Footnote/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0004", "section": "1 Introduction", "page_start": 2, "page_end": 2, "type": "FigureGroup", "text": "Figure 2: We compare pass@k curves of RLVR-trained models, with and without QUESTA. As a controlled experiment, we perform RL training using either easy or hard prompts. Standard RL on easy prompts (red) shows clear degradation in pass@k as k increases compared to the base model (blue). Training on hard prompts (green) improves pass@k, but comes at the cost of substantially longer training. This motivates our development of QUESTA, which scaffolds hard problems to improve training efficiency and delivers consistently stronger results: the RL+QUESTA model (orange) stays above standard RL (red) across all k, while also preserving or improving performance at larger k relative to RL trained with hard prompts.", "source": "marker_v2", "marker_block_id": "/page/1/FigureGroup/99"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0005", "section": "1 Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "2025), have exhibited exceptional performance on high-complexity reasoning tasks spanning mathematics, programming, and formal logic. Recent advances in the field have increasingly prioritized reinforcement learning paradigms (RL), among which Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a scalable and efficient approach to enhancing reasoning capabilities. Using automatically verifiable signals, RLVR enables alignment between model output and objective correctness, thus addressing a critical limitation of traditional RL for reasoning.", "source": "marker_v2", "marker_block_id": "/page/1/Text/3"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0006", "section": "1 Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "However, the community remains divided on a fundamental question regarding RLVR: does it expand the model's intrinsic reasoning capacity, or merely exploit pre-existing knowledge encoded in the base model? Recent research (Yue et al., 2025; Liu et al., 2025; Zhao et al., 2025) show that while state-of-the-art RL methods (e.g., GRPO, DAPO) (Guo et al., 2025; Yu et al., 2025; An et al., 2025) can enhance the pass@1 metric by reinforcing high-reward completions, they encounter significant limitations when tackling high-difficulty tasks where the base model performs poorly. This phenomenon differs from that observed in Supervised Fine-Tuning (SFT) Luo et al. (2025a). Within the SFT paradigm, enhancing the diversity of problem difficulty serves as a critical factor, as it can effectively improve the model's performance on downstream tasks. However, in the framework of RLVR, the inclusion of easy prompts tends to undermine the model's inherent reasoning capabilities.", "source": "marker_v2", "marker_block_id": "/page/1/Text/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0007", "section": "1 Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "One insightful explanation (Cui et al., 2025; Wang et al., 2025a) for the drop suggests that model overfits on correct solutions and hence causes entropy collapse, limiting its ability to explore. To validate this, we design a controlled setup that separates prompts into easy and hard groups. When applying RLVR on the Nemotron 1.5B model (Moshkov et al., 2025) with the OpenR1 dataset, we find that training on easy prompts leads to a clear decline in pass@k accuracy (Figure 2).", "source": "marker_v2", "marker_block_id": "/page/1/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0008", "section": "1 Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "Given these findings, we observe that training with hard prompts is more beneficial than with easy ones. Yet, RL training on hard problems tends to be much slower, as sparse reward signals and limited sample efficiency hinder progress. The key challenge, then, is how to structure the learning process to fully expand reasoning capabilities while mitigating the inefficiency of RL on hard tasks . To this end, we introduce QUESTA: a parsimonious and efficient strategy that dynamically adjusts problem difficulty during RL training. The core contributions of this work are threefold:", "source": "marker_v2", "marker_block_id": "/page/1/Text/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0009", "section": "1 Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "• We notice that the evolution of model capacity in RLVR critically depends on dataset difficulty, underscoring the importance of training on hard problems to expand reasoning ability.", "source": "marker_v2", "marker_block_id": "/page/1/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0010", "section": "1 Introduction", "page_start": 3, "page_end": 3, "type": "FigureGroup", "text": "Figure 3: Comparison of RL training dynamics: Training with only hard problems (green) makes progress very slowly due to sparse rewards, while our method with partial solutions (orange) accelerates training and consistently achieves higher accuracy across training steps.", "source": "marker_v2", "marker_block_id": "/page/2/FigureGroup/102"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0011", "section": "1 Introduction", "page_start": 3, "page_end": 3, "type": "ListGroup", "text": "We introduce QUESTA, an efficient procedure that controls difficulty by augmenting hard problems with partial solutions. This approach provides a smooth curriculum within RL training and makes high-difficulty tasks more tractable. Through our fully open-sourced training pipeline, QUESTA consistently improves pass@1 and pass@k, enabling 1.5B-parameter models to reach new state-of-the-art performance—72.5% on AIME24, 62.3% on AIME25, and 41.7% on HMMT25 (Table 1). Our theoretical analysis in Section 4 explains why partial-solution augmentation accelerates RL training: by decomposing problems into intermediate steps, the method yields denser reward signals and improves sample efficiency, while still driving the model to master the hardest problems.", "source": "marker_v2", "marker_block_id": "/page/2/ListGroup/103"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0012", "section": "2 Tradeoffs between Reasoning Capacity and Learning Efficiency", "page_start": 3, "page_end": 3, "type": "Text", "text": "Given the ongoing debate on whether reinforcement learning enhances the reasoning capacity of language models, we design a controlled experiment to study how dataset difficulty changes model performances measured by pass@k accuracy. Specifically, we filter out easy problems and hard problems from the 220K OpenR1 dataset, base on model's success rate, each containing around 4K data. We then run RL with GRPO for one thousand steps. This setup allows us to isolate how the choice of prompt difficulty impacts the model's reasoning capacity. In Figure 2 and Figure 3, we provide pass@k comparison and the learning dynamics, we make two observations.", "source": "marker_v2", "marker_block_id": "/page/2/Text/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0013", "section": "2 Tradeoffs between Reasoning Capacity and Learning Efficiency", "page_start": 3, "page_end": 3, "type": "Text", "text": "RL with Easy Prompts Hurts pass@k and Reasoning Capacity. Training on easy or already-solvable problems leads to overfitting on shallow patterns, reinforcing confidence rather than expanding reasoning capacity. While pass@1 may rise, output diversity declines and performance on harder benchmarks deteriorates, with pass@k dropping at larger k (see Figure 2). This suggests that the model exploits familiar solution modes instead of exploring new trajectories. To truly expand capacity, RL training should focus on hard problems, where the policy is forced to explore and acquire novel solution strategies.", "source": "marker_v2", "marker_block_id": "/page/2/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0014", "section": "2 Tradeoffs between Reasoning Capacity and Learning Efficiency", "page_start": 3, "page_end": 3, "type": "Text", "text": "RL with Hard Prompts Leads to Slow Learning. Training on hard prompts directly targets the reasoning capacity of the model, but the learning process is much slower (see Figure 3) and less sample-efficient. The difficulty arises because RL rewards on these problems are sparse, providing limited gradient signals for policy improvement. We formallize the underlying reason in Section 4 and in Theorem 4.4.", "source": "marker_v2", "marker_block_id": "/page/2/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0015", "section": "2 Tradeoffs between Reasoning Capacity and Learning Efficiency", "page_start": 3, "page_end": 3, "type": "Text", "text": "In practice, not all questions in the training set Q are equally difficult, and one might hope that training on easier examples could generalize to harder ones. However, empirical evidence suggests that RL-based training exhibits a bi-modal pattern in success rates (An et al., 2025): by the end of training, models tend to either solve a question reliably or fail entirely (see Figure 6). This implies that once a question falls outside the model's capacity set, the RL algorithm is unlikely to recover.", "source": "marker_v2", "marker_block_id": "/page/2/Text/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0016", "section": "2 Tradeoffs between Reasoning Capacity and Learning Efficiency", "page_start": 4, "page_end": 4, "type": "Text", "text": "Together, these results highlight a tension: easy prompts dilute reasoning capacity, while hard prompts stall learning altogether. This motivates the need for strategies that can retain the benefits of hard problems while mitigating the inefficiency caused by sparse rewards. To this end, we introduce partial solutions that break a complex question into smaller, more approachable pieces. Theoretical analysis (Theorem 4.6) suggests that appending part of the solutions as hint can greatly improve RL efficiency.", "source": "marker_v2", "marker_block_id": "/page/3/Text/1"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0017", "section": "2 Tradeoffs between Reasoning Capacity and Learning Efficiency", "page_start": 4, "page_end": 4, "type": "Text", "text": "Empirically, we simply choose the hint to be a part of the solution of the original question q and observe faster learning in Figure 3. Surprisingly, even if we don't explicitly train the model to generate the hint, the model's capacity without hint still continues to improve and lead to steady improvement in problems out of reach in standard RL training (see Table 3). We elaborate on implementation details in the next section.", "source": "marker_v2", "marker_block_id": "/page/3/Text/2"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0018", "section": "3 Questa: Question Augmentation with Partial Solutions", "page_start": 4, "page_end": 4, "type": "Text", "text": "QUESTA is a modular augmentation framework designed to inject partial solution sketches into prompts during reinforcement learning (RL) training. It adresses scenarios where the base model fails to generate correct completions—conditions that typically result in sparse reward signals. Distinct from approaches that modify reward functions or optimization algorithms, QUESTA operates at the input level: it transforms original training prompts into more tractable variants, thereby exposing intermediate reasoning steps to the model.", "source": "marker_v2", "marker_block_id": "/page/3/Text/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0019", "section": "Original Prompt", "page_start": 4, "page_end": 4, "type": "Text", "text": "Let \\mathbb N be the set of positive integers. A function f:\\mathbb N\\to\\mathbb N satisfies the equation", "source": "marker_v2", "marker_block_id": "/page/3/Text/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0020", "section": "Original Prompt", "page_start": 4, "page_end": 4, "type": "Equation", "text": "f(f(\\dots f(n)\\dots)) = \\frac{n^2}{f(f(n))} with f(n) applications of f ,", "source": "marker_v2", "marker_block_id": "/page/3/Equation/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0021", "section": "Original Prompt", "page_start": 4, "page_end": 4, "type": "Text", "text": "for all positive integers n. Given this information, determine all possible values of f(1000).", "source": "marker_v2", "marker_block_id": "/page/3/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0022", "section": "Augmented Prompt", "page_start": 4, "page_end": 4, "type": "Text", "text": "Let \\mathbb N be the set of positive integers. The function f:\\mathbb N\\to\\mathbb N satisfies the equation", "source": "marker_v2", "marker_block_id": "/page/3/Text/11"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0023", "section": "Augmented Prompt", "page_start": 4, "page_end": 4, "type": "Equation", "text": "f(f(\\dots f(n)\\dots)) = \\frac{n^2}{f(f(n))} \\quad \\text{with } f(n) \\text{ applications of } f,", "source": "marker_v2", "marker_block_id": "/page/3/Equation/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0024", "section": "Augmented Prompt", "page_start": 4, "page_end": 4, "type": "Text", "text": "for all positive integers n. Given this information, determine all possible values of f(1000).", "source": "marker_v2", "marker_block_id": "/page/3/Text/13"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0025", "section": "Hint: Partial Solution", "page_start": 4, "page_end": 4, "type": "Text", "text": "Analysis shows that f must be an involution, meaning f(f(n)) = n for all n, and it fixes all odd positive integers, so f(n) = n for odd n. For even positive integers, f either fixes the number or swaps it with another even positive integer in a 2-cycle.", "source": "marker_v2", "marker_block_id": "/page/3/Text/15"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0026", "section": "Hint: Partial Solution", "page_start": 4, "page_end": 4, "type": "Text", "text": "Please reason step by step, and put your final answer within \\boxed{}.", "source": "marker_v2", "marker_block_id": "/page/3/Text/16"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0027", "section": "Hint: Partial Solution", "page_start": 4, "page_end": 4, "type": "Caption", "text": "Figure 4: QUESTA augments each original question in the dataset by prepending the first p% of the solution sketch. In our experiments, we apply augmentation using the solution block rather than the reasoning chain-of-thought. The hint percentage p is computed as the ratio of tokens used as hints to the total number of tokens in the solution sketch.", "source": "marker_v2", "marker_block_id": "/page/3/Caption/17"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0028", "section": "Hint: Partial Solution", "page_start": 4, "page_end": 4, "type": "Text", "text": "Question Augmentation Mechanism For a given problem x with an n-step solution trajectory y=(y_1,y_2,\\ldots,y_n) , QUESTA constructs a set of augmented prompts \\{\\tilde{x}^{(p)}\\} , where each \\tilde{x}^{(p)} appends the first p steps of the solution as a prefix to the original question. The parameter p (e.g., p=50% or 25%) quantifies the proportion of the solution revealed, thereby enabling precise control over the difficulty of the augmented prompt.", "source": "marker_v2", "marker_block_id": "/page/3/Text/18"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0029", "section": "Hint: Partial Solution", "page_start": 4, "page_end": 4, "type": "Text", "text": "In our empirical evaluations, we employed the OpenR1-Math-220K dataset (Open-R1 Team, 2025)—a supervised fine-tuning (SFT) corpus containing solution trajectories generated by DEEPSEEK-R1. Each instance in this dataset comprises a detailed chain-of-thought (CoT) section followed by a final solution block. For augmentation, we extracted the final solution (omitting speculative reasoning", "source": "marker_v2", "marker_block_id": "/page/3/Text/19"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0030", "section": "Hint: Partial Solution", "page_start": 5, "page_end": 5, "type": "Text", "text": "within the CoT section). The solution was then truncated at a predefined percentage p and prepended to the original question, yielding the augmented prompt used in RL training, as shwon in Figure 4.", "source": "marker_v2", "marker_block_id": "/page/4/Text/1"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0031", "section": "Hint: Partial Solution", "page_start": 5, "page_end": 5, "type": "Text", "text": "Targeting High-Difficulty Problems QUESTA is applied exclusively to prompts where the base model's pass rate is close to zero. Using the OpenR1-Math-220K dataset, we first employ lightweight heuristic filters to reduce the full 220K problems to 26K of the hardest candidates. These problems are then augmented with partial-solution prefixes where we conduct a second difficulty screening: sample multiple completions from the model for each augmented prompt, and only those instances with consistently low pass rates are retained. This two-stage filtering pipeline yields a final pool of no more than 10K problems, ensuring that augmentation resources are concentrated on the most challenging cases where the base model needs additional guidance and scaffolding.", "source": "marker_v2", "marker_block_id": "/page/4/Text/2"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0032", "section": "Hint: Partial Solution", "page_start": 5, "page_end": 5, "type": "Text", "text": "Integrating with RL Pipelines QUESTA exhibits orthogonality to underlying RL algorithms, enabling seamless integration into existing training pipelines (e.g., GRPO (Shao et al., 2024), DAPO (Yu et al., 2025)) without modifications. Specifically, integration requires only replacing the original rollout dataset with the augmented dataset, while retaining the original reward function and policy update mechanism. To further exploit this input-level flexibility, we extended QUESTA with an iterative curriculum RL paradigm:", "source": "marker_v2", "marker_block_id": "/page/4/Text/3"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0033", "section": "Hint: Partial Solution", "page_start": 5, "page_end": 5, "type": "ListGroup", "text": "1. First, augment the dataset with p = 50%, apply the difficulty filtering with the augmented prompt, and conduct reinforcement learning training until the performance saturates. 2. Second, reduce the augmentation from p = 50% to p = 25%, i.e. provide fewer hints. Again, we apply the difficulty filtering, and conduct reinforcement learning training until convergence.", "source": "marker_v2", "marker_block_id": "/page/4/ListGroup/152"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0034", "section": "Hint: Partial Solution", "page_start": 5, "page_end": 5, "type": "Text", "text": "Here, the rationale for the choice of p is provided in Appendix B.6. By keeping the training signals strong at each stage, the method speeds up convergence on difficult tasks and makes QUESTA a simple, plug-and-play approach for curriculum-based RL.", "source": "marker_v2", "marker_block_id": "/page/4/Text/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0035", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 5, "page_end": 5, "type": "Text", "text": "In this section, we present a theoretical perspective on how question augmentation improves the efficiency of reinforcement learning. Our central thesis is that the primary bottleneck in RL-based reasoning lies in the difficulty of discovering successful trajectories within a finite sampling budget. Question augmentation addresses this challenge by reshaping the learnability landscape —making hard problems more discoverable by increasing the likelihood of encountering correct trajectories.", "source": "marker_v2", "marker_block_id": "/page/4/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0036", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 5, "page_end": 5, "type": "Text", "text": "Motivated by experiments which quantify model capcity with pass@k accuracy, we introduce the following notions of solution set (Definition 4.1) and model capacity set (Definition 4.2) for a given question q and model \\mu . Let \\mathcal V be the vocaboluary set, and let P_{\\mu}(q,\\tau) denote the probability that a language model \\mu generates trajectory \\tau \\in \\mathcal V^* when conditioned on input question q \\in \\mathcal V^* .", "source": "marker_v2", "marker_block_id": "/page/4/Text/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0037", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 5, "page_end": 5, "type": "Text", "text": "Definition 4.1 (Solution Set). Given a question q and a binary reward function R: \\mathcal{V}^* \\times \\mathcal{V}^* \\to \\{0, 1\\} , the solution set is defined as:", "source": "marker_v2", "marker_block_id": "/page/4/Text/10"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0038", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 5, "page_end": 5, "type": "Equation", "text": "S(q) = \\{ \\tau \\in \\mathcal{V}^* \\mid R(q, \\tau) = 1 \\}.", "source": "marker_v2", "marker_block_id": "/page/4/Equation/11"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0039", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 5, "page_end": 5, "type": "Text", "text": "Definition 4.2 (Model Capacity Set). Given a probability threshold \\delta_p > 0 , a language model \\mu , and a question q, define the model capacity set C(q, \\delta_p) as the smallest set of trajectories whose total probability mass is at least 1 - \\delta_p :", "source": "marker_v2", "marker_block_id": "/page/4/Text/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0040", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 5, "page_end": 5, "type": "Equation", "text": "C(q, \\delta_p) = \\arg\\min_{S \\subseteq \\mathcal{V}^*} \\left\\{ |S| \\left| \\sum_{\\tau \\in S} P_{\\mu}(q, \\tau) \\ge 1 - \\delta_p \\right\\}. \\right.", "source": "marker_v2", "marker_block_id": "/page/4/Equation/13"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0041", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 5, "page_end": 5, "type": "Text", "text": "The Model Capacity Set C(q, \\delta_p) intuitively captures the set of most likely output trajectories that the model \\mu can generate for a given input q, up to a small probability threshold \\delta_p .", "source": "marker_v2", "marker_block_id": "/page/4/Text/14"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0042", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 5, "page_end": 5, "type": "Text", "text": "This formalization leads to a critical insight: if the model's capacity set fails to intersect with the solution set—meaning the model is unlikely to generate any correct completions—then the RL process cannot make progress. To articulate this more formally, we begin by stating a standard assumption satisfied by many popular RL algorithms, such as DAPO and online GRPO:", "source": "marker_v2", "marker_block_id": "/page/4/Text/15"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0043", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "Assumption 4.3 (Null Gradient from Zero-Reinforcement). The RL algorithm does not update the model weights if none of the sampled rollouts receives a positive reward (i.e., reward = 1).", "source": "marker_v2", "marker_block_id": "/page/5/Text/1"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0044", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "Under this assumption, we easily the following lower bound, which states that if all training questions are unreachable within the model's capacity set, the RL process is likely to stall entirely:", "source": "marker_v2", "marker_block_id": "/page/5/Text/2"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0045", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "Theorem 4.4 (Lower Bound on RL Learnability under Solution Inaccessibility). Given a probability threshold \\delta_p > 0 , if for every question q \\in \\mathcal{Q} , the model capacity set C(q, \\delta_p) does not intersect with the solution set \\mathcal{S}(q) , i.e.,", "source": "marker_v2", "marker_block_id": "/page/5/Text/3"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0046", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Equation", "text": "C(q, \\delta_p) \\cap \\mathcal{S}(q) = \\emptyset, \\quad \\forall q \\in \\mathcal{Q},", "source": "marker_v2", "marker_block_id": "/page/5/Equation/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0047", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "then under Assumption 4.3, when training RL for T steps with B samples per step such that TB = \\Theta(1/\\delta_p) , there is a constant probability that the RL algorithm will not update the model.", "source": "marker_v2", "marker_block_id": "/page/5/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0048", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "To overcome this limitation, our method QUESTA provides a simple yet effective solution: augment each question in \\mathcal Q with a partial solution to improve the chances of sampling informative trajectories. Formally, we assume the existence of a hint h_q for every question q \\in \\mathcal Q that can guide the model toward discovering a valid completion.", "source": "marker_v2", "marker_block_id": "/page/5/Text/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0049", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "Definition 4.5 (Question Augmentation). For every question q \\in \\mathcal{Q} , hint h_q \\in \\mathcal{V}^* satisfies that for \\delta'_p = \\delta_p^{1/2 - \\epsilon} for some \\epsilon > 0 :", "source": "marker_v2", "marker_block_id": "/page/5/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0050", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "ListGroup", "text": "the hint h_q can be generated with a non-neglible probability: P_{\\mu}(h_q|q) \\geq \\delta_p' there exists a solution to the hinted problem s_q \\in \\mathcal{S}(q) such that s_q can be generated with high probability after s_q , i.e.", "source": "marker_v2", "marker_block_id": "/page/5/ListGroup/160"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0051", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Equation", "text": "P_{\\mu}(s_q|(q,h_q)) = \\delta'_{p}. \\quad R(q,h \\oplus s_q) = 1.", "source": "marker_v2", "marker_block_id": "/page/5/Equation/10"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0052", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "The hint h_q can exist for every question even when the model's capacity set C(q, \\delta_p) does not intersect with the solution set \\mathcal{S}(q) . For instance, if every solution can be decomposed into two steps, and the model can generate each step correctly with probability \\delta_p' = \\sqrt{o(\\delta_p)} , then the possibility of generating two steps correctly at the same time is only o(\\delta_p) .", "source": "marker_v2", "marker_block_id": "/page/5/Text/11"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0053", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "This implies that the sampling budget needed with a hint is asymptotically almost the square root of the budget required without it (\\Theta(1/\\delta_p)) , as given in Theorem 4.4. We further provide a learnability result where we assume the policy is parameterized by a softmax policy parameterization in a classical tabular RL setup.", "source": "marker_v2", "marker_block_id": "/page/5/Text/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0054", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "Theorem 4.6 (Informal Upper Bound on RL Learnability with Hint). If we have a hint h_q for every question q \\in \\mathcal{Q} (Def. 4.5), then there exists an RL algorithm that can output a policy \\pi_\\theta such that \\mathbb{E}_{q \\sim \\mathrm{Uniform}(\\mathcal{Q})}[\\mathbb{P}_{\\tau \\sim \\pi_\\theta(\\cdot|q)}(\\tau \\in \\mathcal{S}(q))] \\geq 0.99 with O(1/\\delta_p') sampling budget with high probability.", "source": "marker_v2", "marker_block_id": "/page/5/Text/13"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0055", "section": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY", "page_start": 6, "page_end": 6, "type": "Text", "text": "Theorem 4.6 provides a theoretical guarantee that the model can reach a high training success rate when partial solution is included. Empirically, we observe the model generalizes well both in-distribution and out-of-distribution to hard questions.", "source": "marker_v2", "marker_block_id": "/page/5/Text/14"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0056", "section": "5 EXPERIMENTS", "page_start": 6, "page_end": 6, "type": "Text", "text": "Dataset. We begin with the OpenR1-Math-220K dataset and use DeepSeek-R1-Distill-1.5B as a weak selection model to filter it down to the 26K hardest items. This set serves as our base prompts. We then use Nemotron-1.5B to sample eight generations per prompt and classify problems into Easy Data (7–8 correct answers) and Hard Data (0–1 correct answers), enabling controlled experiments introduced in Section 2. The exact prompt template is provided in Appendix B.8", "source": "marker_v2", "marker_block_id": "/page/5/Text/16"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0057", "section": "5 EXPERIMENTS", "page_start": 6, "page_end": 6, "type": "Text", "text": "Data Augmentation (QUESTA). To improve the tractability of the problems, we apply QUESTA to prepend the prompt with partial solutions, i.e. first p% of the full solution in the SFT data provided in the OpenR1-Math-220K dataset. After augmentation, we use the initial model at RL training, either Nemotron-1.5B or DeepScaleR-1.5B, to sample 8 generations per augmented prompts and select samples with 0–4 correct predictions. Full details are provided in Appendix B.5. These high-variance cases provide stronger learning signals and make the training process more effective.", "source": "marker_v2", "marker_block_id": "/page/5/Text/17"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0058", "section": "5 EXPERIMENTS", "page_start": 7, "page_end": 7, "type": "TableGroup", "text": "Table 1: Performance comparison (Pass@1, averaged over 32 samples) across maths benchmarks. The best results among the 1.5B models are highlighted in bold. Larger models are shown in gray as reference points. Reported results for DeepSeek-R1-Distill and Qwen3 are taken from their official documentation (Guo et al., 2025; Yang et al., 2025) , while the rest are self-evaluated. Our QUESTA-Nemotron-1.5B achieves state-of-the-art performance among 1.5B models and, notably, matches or even exceeds the performance of DeepSeek-R1-Distill-32B across several benchmarks, despite being over 20× smaller in parameter count. This demonstrates the effectiveness of QUESTA in enhancing small model capabilities through targeted training. Model AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 Avg DeepSeek-R1-Distill-1.5B 28.7 22.3 12.0 52.4 31.8 29.44 Qwen3-1.7B 48.3 36.8 22.19 56.13 44.06 41.50 DeepSeek-R1-Distill-32B 72.6 51.8 33 65.0 68 58.08 Qwen3-8B 76.0 67.3 44.79 68.56 68.33 64.99 Nemotron-1.5B 61.77 49.50 31.56 64.62 58.23 53.14 QUESTA-Nemotron-1.5B 72.50 62.29 41.67 70.36 69.48 63.26", "source": "marker_v2", "marker_block_id": "/page/6/TableGroup/400"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0059", "section": "5 EXPERIMENTS", "page_start": 7, "page_end": 7, "type": "Text", "text": "Training Setup. We use AReaL (Fu et al., 2025) as our RL training framework, applying the GRPO algorithm (Guo et al., 2025) without the Kullback–Leibler (KL) divergence loss. Following DAPO (Yu et al., 2025) , we also dynamically filter out prompts that are either all correct or all incorrect during rollouts. During training, we sample n = 16 responses per prompt with a maximum prompt length of 8192 tokens and a maximum generation length of 24000 tokens, using a sampling temperature of 1.0 and clipping hyperparameters with εlow = εhigh = 0.2. The batch size is 128 with a mini-batch size of 1, equivalent to 128 gradient updates per rollout step. Optimization is performed with AdamW (Kingma & Ba, 2017; Loshchilov & Hutter, 2019) using a constant learning rate of 2 × 10 − 5 . Experiments are conducted on eight NVIDIA H800 (80GB) nodes. Full details of our training method are provided in Appendix B.1.", "source": "marker_v2", "marker_block_id": "/page/6/Text/3"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0060", "section": "5 EXPERIMENTS", "page_start": 7, "page_end": 7, "type": "Text", "text": "Evaluation Setup. For each problem in the evaluation benchmarks, we generate 32 samples and report pass@1 results. Generation uses a sampling temperature of 0.7 and a top-p value of 0.95, with k = 32 responses per question unless otherwise specified. It is important to note that while partial solutions were incorporated during training, no partial solutions are provided at evaluation time.", "source": "marker_v2", "marker_block_id": "/page/6/Text/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0061", "section": "5.1 EXPERIMENTAL RESULTS", "page_start": 7, "page_end": 7, "type": "Text", "text": "Key Results. Table 1 reports results on challenging math benchmarks. QUESTA yields substantial gains for Nemotron-1.5B, achieving an average improvement of 10% over its baseline and a particularly strong +13% on AIME25. These improvements are consistent across all benchmarks, highlighting the effectiveness of our approach in enhancing problem-solving robustness.", "source": "marker_v2", "marker_block_id": "/page/6/Text/6"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0062", "section": "5.1 EXPERIMENTAL RESULTS", "page_start": 7, "page_end": 7, "type": "Text", "text": "Compared to other models, QUESTA-Nemotron-1.5B consistently outperforms peers of similar scale, such as DeepSeek-R1-Distill-1.5B and Qwen3-1.7B, and even surpasses larger models like DeepSeek-R1-Distill-32B across all benchmarks. On AIME25 in particular, it exceeds DeepSeek-R1-Distill-32B by a substantial margin of +11%. Against the stronger Qwen3-8B, QUESTA-Nemotron-1.5B remains competitive despite operating at a fraction of the parameter scale.", "source": "marker_v2", "marker_block_id": "/page/6/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0063", "section": "5.1 EXPERIMENTAL RESULTS", "page_start": 7, "page_end": 7, "type": "Text", "text": "Training Dynamics. Figure 5 summarizes the training dynamics of QUESTA-Nemotron-1.5B. A positive correlation is observed between average response length and model accuracy, reflecting common trends in RL training. Notably, with QUESTA, the entropy during RL training remains stable and does not exhibit significant collapse.", "source": "marker_v2", "marker_block_id": "/page/6/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0064", "section": "5.1 EXPERIMENTAL RESULTS", "page_start": 7, "page_end": 7, "type": "Text", "text": "Pass@k Analysis. Our evaluation follows the standard pass@k methodology, consistent with DeepSeek-R1 (Guo et al., 2025) , with further details provided in Appendix B.2. In contrast to recent findings that RL-based training can reduce pass@k at larger k values (Yue et al., 2025; Liu et al., 2025) , our results show that QUESTA preserves—and in many cases modestly improves—performance across a broad range of k. As shown in Figure 2, incorporating partial-solution hints within a two-stage curriculum yields consistent gains across models, without the degradation in pass@k often observed under standard RL training. These results indicate that QUESTA enhances both the quality and diversity of candidate solutions, rather than overfitting to a single best trajectory.", "source": "marker_v2", "marker_block_id": "/page/6/Text/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0065", "section": "5.1 EXPERIMENTAL RESULTS", "page_start": 8, "page_end": 8, "type": "FigureGroup", "text": "Figure 5: Training dynamics of QUESTA-Nemotron-1.5B. The first and second charts show the progression of average response length and average reward across rollout samples during the RL process, both of which steadily increase over time. The third chart presents the average entropy. Interestingly, the entropy increases over time, suggesting that QUESTA does not suffer from entropy collapse and instead encourages diverse and exploratory behavior.", "source": "marker_v2", "marker_block_id": "/page/7/FigureGroup/101"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0066", "section": "5.1 EXPERIMENTAL RESULTS", "page_start": 8, "page_end": 8, "type": "FigureGroup", "text": "Figure 6: Pass Rate Distribution on Training Prompts. We compare the success rate on the 26K training set before and after RL, using the average pass rate over 8 samples per question. Although partial solutions are included during QUESTA training, no hints are provided during this evaluation. This setup isolates the true impact of QUESTA by assessing its ability to improve performance on problems without hints. QUESTA significantly reduces the number of unsolved or partially solved problems in the training set, especially for hard ones where initial model solves only 0/8 or 1/8 times.", "source": "marker_v2", "marker_block_id": "/page/7/FigureGroup/102"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0067", "section": "5.1 EXPERIMENTAL RESULTS", "page_start": 8, "page_end": 8, "type": "Text", "text": "Generalization at Test Time when Hints are Removed. A natural question arises from our approach: since we add partial solutions during RL training, does this improvement persist when hints are removed at evaluation time? To answer this, Figure 6 compares the pre- and post-RL models on the 26K training prompt set, evaluated without any hints. The distribution clearly shifts away from the 0/8–1/8 bins toward higher pass rates, indicating that the model solves a larger fraction of problems even without access to partial solutions. On the evaluation AIME benchmarks, Table 2 further demonstrates that QUESTA expands coverage at Pass@32: for Nemotron-1.5B, the number of unsolved problems drops from 5 to 2 on AIME24 (newly solved indices 2, 13, 29) and from 6 to 3 on AIME25 (newly solved indices 9, 13, 27). Taken together, these results show that our method generalizes well beyond the training setting and helps solve hard problems that are otherwise inaccessible without partial-solution guidance.", "source": "marker_v2", "marker_block_id": "/page/7/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0068", "section": "5.2 Further Ablations", "page_start": 8, "page_end": 8, "type": "Text", "text": "Ablation with Difficulty Curriculum. We first motivate the choice of a two-stage curriculum: RL on Partial-50 followed by RL on Partial-25 . From a modeling standpoint, the most appropriate inference distribution for the model should be the original (no-hint) distribution. Hence, during training we should gradually reduce reliance on hints to align the learned policy with the evaluation distribution. This motivates decreasing the partial ratio over time so that the model transitions from scaffolded reasoning to autonomous reasoning.", "source": "marker_v2", "marker_block_id": "/page/7/Text/7"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0069", "section": "5.2 Further Ablations", "page_start": 9, "page_end": 9, "type": "TableGroup", "text": "Table 2: Indices of unsolved problems at Pass@32 on AIME24 and AIME25 (with indices ranging from 0-29). Our method, QUESTA, consistently improve the model capacity on hard cases where the initial model is unable to solve, improving overall coverage at Pass@32. Models AIME24 Unsodlved Indices AIME25 Unsolved Indices Nemotron-1.5B 2, 3, 13, 21, 29 9, 12, 13, 14, 27, 29 QUESTA-Nemotron-1.5B 3, 21 12, 14, 29 Table 3: Ablation Study on the Impact of Curriculum Design. This table demonstrates the importance of curriculum learning in improving model performance. The model QUESTA-Nemotron-1.5B-50 was trained entirely with Partial-50 data for 2000 steps, while QUESTA-Nemotron-1.5B followed a curriculum learning approach, starting with 100 steps of Partial-50 data followed by 1900 steps of Partial-25 data. As seen in the table, the curriculum learning approach (QUESTA-Nemotron-1.5B) outperforms training with only Partial-50 data (QUESTA-Nemotron-1.5B-50). Extension with Partial-50-Partial-25-Partial-0 did not yield significant improvements, and thus, are not included in the table.", "source": "marker_v2", "marker_block_id": "/page/8/TableGroup/82"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0070", "section": "5.2 Further Ablations", "page_start": 9, "page_end": 9, "type": "Table", "text": "Model | AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 | Avg Nemotron-1.5B QUESTA-Nemotron-1.5B-50 QUESTA-Nemotron-1.5B 61.77 49.50 31.56 64.62 58.23 53.14 67.18 59.38 39.17 69.41 66.15 60.26 72.50 62.29 41.67 70.36 69.48 63.26", "source": "marker_v2", "marker_block_id": "/page/8/Table/4"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0071", "section": "5.2 Further Ablations", "page_start": 9, "page_end": 9, "type": "Text", "text": "Empirically, Table 3 shows that, under the same 2000-step budget, the curriculum Partial-50 Partial-25 learns substantially better than training on Partial-50 alone. We cap the Partial-50 stage at 100 steps, after which we switch to Partial-25 . As shown in Figure 11, entropy for QUESTA-Nemotron-1.5B-50 begins to decline beyond 100 steps, so transitioning at this point prevents overconfidence and sustains training stability. We have also tried extending the curriculum from Partial-25 to Partial-0 in our experiments, but observed no gains and no increase in response length (see Figure 12).", "source": "marker_v2", "marker_block_id": "/page/8/Text/5"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0072", "section": "5.2 Further Ablations", "page_start": 9, "page_end": 9, "type": "TableGroup", "text": "Table 4: Performance comparison (Pass@1, averaged over 32 samples) between Nemotron-1.5B and QUESTA-Nemotron-1.5B (By OpenMathReasoning). The two models achieve comparable results, with the version trained on OpenR1 performing slightly better overall. Model AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 | Avg Nemotron-1.5B QUESTA-50 (with OpenMathReasoning) QUESTA-50 (with OpenR1) 61.77 49.50 31.56 64.62 58.23 | 53.14 66.46 58.54 36.35 66.06 63.13 | 58.11 67.18 59.38 39.17 69.41 66.15 | 60.26", "source": "marker_v2", "marker_block_id": "/page/8/TableGroup/83"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0073", "section": "5.2 Further Ablations", "page_start": 9, "page_end": 9, "type": "Text", "text": "Ablation with Different Dataset. We also evaluated QUESTA on OpenMathReasoning Moshkov et al. (2025), selecting the 60K questions with pass_rate_72b_tir of 0 or 1/32. Due to time constraints, we trained only the first stage of QUESTA with 50% partial solutions. Table 4 shows that QUESTA-Nemotron-1.5B-50 achieves similar performance as using the OpenR1 dataset. This indicates that our approach generalizes across datasets.", "source": "marker_v2", "marker_block_id": "/page/8/Text/8"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0074", "section": "5.2 Further Ablations", "page_start": 9, "page_end": 9, "type": "Text", "text": "Other Ablations. We also conduct an extensive set of comparative experiments and ablation studies, with detailed results provided in Appendix D. These include an ablation of QUESTA without hints (Appendix D.1), experiments with different model backbones (Appendix D.2), and the full set of pass rates and training curves for additional models (Appendix D.3).", "source": "marker_v2", "marker_block_id": "/page/8/Text/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0075", "section": "6 CONCLUSIONS", "page_start": 9, "page_end": 9, "type": "Text", "text": "In this work we introduced QUESTA, a lightweight data-centric framework that augments hard prompts with partial-solution hints during RL training. Without altering model architecture or reward design, QUESTA sets new state-of-the-art results for 1.5 B-scale models on AIME24, AIME25 and HMMT25. Further, we theoretically demonstrate how question augmentation can improve sample", "source": "marker_v2", "marker_block_id": "/page/8/Text/11"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0076", "section": "6 CONCLUSIONS", "page_start": 10, "page_end": 10, "type": "Text", "text": "efficiency. Our analysis shows that the method can potentially be generalized to other domains such as competitive coding, software engineering or other agentic tasks. Designing proper question augmentation pipelines for theses new tasks can be an important and interesting future direction.", "source": "marker_v2", "marker_block_id": "/page/9/Text/1"}

iclr26/3MifB0f7qR/marker_meta.json

@@ -0,0 +1,2151 @@
+{
+  "table_of_contents": [
+    {
+      "title": "QUESTA: EXPANDING REASONING CAPACITY IN LLMS VIA QUESTION AUGMENTATION",
+      "heading_level": null,
+      "page_id": 0,
+      "polygon": [
+        [
+          106.3828125,
+          79.27734375
+        ],
+        [
+          504.0,
+          79.27734375
+        ],
+        [
+          504.0,
+          114.08203125
+        ],
+        [
+          106.3828125,
+          114.08203125
+        ]
+      ]
+    },
+    {
+      "title": "ABSTRACT",
+      "heading_level": null,
+      "page_id": 0,
+      "polygon": [
+        [
+          276.416015625,
+          198.38671875
+        ],
+        [
+          334.5,
+          198.38671875
+        ],
+        [
+          334.5,
+          209.25
+        ],
+        [
+          276.416015625,
+          209.25
+        ]
+      ]
+    },
+    {
+      "title": "1 Introduction",
+      "heading_level": null,
+      "page_id": 0,
+      "polygon": [
+        [
+          108.7734375,
+          650.25
+        ],
+        [
+          206.25,
+          650.25
+        ],
+        [
+          206.25,
+          660.12890625
+        ],
+        [
+          108.7734375,
+          660.12890625
+        ]
+      ]
+    },
+    {
+      "title": "2 Tradeoffs between Reasoning Capacity and Learning Efficiency",
+      "heading_level": null,
+      "page_id": 2,
+      "polygon": [
+        [
+          107.578125,
+          413.40234375
+        ],
+        [
+          505.5,
+          413.40234375
+        ],
+        [
+          505.5,
+          423.75
+        ],
+        [
+          107.578125,
+          423.75
+        ]
+      ]
+    },
+    {
+      "title": "3 Questa: Question Augmentation with Partial Solutions",
+      "heading_level": null,
+      "page_id": 3,
+      "polygon": [
+        [
+          108.7734375,
+          210.75
+        ],
+        [
+          459.0,
+          210.75
+        ],
+        [
+          459.0,
+          221.58984375
+        ],
+        [
+          108.7734375,
+          221.58984375
+        ]
+      ]
+    },
+    {
+      "title": "Original Prompt",
+      "heading_level": null,
+      "page_id": 3,
+      "polygon": [
+        [
+          157.78125,
+          309.0
+        ],
+        [
+          217.5,
+          309.0
+        ],
+        [
+          217.5,
+          318.0
+        ],
+        [
+          157.78125,
+          318.0
+        ]
+      ]
+    },
+    {
+      "title": "Augmented Prompt",
+      "heading_level": null,
+      "page_id": 3,
+      "polygon": [
+        [
+          158.37890625,
+          418.5
+        ],
+        [
+          227.25,
+          418.5
+        ],
+        [
+          227.25,
+          428.09765625
+        ],
+        [
+          158.37890625,
+          428.09765625
+        ]
+      ]
+    },
+    {
+      "title": "## Hint: Partial Solution",
+      "heading_level": null,
+      "page_id": 3,
+      "polygon": [
+        [
+          161.3671875,
+          496.93359375
+        ],
+        [
+          248.25,
+          496.93359375
+        ],
+        [
+          248.25,
+          505.5
+        ],
+        [
+          161.3671875,
+          505.5
+        ]
+      ]
+    },
+    {
+      "title": "4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY",
+      "heading_level": null,
+      "page_id": 4,
+      "polygon": [
+        [
+          106.3828125,
+          394.83984375
+        ],
+        [
+          449.25,
+          393.75
+        ],
+        [
+          449.25,
+          403.5
+        ],
+        [
+          106.3828125,
+          405.0
+        ]
+      ]
+    },
+    {
+      "title": "5 EXPERIMENTS",
+      "heading_level": null,
+      "page_id": 5,
+      "polygon": [
+        [
+          107.25,
+          576.75
+        ],
+        [
+          200.25,
+          576.75
+        ],
+        [
+          200.25,
+          586.5
+        ],
+        [
+          107.25,
+          586.5
+        ]
+      ]
+    },
+    {
+      "title": "5.1 EXPERIMENTAL RESULTS",
+      "heading_level": null,
+      "page_id": 6,
+      "polygon": [
+        [
+          108.2490005493164,
+          459.80859375
+        ],
+        [
+          240.2578125,
+          459.80859375
+        ],
+        [
+          240.2578125,
+          470.0510559082031
+        ],
+        [
+          108.2490005493164,
+          470.0510559082031
+        ]
+      ]
+    },
+    {
+      "title": "5.2 Further Ablations",
+      "heading_level": null,
+      "page_id": 7,
+      "polygon": [
+        [
+          106.5,
+          648.52734375
+        ],
+        [
+          225.75,
+          648.52734375
+        ],
+        [
+          225.75,
+          657.75
+        ],
+        [
+          106.5,
+          657.75
+        ]
+      ]
+    },
+    {
+      "title": "6 CONCLUSIONS",
+      "heading_level": null,
+      "page_id": 8,
+      "polygon": [
+        [
+          106.98046875,
+          666.0
+        ],
+        [
+          201.75,
+          666.0
+        ],
+        [
+          201.75,
+          675.75
+        ],
+        [
+          106.98046875,
+          675.75
+        ]
+      ]
+    },
+    {
+      "title": "7 ETHICS STATEMENT",
+      "heading_level": null,
+      "page_id": 9,
+      "polygon": [
+        [
+          108.7734375,
+          131.87109375
+        ],
+        [
+          228.75,
+          131.87109375
+        ],
+        [
+          228.75,
+          141.0
+        ],
+        [
+          108.7734375,
+          141.0
+        ]
+      ]
+    },
+    {
+      "title": "8 Reproducibility Statement",
+      "heading_level": null,
+      "page_id": 9,
+      "polygon": [
+        [
+          108.7734375,
+          244.79296875
+        ],
+        [
+          287.25,
+          244.79296875
+        ],
+        [
+          287.25,
+          254.25
+        ],
+        [
+          108.7734375,
+          254.25
+        ]
+      ]
+    },
+    {
+      "title": "9 ACKNOWLEDGEMENTS AND DISCLOSURE OF FUNDING",
+      "heading_level": null,
+      "page_id": 9,
+      "polygon": [
+        [
+          106.3828125,
+          326.77734375
+        ],
+        [
+          407.25,
+          326.77734375
+        ],
+        [
+          407.25,
+          336.75
+        ],
+        [
+          106.3828125,
+          336.75
+        ]
+      ]
+    },
+    {
+      "title": "REFERENCES",
+      "heading_level": null,
+      "page_id": 9,
+      "polygon": [
+        [
+          106.98046875,
+          390.0
+        ],
+        [
+          175.5,
+          390.0
+        ],
+        [
+          175.5,
+          399.0
+        ],
+        [
+          106.98046875,
+          399.0
+        ]
+      ]
+    },
+    {
+      "title": "A THE USE OF LARGE LANGUAGE MODELS (LLMS)",
+      "heading_level": null,
+      "page_id": 13,
+      "polygon": [
+        [
+          107.578125,
+          81.59765625
+        ],
+        [
+          385.5,
+          81.59765625
+        ],
+        [
+          385.5,
+          93.0
+        ],
+        [
+          107.578125,
+          93.0
+        ]
+      ]
+    },
+    {
+      "title": "B IMPLEMENTATION DETAILS",
+      "heading_level": null,
+      "page_id": 13,
+      "polygon": [
+        [
+          106.98046875,
+          145.01953125
+        ],
+        [
+          269.25,
+          145.01953125
+        ],
+        [
+          269.25,
+          153.75
+        ],
+        [
+          106.98046875,
+          153.75
+        ]
+      ]
+    },
+    {
+      "title": "B.1 RLVR ALGORITHMS",
+      "heading_level": null,
+      "page_id": 13,
+      "polygon": [
+        [
+          106.98046875,
+          167.44921875
+        ],
+        [
+          222.75,
+          167.44921875
+        ],
+        [
+          222.75,
+          177.0
+        ],
+        [
+          106.98046875,
+          177.0
+        ]
+      ]
+    },
+    {
+      "title": "B.2 LOW-VARIANCE PASS@K ESTIMATION",
+      "heading_level": null,
+      "page_id": 13,
+      "polygon": [
+        [
+          106.5,
+          437.25
+        ],
+        [
+          297.75,
+          437.25
+        ],
+        [
+          297.75,
+          446.66015625
+        ],
+        [
+          106.5,
+          446.66015625
+        ]
+      ]
+    },
+    {
+      "title": "B.3 MORE RELATED WORKS",
+      "heading_level": null,
+      "page_id": 13,
+      "polygon": [
+        [
+          107.279296875,
+          636.15234375
+        ],
+        [
+          238.5,
+          636.15234375
+        ],
+        [
+          238.5,
+          648.0
+        ],
+        [
+          107.279296875,
+          648.0
+        ]
+      ]
+    },
+    {
+      "title": "B.4 BENCHMARKS",
+      "heading_level": null,
+      "page_id": 14,
+      "polygon": [
+        [
+          106.3828125,
+          329.09765625
+        ],
+        [
+          196.5,
+          329.09765625
+        ],
+        [
+          196.5,
+          338.25
+        ],
+        [
+          106.3828125,
+          338.25
+        ]
+      ]
+    },
+    {
+      "title": "B.5 TRAINING DATASET",
+      "heading_level": null,
+      "page_id": 14,
+      "polygon": [
+        [
+          106.5,
+          469.08984375
+        ],
+        [
+          219.75,
+          469.08984375
+        ],
+        [
+          219.75,
+          480.0
+        ],
+        [
+          106.5,
+          480.0
+        ]
+      ]
+    },
+    {
+      "title": "B.6 THE RATIONALE FOR THE CHOICE OF p",
+      "heading_level": null,
+      "page_id": 15,
+      "polygon": [
+        [
+          107.578125,
+          83.53125
+        ],
+        [
+          298.828125,
+          83.53125
+        ],
+        [
+          298.828125,
+          94.2310791015625
+        ],
+        [
+          107.578125,
+          94.2310791015625
+        ]
+      ]
+    },
+    {
+      "title": "B.7 EVALUATION SETUP",
+      "heading_level": null,
+      "page_id": 15,
+      "polygon": [
+        [
+          108.2490005493164,
+          413.40234375
+        ],
+        [
+          219.40072631835938,
+          413.40234375
+        ],
+        [
+          219.40072631835938,
+          424.7301025390625
+        ],
+        [
+          108.2490005493164,
+          424.7301025390625
+        ]
+      ]
+    },
+    {
+      "title": "B.8 DETAIL ON PROMPT TEMPLATE",
+      "heading_level": null,
+      "page_id": 15,
+      "polygon": [
+        [
+          108.2490005493164,
+          509.30859375
+        ],
+        [
+          267.2513732910156,
+          509.30859375
+        ],
+        [
+          267.2513732910156,
+          521.0150756835938
+        ],
+        [
+          108.2490005493164,
+          521.0150756835938
+        ]
+      ]
+    },
+    {
+      "title": "DeepScaleR Coding's Inference:\n<|User|>{input}<|Assistant|><think>",
+      "heading_level": null,
+      "page_id": 15,
+      "polygon": [
+        [
+          109.96875,
+          529.8046875
+        ],
+        [
+          492.46875,
+          529.8046875
+        ],
+        [
+          492.46875,
+          559.1953125
+        ],
+        [
+          109.96875,
+          559.1953125
+        ]
+      ]
+    },
+    {
+      "title": "DeepScaleR Others' Inference:",
+      "heading_level": null,
+      "page_id": 15,
+      "polygon": [
+        [
+          112.359375,
+          571.4505310058594
+        ],
+        [
+          245.38922119140625,
+          571.4505310058594
+        ],
+        [
+          245.38922119140625,
+          582.78515625
+        ],
+        [
+          112.359375,
+          582.78515625
+        ]
+      ]
+    },
+    {
+      "title": "Nemotrion Coding's Inference:",
+      "heading_level": null,
+      "page_id": 15,
+      "polygon": [
+        [
+          112.060546875,
+          620.0135192871094
+        ],
+        [
+          245.58847045898438,
+          620.0135192871094
+        ],
+        [
+          245.58847045898438,
+          629.9761199951172
+        ],
+        [
+          112.060546875,
+          629.9761199951172
+        ]
+      ]
+    },
+    {
+      "title": "Nemotrion Others' Inference:",
+      "heading_level": null,
+      "page_id": 15,
+      "polygon": [
+        [
+          113.32499694824219,
+          682.55859375
+        ],
+        [
+          240.4079132080078,
+          682.55859375
+        ],
+        [
+          240.4079132080078,
+          694.4391098022461
+        ],
+        [
+          113.32499694824219,
+          694.4391098022461
+        ]
+      ]
+    },
+    {
+      "title": "Training prompt with partial solutions (math RL):",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
+        [
+          111.75,
+          120.0
+        ],
+        [
+          327.75,
+          120.0
+        ],
+        [
+          327.75,
+          130.32421875
+        ],
+        [
+          111.75,
+          130.32421875
+        ]
+      ]
+    },
+    {
+      "title": "C THEORY",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
+        [
+          107.25,
+          182.91796875
+        ],
+        [
+          171.75,
+          182.91796875
+        ],
+        [
+          171.75,
+          192.0
+        ],
+        [
+          107.25,
+          192.0
+        ]
+      ]
+    },
+    {
+      "title": "C.1 Proofs",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
+        [
+          106.5,
+          206.89453125
+        ],
+        [
+          167.25,
+          206.89453125
+        ],
+        [
+          167.25,
+          215.25
+        ],
+        [
+          106.5,
+          215.25
+        ]
+      ]
+    },
+    {
+      "title": "D ADDITIONAL EXPERIMENTAL RESULTS",
+      "heading_level": null,
+      "page_id": 18,
+      "polygon": [
+        [
+          108.7734375,
+          81.75
+        ],
+        [
+          330.0,
+          81.75
+        ],
+        [
+          330.0,
+          92.25
+        ],
+        [
+          108.7734375,
+          92.25
+        ]
+      ]
+    },
+    {
+      "title": "D.1 ABLATION STUDY WITHOUT HINT",
+      "heading_level": null,
+      "page_id": 18,
+      "polygon": [
+        [
+          106.5,
+          107.25
+        ],
+        [
+          279.0,
+          107.25
+        ],
+        [
+          279.0,
+          116.25
+        ],
+        [
+          106.5,
+          116.25
+        ]
+      ]
+    },
+    {
+      "title": "D.2 ABLATION STUDY WITH DIFFERENT MODELS",
+      "heading_level": null,
+      "page_id": 19,
+      "polygon": [
+        [
+          106.5,
+          229.5
+        ],
+        [
+          324.75,
+          229.5
+        ],
+        [
+          324.75,
+          237.75
+        ],
+        [
+          106.5,
+          237.75
+        ]
+      ]
+    },
+    {
+      "title": "D.3 SUPPLEMENTAL EXPERIMENT DETAIL",
+      "heading_level": null,
+      "page_id": 20,
+      "polygon": [
+        [
+          106.5,
+          507.0
+        ],
+        [
+          297.75,
+          507.75
+        ],
+        [
+          297.75,
+          517.81640625
+        ],
+        [
+          106.5,
+          517.81640625
+        ]
+      ]
+    }
+  ],
+  "page_stats": [
+    {
+      "page_id": 0,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          55
+        ],
+        [
+          "Span",
+          43
+        ],
+        [
+          "Footnote",
+          4
+        ],
+        [
+          "SectionHeader",
+          3
+        ],
+        [
+          "Text",
+          3
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Figure",
+          1
+        ],
+        [
+          "Caption",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "FigureGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 1,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          70
+        ],
+        [
+          "Span",
+          20
+        ],
+        [
+          "Text",
+          5
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Figure",
+          1
+        ],
+        [
+          "Caption",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "FigureGroup",
+          1
+        ],
+        [
+          "Reference",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 2,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          67
+        ],
+        [
+          "Span",
+          24
+        ],
+        [
+          "Text",
+          4
+        ],
+        [
+          "ListItem",
+          2
+        ],
+        [
+          "Reference",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Figure",
+          1
+        ],
+        [
+          "Caption",
+          1
+        ],
+        [
+          "SectionHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "FigureGroup",
+          1
+        ],
+        [
+          "ListGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 3,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Span",
+          70
+        ],
+        [
+          "Line",
+          62
+        ],
+        [
+          "Text",
+          11
+        ],
+        [
+          "SectionHeader",
+          4
+        ],
+        [
+          "Equation",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Picture",
+          1
+        ],
+        [
+          "Caption",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "Reference",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 4,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Span",
+          75
+        ],
+        [
+          "Line",
+          60
+        ],
+        [
+          "Text",
+          10
+        ],
+        [
+          "Reference",
+          4
+        ],
+        [
+          "ListItem",
+          2
+        ],
+        [
+          "Equation",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "SectionHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "ListGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 5,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Span",
+          78
+        ],
+        [
+          "Line",
+          63
+        ],
+        [
+          "Text",
+          12
+        ],
+        [
+          "Reference",
+          3
+        ],
+        [
+          "Equation",
+          2
+        ],
+        [
+          "ListItem",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "SectionHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "ListGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 6,
+      "text_extraction_method": "pdftext",
+      "block_counts": [
+        [
+          "Span",
+          335
+        ],
+        [
+          "Line",
+          54
+        ],
+        [
+          "TableCell",
+          49
+        ],
+        [
+          "Text",
+          6
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Caption",
+          1
+        ],
+        [
+          "Table",
+          1
+        ],
+        [
+          "SectionHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "TableGroup",
+          1
+        ],
+        [
+          "Reference",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 7,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          78
+        ],
+        [
+          "Span",
+          14
+        ],
+        [
+          "Figure",
+          2
+        ],
+        [
+          "Caption",
+          2
+        ],
+        [
+          "Text",
+          2
+        ],
+        [
+          "FigureGroup",
+          2
+        ],
+        [
+          "Reference",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "SectionHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 8,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "TableCell",
+          61
+        ],
+        [
+          "Line",
+          44
+        ],
+        [
+          "Span",
+          25
+        ],
+        [
+          "Text",
+          4
+        ],
+        [
+          "Caption",
+          3
+        ],
+        [
+          "Table",
+          3
+        ],
+        [
+          "Reference",
+          3
+        ],
+        [
+          "TableGroup",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "SectionHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 9,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          64
+        ],
+        [
+          "Span",
+          22
+        ],
+        [
+          "Text",
+          12
+        ],
+        [
+          "Reference",
+          8
+        ],
+        [
+          "SectionHeader",
+          4
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 10,
+      "text_extraction_method": "pdftext",
+      "block_counts": [
+        [
+          "Span",
+          106
+        ],
+        [
+          "Line",
+          49
+        ],
+        [
+          "ListItem",
+          18
+        ],
+        [
+          "Reference",
+          18
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "ListGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 11,
+      "text_extraction_method": "pdftext",
+      "block_counts": [
+        [
+          "Span",
+          129
+        ],
+        [
+          "Line",
+          52
+        ],
+        [
+          "ListItem",
+          17
+        ],
+        [
+          "Reference",
+          17
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "ListGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 12,
+      "text_extraction_method": "pdftext",
+      "block_counts": [
+        [
+          "Span",
+          23
+        ],
+        [
+          "Line",
+          9
+        ],
+        [
+          "ListItem",
+          3
+        ],
+        [
+          "Reference",
+          3
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "ListGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 13,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          57
+        ],
+        [
+          "Span",
+          38
+        ],
+        [
+          "Text",
+          8
+        ],
+        [
+          "SectionHeader",
+          5
+        ],
+        [
+          "Equation",
+          5
+        ],
+        [
+          "Reference",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 14,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          67
+        ],
+        [
+          "Span",
+          31
+        ],
+        [
+          "Text",
+          7
+        ],
+        [
+          "Reference",
+          3
+        ],
+        [
+          "SectionHeader",
+          2
+        ],
+        [
+          "Footnote",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 15,
+      "text_extraction_method": "pdftext",
+      "block_counts": [
+        [
+          "Span",
+          201
+        ],
+        [
+          "TableCell",
+          50
+        ],
+        [
+          "Line",
+          47
+        ],
+        [
+          "Text",
+          12
+        ],
+        [
+          "SectionHeader",
+          7
+        ],
+        [
+          "Reference",
+          3
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Caption",
+          1
+        ],
+        [
+          "Table",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "TableGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 16,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Span",
+          79
+        ],
+        [
+          "Line",
+          60
+        ],
+        [
+          "Text",
+          16
+        ],
+        [
+          "Equation",
+          7
+        ],
+        [
+          "SectionHeader",
+          3
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "Reference",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 17,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Span",
+          88
+        ],
+        [
+          "Line",
+          56
+        ],
+        [
+          "Text",
+          15
+        ],
+        [
+          "Equation",
+          6
+        ],
+        [
+          "ListItem",
+          2
+        ],
+        [
+          "Reference",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "ListGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 18,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          65
+        ],
+        [
+          "TableCell",
+          35
+        ],
+        [
+          "Span",
+          19
+        ],
+        [
+          "Reference",
+          4
+        ],
+        [
+          "SectionHeader",
+          2
+        ],
+        [
+          "Caption",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Table",
+          1
+        ],
+        [
+          "Figure",
+          1
+        ],
+        [
+          "Text",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "TableGroup",
+          1
+        ],
+        [
+          "FigureGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 19,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          54
+        ],
+        [
+          "TableCell",
+          42
+        ],
+        [
+          "Span",
+          37
+        ],
+        [
+          "Reference",
+          4
+        ],
+        [
+          "Text",
+          3
+        ],
+        [
+          "Caption",
+          3
+        ],
+        [
+          "Table",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "SectionHeader",
+          1
+        ],
+        [
+          "Figure",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "TableGroup",
+          1
+        ],
+        [
+          "FigureGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 20,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          63
+        ],
+        [
+          "Span",
+          22
+        ],
+        [
+          "Text",
+          3
+        ],
+        [
+          "Figure",
+          2
+        ],
+        [
+          "Caption",
+          2
+        ],
+        [
+          "FigureGroup",
+          2
+        ],
+        [
+          "Reference",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "SectionHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 21,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          86
+        ],
+        [
+          "Span",
+          6
+        ],
+        [
+          "Figure",
+          4
+        ],
+        [
+          "Caption",
+          4
+        ],
+        [
+          "FigureGroup",
+          4
+        ],
+        [
+          "Reference",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 22,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          54
+        ],
+        [
+          "Span",
+          4
+        ],
+        [
+          "Figure",
+          2
+        ],
+        [
+          "Caption",
+          2
+        ],
+        [
+          "FigureGroup",
+          2
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 23,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          51
+        ],
+        [
+          "Span",
+          3
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Figure",
+          1
+        ],
+        [
+          "Caption",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "FigureGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    },
+    {
+      "page_id": 24,
+      "text_extraction_method": "surya",
+      "block_counts": [
+        [
+          "Line",
+          51
+        ],
+        [
+          "Span",
+          3
+        ],
+        [
+          "PageHeader",
+          1
+        ],
+        [
+          "Figure",
+          1
+        ],
+        [
+          "Caption",
+          1
+        ],
+        [
+          "PageFooter",
+          1
+        ],
+        [
+          "FigureGroup",
+          1
+        ]
+      ],
+      "block_metadata": {
+        "llm_request_count": 0,
+        "llm_error_count": 0,
+        "llm_tokens_used": 0,
+        "previous_text": "",
+        "previous_type": "",
+        "previous_order": 0
+      }
+    }
+  ],
+  "debug_data_path": "debug_data/3MifB0f7qR"
+}

iclr26/3MifB0f7qR/model_text_v3.txt

@@ -0,0 +1,230 @@
+[p. 1 | section: ABSTRACT | type: Text]
+Reinforcement learning (RL) has emerged as a central paradigm for training large language models (LLMs) in reasoning tasks. Yet recent studies (Yue et al., 2025; Liu et al., 2025) question RL's ability to incentivize reasoning capacity beyond the base model. This raises a key challenge: how can RL be adapted to solve harder reasoning problems more effectively? To address this challenge, we propose a simple yet effective strategy via Question Augmentation: introduce partial solutions during training to reduce problem difficulty and provide more informative learning signals. Our method, QuestA, when applied during RL training on math reasoning tasks, not only improves pass@1 but also pass@k—particularly on problems where standard RL struggles to make progress. This enables continual improvement over strong open-source models such as DEEPSCALER and OPENMATH NEMOTRON, further enhancing their reasoning capabilities. We achieve new state-of-the-art results on math benchmarks using 1.5B-parameter models: 72.50% (+10.73%) on AIME24, 62.29% (+12.79%) on AIME25, and 41.67% (+10.11%) on HMMT25. Code, data and model are available at
+
+[p. 1 | section: ABSTRACT | type: FigureGroup]
+Figure 1: QUESTA is a data augmentation method that injects partial solutions to effectively scaffold RL training on hard reasoning problems. We construct 26K high-quality augmented prompts from challenging instances in OpenR1 (Open-R1 Team, 2025), and fine-tune models using 32K-context-length RL. When applied to Nemotron-1.5B, QUESTA delivers substantial performance gains—achieving new state-of-the-art results across all math benchmarks for 1.5B-parameter models.
+
+[p. 1 | section: 1 Introduction | type: Text]
+Frontier large language models (LLMs), including OpenAI-O1, O3 (Jaech et al., 2024), DeepSeek-R1 (Guo et al., 2025), Qwen3 (Yang et al., 2025), and Gemini 2.5 (Gemini Team, Google DeepMind,
+
+[p. 1 | section: 1 Introduction | type: Footnote]
+& lt;sup>‡</sup>This work is independent of and outside of the work at Amazon.
+
+[p. 2 | section: 1 Introduction | type: FigureGroup]
+Figure 2: We compare pass@k curves of RLVR-trained models, with and without QUESTA. As a controlled experiment, we perform RL training using either easy or hard prompts. Standard RL on easy prompts (red) shows clear degradation in pass@k as k increases compared to the base model (blue). Training on hard prompts (green) improves pass@k, but comes at the cost of substantially longer training. This motivates our development of QUESTA, which scaffolds hard problems to improve training efficiency and delivers consistently stronger results: the RL+QUESTA model (orange) stays above standard RL (red) across all k, while also preserving or improving performance at larger k relative to RL trained with hard prompts.
+
+[p. 2 | section: 1 Introduction | type: Text]
+2025), have exhibited exceptional performance on high-complexity reasoning tasks spanning mathematics, programming, and formal logic. Recent advances in the field have increasingly prioritized reinforcement learning paradigms (RL), among which Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a scalable and efficient approach to enhancing reasoning capabilities. Using automatically verifiable signals, RLVR enables alignment between model output and objective correctness, thus addressing a critical limitation of traditional RL for reasoning.
+
+[p. 2 | section: 1 Introduction | type: Text]
+However, the community remains divided on a fundamental question regarding RLVR: does it expand the model's intrinsic reasoning capacity, or merely exploit pre-existing knowledge encoded in the base model? Recent research (Yue et al., 2025; Liu et al., 2025; Zhao et al., 2025) show that while state-of-the-art RL methods (e.g., GRPO, DAPO) (Guo et al., 2025; Yu et al., 2025; An et al., 2025) can enhance the pass@1 metric by reinforcing high-reward completions, they encounter significant limitations when tackling high-difficulty tasks where the base model performs poorly. This phenomenon differs from that observed in Supervised Fine-Tuning (SFT) Luo et al. (2025a). Within the SFT paradigm, enhancing the diversity of problem difficulty serves as a critical factor, as it can effectively improve the model's performance on downstream tasks. However, in the framework of RLVR, the inclusion of easy prompts tends to undermine the model's inherent reasoning capabilities.
+
+[p. 2 | section: 1 Introduction | type: Text]
+One insightful explanation (Cui et al., 2025; Wang et al., 2025a) for the drop suggests that model overfits on correct solutions and hence causes entropy collapse, limiting its ability to explore. To validate this, we design a controlled setup that separates prompts into easy and hard groups. When applying RLVR on the Nemotron 1.5B model (Moshkov et al., 2025) with the OpenR1 dataset, we find that training on easy prompts leads to a clear decline in pass@k accuracy (Figure 2).
+
+[p. 2 | section: 1 Introduction | type: Text]
+Given these findings, we observe that training with hard prompts is more beneficial than with easy ones. Yet, RL training on hard problems tends to be much slower, as sparse reward signals and limited sample efficiency hinder progress. The key challenge, then, is how to structure the learning process to fully expand reasoning capabilities while mitigating the inefficiency of RL on hard tasks . To this end, we introduce QUESTA: a parsimonious and efficient strategy that dynamically adjusts problem difficulty during RL training. The core contributions of this work are threefold:
+
+[p. 2 | section: 1 Introduction | type: Text]
+• We notice that the evolution of model capacity in RLVR critically depends on dataset difficulty, underscoring the importance of training on hard problems to expand reasoning ability.
+
+[p. 3 | section: 1 Introduction | type: FigureGroup]
+Figure 3: Comparison of RL training dynamics: Training with only hard problems (green) makes progress very slowly due to sparse rewards, while our method with partial solutions (orange) accelerates training and consistently achieves higher accuracy across training steps.
+
+[p. 3 | section: 1 Introduction | type: ListGroup]
+We introduce QUESTA, an efficient procedure that controls difficulty by augmenting hard problems with partial solutions. This approach provides a smooth curriculum within RL training and makes high-difficulty tasks more tractable. Through our fully open-sourced training pipeline, QUESTA consistently improves pass@1 and pass@k, enabling 1.5B-parameter models to reach new state-of-the-art performance—72.5% on AIME24, 62.3% on AIME25, and 41.7% on HMMT25 (Table 1). Our theoretical analysis in Section 4 explains why partial-solution augmentation accelerates RL training: by decomposing problems into intermediate steps, the method yields denser reward signals and improves sample efficiency, while still driving the model to master the hardest problems.
+
+[p. 3 | section: 2 Tradeoffs between Reasoning Capacity and Learning Efficiency | type: Text]
+Given the ongoing debate on whether reinforcement learning enhances the reasoning capacity of language models, we design a controlled experiment to study how dataset difficulty changes model performances measured by pass@k accuracy. Specifically, we filter out easy problems and hard problems from the 220K OpenR1 dataset, base on model's success rate, each containing around 4K data. We then run RL with GRPO for one thousand steps. This setup allows us to isolate how the choice of prompt difficulty impacts the model's reasoning capacity. In Figure 2 and Figure 3, we provide pass@k comparison and the learning dynamics, we make two observations.
+
+[p. 3 | section: 2 Tradeoffs between Reasoning Capacity and Learning Efficiency | type: Text]
+RL with Easy Prompts Hurts pass@k and Reasoning Capacity. Training on easy or already-solvable problems leads to overfitting on shallow patterns, reinforcing confidence rather than expanding reasoning capacity. While pass@1 may rise, output diversity declines and performance on harder benchmarks deteriorates, with pass@k dropping at larger k (see Figure 2). This suggests that the model exploits familiar solution modes instead of exploring new trajectories. To truly expand capacity, RL training should focus on hard problems, where the policy is forced to explore and acquire novel solution strategies.
+
+[p. 3 | section: 2 Tradeoffs between Reasoning Capacity and Learning Efficiency | type: Text]
+RL with Hard Prompts Leads to Slow Learning. Training on hard prompts directly targets the reasoning capacity of the model, but the learning process is much slower (see Figure 3) and less sample-efficient. The difficulty arises because RL rewards on these problems are sparse, providing limited gradient signals for policy improvement. We formallize the underlying reason in Section 4 and in Theorem 4.4.
+
+[p. 3 | section: 2 Tradeoffs between Reasoning Capacity and Learning Efficiency | type: Text]
+In practice, not all questions in the training set Q are equally difficult, and one might hope that training on easier examples could generalize to harder ones. However, empirical evidence suggests that RL-based training exhibits a bi-modal pattern in success rates (An et al., 2025): by the end of training, models tend to either solve a question reliably or fail entirely (see Figure 6). This implies that once a question falls outside the model's capacity set, the RL algorithm is unlikely to recover.
+
+[p. 4 | section: 2 Tradeoffs between Reasoning Capacity and Learning Efficiency | type: Text]
+Together, these results highlight a tension: easy prompts dilute reasoning capacity, while hard prompts stall learning altogether. This motivates the need for strategies that can retain the benefits of hard problems while mitigating the inefficiency caused by sparse rewards. To this end, we introduce partial solutions that break a complex question into smaller, more approachable pieces. Theoretical analysis (Theorem 4.6) suggests that appending part of the solutions as hint can greatly improve RL efficiency.
+
+[p. 4 | section: 2 Tradeoffs between Reasoning Capacity and Learning Efficiency | type: Text]
+Empirically, we simply choose the hint to be a part of the solution of the original question q and observe faster learning in Figure 3. Surprisingly, even if we don't explicitly train the model to generate the hint, the model's capacity without hint still continues to improve and lead to steady improvement in problems out of reach in standard RL training (see Table 3). We elaborate on implementation details in the next section.
+
+[p. 4 | section: 3 Questa: Question Augmentation with Partial Solutions | type: Text]
+QUESTA is a modular augmentation framework designed to inject partial solution sketches into prompts during reinforcement learning (RL) training. It adresses scenarios where the base model fails to generate correct completions—conditions that typically result in sparse reward signals. Distinct from approaches that modify reward functions or optimization algorithms, QUESTA operates at the input level: it transforms original training prompts into more tractable variants, thereby exposing intermediate reasoning steps to the model.
+
+[p. 4 | section: Original Prompt | type: Text]
+Let \mathbb N be the set of positive integers. A function f:\mathbb N\to\mathbb N satisfies the equation
+
+[p. 4 | section: Original Prompt | type: Equation]
+f(f(\dots f(n)\dots)) = \frac{n^2}{f(f(n))} with f(n) applications of f ,
+
+[p. 4 | section: Original Prompt | type: Text]
+for all positive integers n. Given this information, determine all possible values of f(1000).
+
+[p. 4 | section: Augmented Prompt | type: Text]
+Let \mathbb N be the set of positive integers. The function f:\mathbb N\to\mathbb N satisfies the equation
+
+[p. 4 | section: Augmented Prompt | type: Equation]
+f(f(\dots f(n)\dots)) = \frac{n^2}{f(f(n))} \quad \text{with } f(n) \text{ applications of } f,
+
+[p. 4 | section: Augmented Prompt | type: Text]
+for all positive integers n. Given this information, determine all possible values of f(1000).
+
+[p. 4 | section: Hint: Partial Solution | type: Text]
+Analysis shows that f must be an involution, meaning f(f(n)) = n for all n, and it fixes all odd positive integers, so f(n) = n for odd n. For even positive integers, f either fixes the number or swaps it with another even positive integer in a 2-cycle.
+
+[p. 4 | section: Hint: Partial Solution | type: Text]
+Please reason step by step, and put your final answer within \boxed{}.
+
+[p. 4 | section: Hint: Partial Solution | type: Caption]
+Figure 4: QUESTA augments each original question in the dataset by prepending the first p% of the solution sketch. In our experiments, we apply augmentation using the solution block rather than the reasoning chain-of-thought. The hint percentage p is computed as the ratio of tokens used as hints to the total number of tokens in the solution sketch.
+
+[p. 4 | section: Hint: Partial Solution | type: Text]
+Question Augmentation Mechanism For a given problem x with an n-step solution trajectory y=(y_1,y_2,\ldots,y_n) , QUESTA constructs a set of augmented prompts \{\tilde{x}^{(p)}\} , where each \tilde{x}^{(p)} appends the first p steps of the solution as a prefix to the original question. The parameter p (e.g., p=50% or 25%) quantifies the proportion of the solution revealed, thereby enabling precise control over the difficulty of the augmented prompt.
+
+[p. 4 | section: Hint: Partial Solution | type: Text]
+In our empirical evaluations, we employed the OpenR1-Math-220K dataset (Open-R1 Team, 2025)—a supervised fine-tuning (SFT) corpus containing solution trajectories generated by DEEPSEEK-R1. Each instance in this dataset comprises a detailed chain-of-thought (CoT) section followed by a final solution block. For augmentation, we extracted the final solution (omitting speculative reasoning
+
+[p. 5 | section: Hint: Partial Solution | type: Text]
+within the CoT section). The solution was then truncated at a predefined percentage p and prepended to the original question, yielding the augmented prompt used in RL training, as shwon in Figure 4.
+
+[p. 5 | section: Hint: Partial Solution | type: Text]
+Targeting High-Difficulty Problems QUESTA is applied exclusively to prompts where the base model's pass rate is close to zero. Using the OpenR1-Math-220K dataset, we first employ lightweight heuristic filters to reduce the full 220K problems to 26K of the hardest candidates. These problems are then augmented with partial-solution prefixes where we conduct a second difficulty screening: sample multiple completions from the model for each augmented prompt, and only those instances with consistently low pass rates are retained. This two-stage filtering pipeline yields a final pool of no more than 10K problems, ensuring that augmentation resources are concentrated on the most challenging cases where the base model needs additional guidance and scaffolding.
+
+[p. 5 | section: Hint: Partial Solution | type: Text]
+Integrating with RL Pipelines QUESTA exhibits orthogonality to underlying RL algorithms, enabling seamless integration into existing training pipelines (e.g., GRPO (Shao et al., 2024), DAPO (Yu et al., 2025)) without modifications. Specifically, integration requires only replacing the original rollout dataset with the augmented dataset, while retaining the original reward function and policy update mechanism. To further exploit this input-level flexibility, we extended QUESTA with an iterative curriculum RL paradigm:
+
+[p. 5 | section: Hint: Partial Solution | type: ListGroup]
+1. First, augment the dataset with p = 50%, apply the difficulty filtering with the augmented prompt, and conduct reinforcement learning training until the performance saturates. 2. Second, reduce the augmentation from p = 50% to p = 25%, i.e. provide fewer hints. Again, we apply the difficulty filtering, and conduct reinforcement learning training until convergence.
+
+[p. 5 | section: Hint: Partial Solution | type: Text]
+Here, the rationale for the choice of p is provided in Appendix B.6. By keeping the training signals strong at each stage, the method speeds up convergence on difficult tasks and makes QUESTA a simple, plug-and-play approach for curriculum-based RL.
+
+[p. 5 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+In this section, we present a theoretical perspective on how question augmentation improves the efficiency of reinforcement learning. Our central thesis is that the primary bottleneck in RL-based reasoning lies in the difficulty of discovering successful trajectories within a finite sampling budget. Question augmentation addresses this challenge by reshaping the learnability landscape —making hard problems more discoverable by increasing the likelihood of encountering correct trajectories.
+
+[p. 5 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Motivated by experiments which quantify model capcity with pass@k accuracy, we introduce the following notions of solution set (Definition 4.1) and model capacity set (Definition 4.2) for a given question q and model \mu . Let \mathcal V be the vocaboluary set, and let P_{\mu}(q,\tau) denote the probability that a language model \mu generates trajectory \tau \in \mathcal V^* when conditioned on input question q \in \mathcal V^* .
+
+[p. 5 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Definition 4.1 (Solution Set). Given a question q and a binary reward function R: \mathcal{V}^* \times \mathcal{V}^* \to \{0, 1\} , the solution set is defined as:
+
+[p. 5 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Equation]
+S(q) = \{ \tau \in \mathcal{V}^* \mid R(q, \tau) = 1 \}.
+
+[p. 5 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Definition 4.2 (Model Capacity Set). Given a probability threshold \delta_p > 0 , a language model \mu , and a question q, define the model capacity set C(q, \delta_p) as the smallest set of trajectories whose total probability mass is at least 1 - \delta_p :
+
+[p. 5 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Equation]
+C(q, \delta_p) = \arg\min_{S \subseteq \mathcal{V}^*} \left\{ |S| \left| \sum_{\tau \in S} P_{\mu}(q, \tau) \ge 1 - \delta_p \right\}. \right.
+
+[p. 5 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+The Model Capacity Set C(q, \delta_p) intuitively captures the set of most likely output trajectories that the model \mu can generate for a given input q, up to a small probability threshold \delta_p .
+
+[p. 5 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+This formalization leads to a critical insight: if the model's capacity set fails to intersect with the solution set—meaning the model is unlikely to generate any correct completions—then the RL process cannot make progress. To articulate this more formally, we begin by stating a standard assumption satisfied by many popular RL algorithms, such as DAPO and online GRPO:
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Assumption 4.3 (Null Gradient from Zero-Reinforcement). The RL algorithm does not update the model weights if none of the sampled rollouts receives a positive reward (i.e., reward = 1).
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Under this assumption, we easily the following lower bound, which states that if all training questions are unreachable within the model's capacity set, the RL process is likely to stall entirely:
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Theorem 4.4 (Lower Bound on RL Learnability under Solution Inaccessibility). Given a probability threshold \delta_p > 0 , if for every question q \in \mathcal{Q} , the model capacity set C(q, \delta_p) does not intersect with the solution set \mathcal{S}(q) , i.e.,
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Equation]
+C(q, \delta_p) \cap \mathcal{S}(q) = \emptyset, \quad \forall q \in \mathcal{Q},
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+then under Assumption 4.3, when training RL for T steps with B samples per step such that TB = \Theta(1/\delta_p) , there is a constant probability that the RL algorithm will not update the model.
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+To overcome this limitation, our method QUESTA provides a simple yet effective solution: augment each question in \mathcal Q with a partial solution to improve the chances of sampling informative trajectories. Formally, we assume the existence of a hint h_q for every question q \in \mathcal Q that can guide the model toward discovering a valid completion.
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Definition 4.5 (Question Augmentation). For every question q \in \mathcal{Q} , hint h_q \in \mathcal{V}^* satisfies that for \delta'_p = \delta_p^{1/2 - \epsilon} for some \epsilon > 0 :
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: ListGroup]
+the hint h_q can be generated with a non-neglible probability: P_{\mu}(h_q|q) \geq \delta_p' there exists a solution to the hinted problem s_q \in \mathcal{S}(q) such that s_q can be generated with high probability after s_q , i.e.
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Equation]
+P_{\mu}(s_q|(q,h_q)) = \delta'_{p}. \quad R(q,h \oplus s_q) = 1.
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+The hint h_q can exist for every question even when the model's capacity set C(q, \delta_p) does not intersect with the solution set \mathcal{S}(q) . For instance, if every solution can be decomposed into two steps, and the model can generate each step correctly with probability \delta_p' = \sqrt{o(\delta_p)} , then the possibility of generating two steps correctly at the same time is only o(\delta_p) .
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+This implies that the sampling budget needed with a hint is asymptotically almost the square root of the budget required without it (\Theta(1/\delta_p)) , as given in Theorem 4.4. We further provide a learnability result where we assume the policy is parameterized by a softmax policy parameterization in a classical tabular RL setup.
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Theorem 4.6 (Informal Upper Bound on RL Learnability with Hint). If we have a hint h_q for every question q \in \mathcal{Q} (Def. 4.5), then there exists an RL algorithm that can output a policy \pi_\theta such that \mathbb{E}_{q \sim \mathrm{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \pi_\theta(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99 with O(1/\delta_p') sampling budget with high probability.
+
+[p. 6 | section: 4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY | type: Text]
+Theorem 4.6 provides a theoretical guarantee that the model can reach a high training success rate when partial solution is included. Empirically, we observe the model generalizes well both in-distribution and out-of-distribution to hard questions.
+
+[p. 6 | section: 5 EXPERIMENTS | type: Text]
+Dataset. We begin with the OpenR1-Math-220K dataset and use DeepSeek-R1-Distill-1.5B as a weak selection model to filter it down to the 26K hardest items. This set serves as our base prompts. We then use Nemotron-1.5B to sample eight generations per prompt and classify problems into Easy Data (7–8 correct answers) and Hard Data (0–1 correct answers), enabling controlled experiments introduced in Section 2. The exact prompt template is provided in Appendix B.8
+
+[p. 6 | section: 5 EXPERIMENTS | type: Text]
+Data Augmentation (QUESTA). To improve the tractability of the problems, we apply QUESTA to prepend the prompt with partial solutions, i.e. first p% of the full solution in the SFT data provided in the OpenR1-Math-220K dataset. After augmentation, we use the initial model at RL training, either Nemotron-1.5B or DeepScaleR-1.5B, to sample 8 generations per augmented prompts and select samples with 0–4 correct predictions. Full details are provided in Appendix B.5. These high-variance cases provide stronger learning signals and make the training process more effective.
+
+[p. 7 | section: 5 EXPERIMENTS | type: TableGroup]
+Table 1: Performance comparison (Pass@1, averaged over 32 samples) across maths benchmarks. The best results among the 1.5B models are highlighted in bold. Larger models are shown in gray as reference points. Reported results for DeepSeek-R1-Distill and Qwen3 are taken from their official documentation (Guo et al., 2025; Yang et al., 2025) , while the rest are self-evaluated. Our QUESTA-Nemotron-1.5B achieves state-of-the-art performance among 1.5B models and, notably, matches or even exceeds the performance of DeepSeek-R1-Distill-32B across several benchmarks, despite being over 20× smaller in parameter count. This demonstrates the effectiveness of QUESTA in enhancing small model capabilities through targeted training. Model AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 Avg DeepSeek-R1-Distill-1.5B 28.7 22.3 12.0 52.4 31.8 29.44 Qwen3-1.7B 48.3 36.8 22.19 56.13 44.06 41.50 DeepSeek-R1-Distill-32B 72.6 51.8 33 65.0 68 58.08 Qwen3-8B 76.0 67.3 44.79 68.56 68.33 64.99 Nemotron-1.5B 61.77 49.50 31.56 64.62 58.23 53.14 QUESTA-Nemotron-1.5B 72.50 62.29 41.67 70.36 69.48 63.26
+
+[p. 7 | section: 5 EXPERIMENTS | type: Text]
+Training Setup. We use AReaL (Fu et al., 2025) as our RL training framework, applying the GRPO algorithm (Guo et al., 2025) without the Kullback–Leibler (KL) divergence loss. Following DAPO (Yu et al., 2025) , we also dynamically filter out prompts that are either all correct or all incorrect during rollouts. During training, we sample n = 16 responses per prompt with a maximum prompt length of 8192 tokens and a maximum generation length of 24000 tokens, using a sampling temperature of 1.0 and clipping hyperparameters with εlow = εhigh = 0.2. The batch size is 128 with a mini-batch size of 1, equivalent to 128 gradient updates per rollout step. Optimization is performed with AdamW (Kingma & Ba, 2017; Loshchilov & Hutter, 2019) using a constant learning rate of 2 × 10 − 5 . Experiments are conducted on eight NVIDIA H800 (80GB) nodes. Full details of our training method are provided in Appendix B.1.
+
+[p. 7 | section: 5 EXPERIMENTS | type: Text]
+Evaluation Setup. For each problem in the evaluation benchmarks, we generate 32 samples and report pass@1 results. Generation uses a sampling temperature of 0.7 and a top-p value of 0.95, with k = 32 responses per question unless otherwise specified. It is important to note that while partial solutions were incorporated during training, no partial solutions are provided at evaluation time.
+
+[p. 7 | section: 5.1 EXPERIMENTAL RESULTS | type: Text]
+Key Results. Table 1 reports results on challenging math benchmarks. QUESTA yields substantial gains for Nemotron-1.5B, achieving an average improvement of 10% over its baseline and a particularly strong +13% on AIME25. These improvements are consistent across all benchmarks, highlighting the effectiveness of our approach in enhancing problem-solving robustness.
+
+[p. 7 | section: 5.1 EXPERIMENTAL RESULTS | type: Text]
+Compared to other models, QUESTA-Nemotron-1.5B consistently outperforms peers of similar scale, such as DeepSeek-R1-Distill-1.5B and Qwen3-1.7B, and even surpasses larger models like DeepSeek-R1-Distill-32B across all benchmarks. On AIME25 in particular, it exceeds DeepSeek-R1-Distill-32B by a substantial margin of +11%. Against the stronger Qwen3-8B, QUESTA-Nemotron-1.5B remains competitive despite operating at a fraction of the parameter scale.
+
+[p. 7 | section: 5.1 EXPERIMENTAL RESULTS | type: Text]
+Training Dynamics. Figure 5 summarizes the training dynamics of QUESTA-Nemotron-1.5B. A positive correlation is observed between average response length and model accuracy, reflecting common trends in RL training. Notably, with QUESTA, the entropy during RL training remains stable and does not exhibit significant collapse.
+
+[p. 7 | section: 5.1 EXPERIMENTAL RESULTS | type: Text]
+Pass@k Analysis. Our evaluation follows the standard pass@k methodology, consistent with DeepSeek-R1 (Guo et al., 2025) , with further details provided in Appendix B.2. In contrast to recent findings that RL-based training can reduce pass@k at larger k values (Yue et al., 2025; Liu et al., 2025) , our results show that QUESTA preserves—and in many cases modestly improves—performance across a broad range of k. As shown in Figure 2, incorporating partial-solution hints within a two-stage curriculum yields consistent gains across models, without the degradation in pass@k often observed under standard RL training. These results indicate that QUESTA enhances both the quality and diversity of candidate solutions, rather than overfitting to a single best trajectory.
+
+[p. 8 | section: 5.1 EXPERIMENTAL RESULTS | type: FigureGroup]
+Figure 5: Training dynamics of QUESTA-Nemotron-1.5B. The first and second charts show the progression of average response length and average reward across rollout samples during the RL process, both of which steadily increase over time. The third chart presents the average entropy. Interestingly, the entropy increases over time, suggesting that QUESTA does not suffer from entropy collapse and instead encourages diverse and exploratory behavior.
+
+[p. 8 | section: 5.1 EXPERIMENTAL RESULTS | type: FigureGroup]
+Figure 6: Pass Rate Distribution on Training Prompts. We compare the success rate on the 26K training set before and after RL, using the average pass rate over 8 samples per question. Although partial solutions are included during QUESTA training, no hints are provided during this evaluation. This setup isolates the true impact of QUESTA by assessing its ability to improve performance on problems without hints. QUESTA significantly reduces the number of unsolved or partially solved problems in the training set, especially for hard ones where initial model solves only 0/8 or 1/8 times.
+
+[p. 8 | section: 5.1 EXPERIMENTAL RESULTS | type: Text]
+Generalization at Test Time when Hints are Removed. A natural question arises from our approach: since we add partial solutions during RL training, does this improvement persist when hints are removed at evaluation time? To answer this, Figure 6 compares the pre- and post-RL models on the 26K training prompt set, evaluated without any hints. The distribution clearly shifts away from the 0/8–1/8 bins toward higher pass rates, indicating that the model solves a larger fraction of problems even without access to partial solutions. On the evaluation AIME benchmarks, Table 2 further demonstrates that QUESTA expands coverage at Pass@32: for Nemotron-1.5B, the number of unsolved problems drops from 5 to 2 on AIME24 (newly solved indices 2, 13, 29) and from 6 to 3 on AIME25 (newly solved indices 9, 13, 27). Taken together, these results show that our method generalizes well beyond the training setting and helps solve hard problems that are otherwise inaccessible without partial-solution guidance.
+
+[p. 8 | section: 5.2 Further Ablations | type: Text]
+Ablation with Difficulty Curriculum. We first motivate the choice of a two-stage curriculum: RL on Partial-50 followed by RL on Partial-25 . From a modeling standpoint, the most appropriate inference distribution for the model should be the original (no-hint) distribution. Hence, during training we should gradually reduce reliance on hints to align the learned policy with the evaluation distribution. This motivates decreasing the partial ratio over time so that the model transitions from scaffolded reasoning to autonomous reasoning.
+
+[p. 9 | section: 5.2 Further Ablations | type: TableGroup]
+Table 2: Indices of unsolved problems at Pass@32 on AIME24 and AIME25 (with indices ranging from 0-29). Our method, QUESTA, consistently improve the model capacity on hard cases where the initial model is unable to solve, improving overall coverage at Pass@32. Models AIME24 Unsodlved Indices AIME25 Unsolved Indices Nemotron-1.5B 2, 3, 13, 21, 29 9, 12, 13, 14, 27, 29 QUESTA-Nemotron-1.5B 3, 21 12, 14, 29 Table 3: Ablation Study on the Impact of Curriculum Design. This table demonstrates the importance of curriculum learning in improving model performance. The model QUESTA-Nemotron-1.5B-50 was trained entirely with Partial-50 data for 2000 steps, while QUESTA-Nemotron-1.5B followed a curriculum learning approach, starting with 100 steps of Partial-50 data followed by 1900 steps of Partial-25 data. As seen in the table, the curriculum learning approach (QUESTA-Nemotron-1.5B) outperforms training with only Partial-50 data (QUESTA-Nemotron-1.5B-50). Extension with Partial-50-Partial-25-Partial-0 did not yield significant improvements, and thus, are not included in the table.
+
+[p. 9 | section: 5.2 Further Ablations | type: Table]
+Model | AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 | Avg Nemotron-1.5B QUESTA-Nemotron-1.5B-50 QUESTA-Nemotron-1.5B 61.77 49.50 31.56 64.62 58.23 53.14 67.18 59.38 39.17 69.41 66.15 60.26 72.50 62.29 41.67 70.36 69.48 63.26
+
+[p. 9 | section: 5.2 Further Ablations | type: Text]
+Empirically, Table 3 shows that, under the same 2000-step budget, the curriculum Partial-50 Partial-25 learns substantially better than training on Partial-50 alone. We cap the Partial-50 stage at 100 steps, after which we switch to Partial-25 . As shown in Figure 11, entropy for QUESTA-Nemotron-1.5B-50 begins to decline beyond 100 steps, so transitioning at this point prevents overconfidence and sustains training stability. We have also tried extending the curriculum from Partial-25 to Partial-0 in our experiments, but observed no gains and no increase in response length (see Figure 12).
+
+[p. 9 | section: 5.2 Further Ablations | type: TableGroup]
+Table 4: Performance comparison (Pass@1, averaged over 32 samples) between Nemotron-1.5B and QUESTA-Nemotron-1.5B (By OpenMathReasoning). The two models achieve comparable results, with the version trained on OpenR1 performing slightly better overall. Model AIME24 AIME25 HMMT FEB 25 Olympiad Bench BRUMO25 | Avg Nemotron-1.5B QUESTA-50 (with OpenMathReasoning) QUESTA-50 (with OpenR1) 61.77 49.50 31.56 64.62 58.23 | 53.14 66.46 58.54 36.35 66.06 63.13 | 58.11 67.18 59.38 39.17 69.41 66.15 | 60.26
+
+[p. 9 | section: 5.2 Further Ablations | type: Text]
+Ablation with Different Dataset. We also evaluated QUESTA on OpenMathReasoning Moshkov et al. (2025), selecting the 60K questions with pass_rate_72b_tir of 0 or 1/32. Due to time constraints, we trained only the first stage of QUESTA with 50% partial solutions. Table 4 shows that QUESTA-Nemotron-1.5B-50 achieves similar performance as using the OpenR1 dataset. This indicates that our approach generalizes across datasets.
+
+[p. 9 | section: 5.2 Further Ablations | type: Text]
+Other Ablations. We also conduct an extensive set of comparative experiments and ablation studies, with detailed results provided in Appendix D. These include an ablation of QUESTA without hints (Appendix D.1), experiments with different model backbones (Appendix D.2), and the full set of pass rates and training curves for additional models (Appendix D.3).
+
+[p. 9 | section: 6 CONCLUSIONS | type: Text]
+In this work we introduced QUESTA, a lightweight data-centric framework that augments hard prompts with partial-solution hints during RL training. Without altering model architecture or reward design, QUESTA sets new state-of-the-art results for 1.5 B-scale models on AIME24, AIME25 and HMMT25. Further, we theoretically demonstrate how question augmentation can improve sample
+
+[p. 10 | section: 6 CONCLUSIONS | type: Text]
+efficiency. Our analysis shows that the method can potentially be generalized to other domains such as competitive coding, software engineering or other agentic tasks. Designing proper question augmentation pipelines for theses new tasks can be an important and interesting future direction.

iclr26/3MifB0f7qR/paper.md

@@ -0,0 +1,647 @@
+
+
+{0}------------------------------------------------
+
+# QUESTA: EXPANDING REASONING CAPACITY IN LLMS VIA QUESTION AUGMENTATION
+
+Jiazheng Li\* $^{\dagger 1,2}$ , Hongzhou Lin\* $^{\ddagger 3}$ , Hong Lu $^{1,2}$ , Kaiyue Wen $^4$ , Zaiwen Yang $^1$ , Jiaxuan Gao $^1$ , Yi Wu $^{1,2}$ , Jingzhao Zhang $^{1,2}$ 
+
+## **ABSTRACT**
+
+Reinforcement learning (RL) has emerged as a central paradigm for training large language models (LLMs) in reasoning tasks. Yet recent studies (Yue et al., 2025; Liu et al., 2025) question RL's ability to incentivize reasoning capacity beyond the base model. This raises a key challenge: how can RL be adapted to solve harder reasoning problems more effectively? To address this challenge, we propose a simple yet effective strategy via Question Augmentation: introduce partial solutions during training to reduce problem difficulty and provide more informative learning signals. Our method, QuestA, when applied during RL training on math reasoning tasks, not only improves pass@1 but also pass@k—particularly on problems where standard RL struggles to make progress. This enables continual improvement over strong open-source models such as DEEPSCALER and OPENMATH NEMOTRON, further enhancing their reasoning capabilities. We achieve new state-of-the-art results on math benchmarks using 1.5B-parameter models: 72.50% (+10.73%) on AIME24, 62.29% (+12.79%) on AIME25, and 41.67% (+10.11%) on HMMT25. Code, data and model are available at https://github.com/foreverlasting1202/QuestA.
+
+![](_page_0_Figure_6.jpeg)
+
+Figure 1: QUESTA is a data augmentation method that injects partial solutions to effectively scaffold RL training on hard reasoning problems. We construct 26K high-quality augmented prompts from challenging instances in OpenR1 (Open-R1 Team, 2025), and fine-tune models using 32K-context-length RL. When applied to Nemotron-1.5B, QUESTA delivers substantial performance gains—achieving new state-of-the-art results across all math benchmarks for 1.5B-parameter models.
+
+#### 1 Introduction
+
+Frontier large language models (LLMs), including OpenAI-O1, O3 (Jaech et al., 2024), DeepSeek-R1 (Guo et al., 2025), Qwen3 (Yang et al., 2025), and Gemini 2.5 (Gemini Team, Google DeepMind,
+
+<sup>&</sup>lt;sup>1</sup>Tsinghua University, <sup>2</sup>Shanghai Qi Zhi Institute, <sup>3</sup>Amazon, <sup>4</sup>Stanford University
+
+<sup>\*</sup>Equal Contribution
+
+<sup>&</sup>lt;sup>†</sup>Contact: Jiazheng at foreverlasting1202@outlook.com, and Jingzhao at jingzhaoz@mail.tsinghua.edu.cn
+
+<sup>&</sup>lt;sup>‡</sup>This work is independent of and outside of the work at Amazon.
+
+{1}------------------------------------------------
+
+<span id="page-1-0"></span>![](_page_1_Figure_1.jpeg)
+
+Figure 2: We compare pass@k curves of RLVR-trained models, with and without QUESTA. As a controlled experiment, we perform RL training using either easy or hard prompts. Standard RL on easy prompts (red) shows clear degradation in pass@k as k increases compared to the base model (blue). Training on hard prompts (green) improves pass@k, but comes at the cost of substantially longer training. This motivates our development of QUESTA, which scaffolds hard problems to improve training efficiency and delivers consistently stronger results: the RL+QUESTA model (orange) stays above standard RL (red) across all k, while also preserving or improving performance at larger k relative to RL trained with hard prompts.
+
+2025), have exhibited exceptional performance on high-complexity reasoning tasks spanning mathematics, programming, and formal logic. Recent advances in the field have increasingly prioritized reinforcement learning paradigms (RL), among which *Reinforcement Learning with Verifiable Rewards* (RLVR) has emerged as a scalable and efficient approach to enhancing reasoning capabilities. Using automatically verifiable signals, RLVR enables alignment between model output and objective correctness, thus addressing a critical limitation of traditional RL for reasoning.
+
+However, the community remains divided on a fundamental question regarding RLVR: does it expand the model's intrinsic reasoning capacity, or merely exploit pre-existing knowledge encoded in the base model? Recent research (Yue et al., 2025; Liu et al., 2025; Zhao et al., 2025) show that while state-of-the-art RL methods (e.g., GRPO, DAPO) (Guo et al., 2025; Yu et al., 2025; An et al., 2025) can enhance the pass@1 metric by reinforcing high-reward completions, they encounter significant limitations when tackling high-difficulty tasks where the base model performs poorly. This phenomenon differs from that observed in Supervised Fine-Tuning (SFT) Luo et al. (2025a). Within the SFT paradigm, enhancing the diversity of problem difficulty serves as a critical factor, as it can effectively improve the model's performance on downstream tasks. However, in the framework of RLVR, the inclusion of easy prompts tends to undermine the model's inherent reasoning capabilities.
+
+One insightful explanation (Cui et al., 2025; Wang et al., 2025a) for the drop suggests that model overfits on correct solutions and hence causes entropy collapse, limiting its ability to explore. To validate this, we design a controlled setup that separates prompts into easy and hard groups. When applying RLVR on the Nemotron 1.5B model (Moshkov et al., 2025) with the OpenR1 dataset, we find that training on easy prompts leads to a clear decline in pass@k accuracy (Figure 2).
+
+Given these findings, we observe that training with hard prompts is more beneficial than with easy ones. Yet, RL training on hard problems tends to be much slower, as sparse reward signals and limited sample efficiency hinder progress. The key challenge, then, is **how to structure the learning process to fully expand reasoning capabilities while mitigating the inefficiency of RL on hard tasks**. To this end, we introduce QUESTA: a parsimonious and efficient strategy that dynamically adjusts problem difficulty during RL training. The core contributions of this work are threefold:
+
+• We notice that the evolution of model capacity in RLVR critically depends on dataset difficulty, underscoring the importance of training on *hard problems* to expand reasoning ability.
+
+{2}------------------------------------------------
+
+<span id="page-2-0"></span>![](_page_2_Figure_1.jpeg)
+
+Figure 3: Comparison of RL training dynamics: Training with only hard problems (green) makes progress very slowly due to sparse rewards, while our method with partial solutions (orange) accelerates training and consistently achieves higher accuracy across training steps.
+
+- We introduce QUESTA, an efficient procedure that controls difficulty by augmenting hard problems with partial solutions. This approach provides a smooth curriculum within RL training and makes high-difficulty tasks more tractable. Through our **fully open-sourced** training pipeline, QUESTA consistently improves pass@1 and pass@k, enabling 1.5B-parameter models to reach new state-of-the-art performance—72.5% on AIME24, 62.3% on AIME25, and 41.7% on HMMT25 (Table 1).
+- Our theoretical analysis in Section 4 explains why partial-solution augmentation accelerates RL training: by decomposing problems into intermediate steps, the method yields denser reward signals and improves sample efficiency, while still driving the model to master the hardest problems.
+
+## <span id="page-2-1"></span>2 Tradeoffs between Reasoning Capacity and Learning Efficiency
+
+Given the ongoing debate on whether reinforcement learning enhances the reasoning capacity of language models, we design a controlled experiment to study how dataset difficulty changes model performances measured by pass@k accuracy. Specifically, we filter out easy problems and hard problems from the 220K OpenR1 dataset, base on model's success rate, each containing around 4K data. We then run RL with GRPO for one thousand steps. This setup allows us to isolate how the choice of prompt difficulty impacts the model's reasoning capacity. In Figure 2 and Figure 3, we provide pass@k comparison and the learning dynamics, we make two observations.
+
+**RL** with Easy Prompts Hurts pass@k and Reasoning Capacity. Training on easy or already-solvable problems leads to overfitting on shallow patterns, reinforcing confidence rather than expanding reasoning capacity. While pass@1 may rise, output diversity declines and performance on harder benchmarks deteriorates, with pass@k dropping at larger k (see Figure 2). This suggests that the model exploits familiar solution modes instead of exploring new trajectories. To truly expand capacity, RL training should focus on hard problems, where the policy is forced to explore and acquire novel solution strategies.
+
+**RL** with Hard Prompts Leads to Slow Learning. Training on hard prompts directly targets the reasoning capacity of the model, but the learning process is much slower (see Figure 3) and less sample-efficient. The difficulty arises because RL rewards on these problems are sparse, providing limited gradient signals for policy improvement. We formallize the underlying reason in Section 4 and in Theorem 4.4.
+
+In practice, not all questions in the training set Q are equally difficult, and one might hope that training on easier examples could generalize to harder ones. However, empirical evidence suggests that RL-based training exhibits a bi-modal pattern in success rates (An et al., 2025): by the end of training, models tend to either solve a question reliably or fail entirely (see Figure 6). This implies that once a question falls outside the model's capacity set, the RL algorithm is unlikely to recover.
+
+{3}------------------------------------------------
+
+Together, these results highlight a tension: *easy prompts dilute reasoning capacity, while hard prompts stall learning altogether.* This motivates the need for strategies that can retain the benefits of hard problems while mitigating the inefficiency caused by sparse rewards. To this end, we introduce partial solutions that break a complex question into smaller, more approachable pieces. Theoretical analysis (Theorem 4.6) suggests that appending part of the solutions as hint can greatly improve RL efficiency.
+
+Empirically, we simply choose the hint to be a part of the solution of the original question q and observe faster learning in Figure 3. Surprisingly, even if we don't explicitly train the model to generate the hint, the model's capacity without hint still continues to improve and lead to steady improvement in problems out of reach in standard RL training (see Table 3). We elaborate on implementation details in the next section.
+
+## 3 Questa: Question Augmentation with Partial Solutions
+
+QUESTA is a modular augmentation framework designed to inject partial solution sketches into prompts during reinforcement learning (RL) training. It adresses scenarios where the base model fails to generate correct completions—conditions that typically result in sparse reward signals. Distinct from approaches that modify reward functions or optimization algorithms, QUESTA operates at the input level: it transforms original training prompts into more tractable variants, thereby exposing intermediate reasoning steps to the model.
+
+#### <span id="page-3-0"></span>**Original Prompt**
+
+Let  $\mathbb N$  be the set of positive integers. A function  $f:\mathbb N\to\mathbb N$  satisfies the equation
+
+$$f(f(\dots f(n)\dots)) = \frac{n^2}{f(f(n))}$$
+ with  $f(n)$  applications of  $f$ ,
+
+for all positive integers n. Given this information, determine all possible values of f(1000).
+
+![](_page_3_Picture_9.jpeg)
+
+#### **Augmented Prompt**
+
+Let  $\mathbb N$  be the set of positive integers. The function  $f:\mathbb N\to\mathbb N$  satisfies the equation
+
+$$f(f(\dots f(n)\dots)) = \frac{n^2}{f(f(n))} \quad \text{with } f(n) \text{ applications of } f,$$
+
+for all positive integers n. Given this information, determine all possible values of f(1000).
+
+#### ## Hint: Partial Solution
+
+Analysis shows that f must be an involution, meaning f(f(n)) = n for all n, and it fixes all odd positive integers, so f(n) = n for odd n. For even positive integers, f either fixes the number or swaps it with another even positive integer in a 2-cycle.
+
+Please reason step by step, and put your final answer within \boxed{}.
+
+Figure 4: QUESTA augments each original question in the dataset by prepending the first p% of the solution sketch. In our experiments, we apply augmentation using the solution block rather than the reasoning chain-of-thought. The hint percentage p is computed as the ratio of tokens used as hints to the total number of tokens in the solution sketch.
+
+Question Augmentation Mechanism For a given problem x with an n-step solution trajectory  $y=(y_1,y_2,\ldots,y_n)$ , QUESTA constructs a set of augmented prompts  $\{\tilde{x}^{(p)}\}$ , where each  $\tilde{x}^{(p)}$  appends the first p steps of the solution as a prefix to the original question. The parameter p (e.g., p=50% or 25%) quantifies the proportion of the solution revealed, thereby enabling precise control over the difficulty of the augmented prompt.
+
+In our empirical evaluations, we employed the OpenR1-Math-220K dataset (Open-R1 Team, 2025)—a supervised fine-tuning (SFT) corpus containing solution trajectories generated by DEEPSEEK-R1. Each instance in this dataset comprises a detailed chain-of-thought (CoT) section followed by a final solution block. For augmentation, we extracted the final solution (omitting speculative reasoning
+
+{4}------------------------------------------------
+
+within the CoT section). The solution was then truncated at a predefined percentage p and prepended to the original question, yielding the augmented prompt used in RL training, as shwon in Figure 4.
+
+**Targeting High-Difficulty Problems** QUESTA is applied exclusively to prompts where the base model's pass rate is close to zero. Using the OpenR1-Math-220K dataset, we first employ lightweight heuristic filters to reduce the full 220K problems to 26K of the hardest candidates. These problems are then augmented with partial-solution prefixes where we conduct a second difficulty screening: sample multiple completions from the model for each augmented prompt, and only those instances with consistently low pass rates are retained. This two-stage filtering pipeline yields a final pool of no more than 10K problems, ensuring that augmentation resources are concentrated on the most challenging cases where the base model needs additional guidance and scaffolding.
+
+**Integrating with RL Pipelines** QUESTA exhibits orthogonality to underlying RL algorithms, enabling seamless integration into existing training pipelines (e.g., GRPO (Shao et al., 2024), DAPO (Yu et al., 2025)) without modifications. Specifically, integration requires only replacing the original rollout dataset with the augmented dataset, while retaining the original reward function and policy update mechanism. To further exploit this input-level flexibility, we extended QUESTA with an iterative curriculum RL paradigm:
+
+- 1. First, augment the dataset with p = 50%, apply the difficulty filtering with the augmented prompt, and conduct reinforcement learning training until the performance saturates.
+- 2. Second, reduce the augmentation from p = 50% to p = 25%, i.e. provide fewer hints. Again, we apply the difficulty filtering, and conduct reinforcement learning training until convergence.
+
+Here, the rationale for the choice of *p* is provided in Appendix B.6. By keeping the training signals strong at each stage, the method speeds up convergence on difficult tasks and makes QUESTA a simple, plug-and-play approach for curriculum-based RL.
+
+## <span id="page-4-0"></span>4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY
+
+In this section, we present a theoretical perspective on how question augmentation improves the efficiency of reinforcement learning. Our central thesis is that the primary bottleneck in RL-based reasoning lies in the difficulty of discovering successful trajectories within a finite sampling budget. Question augmentation addresses this challenge by reshaping the *learnability landscape*—making hard problems more discoverable by increasing the likelihood of encountering correct trajectories.
+
+Motivated by experiments which quantify model capcity with pass@k accuracy, we introduce the following notions of *solution set* (Definition 4.1) and *model capacity set* (Definition 4.2) for a given question q and model  $\mu$ . Let  $\mathcal V$  be the vocaboluary set, and let  $P_{\mu}(q,\tau)$  denote the probability that a language model  $\mu$  generates trajectory  $\tau \in \mathcal V^*$  when conditioned on input question  $q \in \mathcal V^*$ .
+
+<span id="page-4-1"></span>**Definition 4.1** (Solution Set). Given a question q and a binary reward function  $R: \mathcal{V}^* \times \mathcal{V}^* \to \{0, 1\}$ , the *solution set* is defined as:
+
+$$S(q) = \{ \tau \in \mathcal{V}^* \mid R(q, \tau) = 1 \}.$$
+
+<span id="page-4-2"></span>**Definition 4.2** (Model Capacity Set). Given a probability threshold  $\delta_p > 0$ , a language model  $\mu$ , and a question q, define the *model capacity set*  $C(q, \delta_p)$  as the smallest set of trajectories whose total probability mass is at least  $1 - \delta_p$ :
+
+$$C(q, \delta_p) = \arg\min_{S \subseteq \mathcal{V}^*} \left\{ |S| \left| \sum_{\tau \in S} P_{\mu}(q, \tau) \ge 1 - \delta_p \right\}. \right.$$
+
+The Model Capacity Set  $C(q, \delta_p)$  intuitively captures the set of most likely output trajectories that the model  $\mu$  can generate for a given input q, up to a small probability threshold  $\delta_p$ .
+
+<span id="page-4-3"></span>This formalization leads to a critical insight: if the model's capacity set fails to intersect with the solution set—meaning the model is unlikely to generate any correct completions—then the RL process cannot make progress. To articulate this more formally, we begin by stating a standard assumption satisfied by many popular RL algorithms, such as DAPO and online GRPO:
+
+{5}------------------------------------------------
+
+**Assumption 4.3** (Null Gradient from Zero-Reinforcement). The RL algorithm does not update the model weights if none of the sampled rollouts receives a positive reward (i.e., reward = 1).
+
+Under this assumption, we easily the following lower bound, which states that if all training questions are unreachable within the model's capacity set, the RL process is likely to stall entirely:
+
+<span id="page-5-0"></span>**Theorem 4.4** (Lower Bound on RL Learnability under Solution Inaccessibility). Given a probability threshold  $\delta_p > 0$ , if for every question  $q \in \mathcal{Q}$ , the model capacity set  $C(q, \delta_p)$  does not intersect with the solution set  $\mathcal{S}(q)$ , i.e.,
+
+$$C(q, \delta_p) \cap \mathcal{S}(q) = \emptyset, \quad \forall q \in \mathcal{Q},$$
+
+then under Assumption 4.3, when training RL for T steps with B samples per step such that  $TB = \Theta(1/\delta_p)$ , there is a constant probability that the RL algorithm will not update the model.
+
+To overcome this limitation, our method QUESTA provides a simple yet effective solution: augment each question in  $\mathcal Q$  with a partial solution to improve the chances of sampling informative trajectories. Formally, we assume the existence of a hint  $h_q$  for every question  $q \in \mathcal Q$  that can guide the model toward discovering a valid completion.
+
+<span id="page-5-2"></span>**Definition 4.5** (Question Augmentation). For every question  $q \in \mathcal{Q}$ , hint  $h_q \in \mathcal{V}^*$  satisfies that for  $\delta'_p = \delta_p^{1/2 - \epsilon}$  for some  $\epsilon > 0$ :
+
+- the hint  $h_q$  can be generated with a non-neglible probability:  $P_{\mu}(h_q|q) \geq \delta_p'$
+- there exists a solution to the hinted problem  $s_q \in \mathcal{S}(q)$  such that  $s_q$  can be generated with high probability after  $s_q$ , i.e.
+
+$$P_{\mu}(s_q|(q,h_q)) = \delta'_{p}. \quad R(q,h \oplus s_q) = 1.$$
+
+The hint  $h_q$  can exist for every question even when the model's capacity set  $C(q, \delta_p)$  does not intersect with the solution set  $\mathcal{S}(q)$ . For instance, if every solution can be decomposed into two steps, and the model can generate each step correctly with probability  $\delta_p' = \sqrt{o(\delta_p)}$ , then the possibility of generating two steps correctly at the same time is only  $o(\delta_p)$ .
+
+This implies that the sampling budget needed with a hint is asymptotically almost the square root of the budget required without it  $(\Theta(1/\delta_p))$ , as given in Theorem 4.4. We further provide a learnability result where we assume the policy is parameterized by a softmax policy parameterization in a classical tabular RL setup.
+
+<span id="page-5-1"></span>**Theorem 4.6** (Informal Upper Bound on RL Learnability with Hint). If we have a hint  $h_q$  for every question  $q \in \mathcal{Q}$  (Def. 4.5), then there exists an RL algorithm that can output a policy  $\pi_\theta$  such that  $\mathbb{E}_{q \sim \mathrm{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \pi_\theta(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99$  with  $O(1/\delta_p')$  sampling budget with high probability.
+
+Theorem 4.6 provides a theoretical guarantee that the model can reach a high training success rate when partial solution is included. Empirically, we observe the model generalizes well both in-distribution and out-of-distribution to hard questions.
+
+#### 5 EXPERIMENTS
+
+**Dataset.** We begin with the OpenR1-Math-220K dataset and use DeepSeek-R1-Distill-1.5B as a weak selection model to filter it down to the 26K hardest items. This set serves as our base prompts. We then use Nemotron-1.5B to sample eight generations per prompt and classify problems into Easy Data (7–8 correct answers) and Hard Data (0–1 correct answers), enabling controlled experiments introduced in Section 2. The exact prompt template is provided in Appendix B.8
+
+**Data Augmentation (QUESTA).** To improve the tractability of the problems, we apply QUESTA to prepend the prompt with partial solutions, i.e. first p% of the full solution in the SFT data provided in the OpenR1-Math-220K dataset. After augmentation, we use the initial model at RL training, either Nemotron-1.5B or DeepScaleR-1.5B, to sample 8 generations per augmented prompts and select samples with 0–4 correct predictions. Full details are provided in Appendix B.5. These high-variance cases provide stronger learning signals and make the training process more effective.
+
+{6}------------------------------------------------
+
+<span id="page-6-0"></span>Table 1: Performance comparison (Pass@1, averaged over 32 samples) across maths benchmarks. The best results among the 1.5B models are highlighted in bold. Larger models are shown in gray as reference points. Reported results for DeepSeek-R1-Distill and Qwen3 are taken from their official documentation [\(Guo et al., 2025;](#page-10-2) [Yang et al., 2025\)](#page-11-2), while the rest are self-evaluated. Our QUESTA-Nemotron-1.5B achieves state-of-the-art performance among 1.5B models and, notably, matches or even exceeds the performance of DeepSeek-R1-Distill-32B across several benchmarks, despite being over 20× smaller in parameter count. This demonstrates the effectiveness of QUESTA in enhancing small model capabilities through targeted training.
+
+| Model                    | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25 | Avg   |
+|--------------------------|--------|--------|-------------|----------------|---------|-------|
+| DeepSeek-R1-Distill-1.5B | 28.7   | 22.3   | 12.0        | 52.4           | 31.8    | 29.44 |
+| Qwen3-1.7B               | 48.3   | 36.8   | 22.19       | 56.13          | 44.06   | 41.50 |
+| DeepSeek-R1-Distill-32B  | 72.6   | 51.8   | 33          | 65.0           | 68      | 58.08 |
+| Qwen3-8B                 | 76.0   | 67.3   | 44.79       | 68.56          | 68.33   | 64.99 |
+| Nemotron-1.5B            | 61.77  | 49.50  | 31.56       | 64.62          | 58.23   | 53.14 |
+| QUESTA-Nemotron-1.5B     | 72.50  | 62.29  | 41.67       | 70.36          | 69.48   | 63.26 |
+
+Training Setup. We use AReaL [\(Fu et al., 2025\)](#page-9-2) as our RL training framework, applying the GRPO algorithm [\(Guo et al., 2025\)](#page-10-2) without the Kullback–Leibler (KL) divergence loss. Following DAPO [\(Yu et al., 2025\)](#page-11-3), we also dynamically filter out prompts that are either all correct or all incorrect during rollouts. During training, we sample n = 16 responses per prompt with a maximum prompt length of 8192 tokens and a maximum generation length of 24000 tokens, using a sampling temperature of 1.0 and clipping hyperparameters with εlow = εhigh = 0.2. The batch size is 128 with a mini-batch size of 1, equivalent to 128 gradient updates per rollout step. Optimization is performed with AdamW [\(Kingma & Ba, 2017;](#page-10-5) [Loshchilov & Hutter, 2019\)](#page-10-6) using a constant learning rate of 2 × 10<sup>−</sup><sup>5</sup> . Experiments are conducted on eight NVIDIA H800 (80GB) nodes. Full details of our training method are provided in Appendix [B.1.](#page-13-0)
+
+Evaluation Setup. For each problem in the evaluation benchmarks, we generate 32 samples and report pass@1 results. Generation uses a sampling temperature of 0.7 and a top-p value of 0.95, with k = 32 responses per question unless otherwise specified. *It is important to note that while partial solutions were incorporated during training, no partial solutions are provided at evaluation time.*
+
+## 5.1 EXPERIMENTAL RESULTS
+
+Key Results. Table [1](#page-6-0) reports results on challenging math benchmarks. QUESTA yields substantial gains for Nemotron-1.5B, achieving an average improvement of 10% over its baseline and a particularly strong +13% on AIME25. These improvements are consistent across all benchmarks, highlighting the effectiveness of our approach in enhancing problem-solving robustness.
+
+Compared to other models, QUESTA-Nemotron-1.5B consistently outperforms peers of similar scale, such as DeepSeek-R1-Distill-1.5B and Qwen3-1.7B, and even surpasses larger models like DeepSeek-R1-Distill-32B across all benchmarks. On AIME25 in particular, it exceeds DeepSeek-R1-Distill-32B by a substantial margin of +11%. Against the stronger Qwen3-8B, QUESTA-Nemotron-1.5B remains competitive despite operating at a fraction of the parameter scale.
+
+Training Dynamics. Figure [5](#page-7-1) summarizes the training dynamics of QUESTA-Nemotron-1.5B. A positive correlation is observed between average response length and model accuracy, reflecting common trends in RL training. Notably, with QUESTA, the entropy during RL training remains stable and does not exhibit significant collapse.
+
+Pass@k Analysis. Our evaluation follows the standard pass@k methodology, consistent with DeepSeek-R1 [\(Guo et al., 2025\)](#page-10-2), with further details provided in Appendix [B.2.](#page-13-1) In contrast to recent findings that RL-based training can reduce pass@k at larger k values [\(Yue et al., 2025;](#page-11-0) [Liu et al., 2025\)](#page-10-0), our results show that QUESTA preserves—and in many cases modestly improves—performance across a broad range of k. As shown in Figure [2,](#page-1-0) incorporating partial-solution hints within a two-stage curriculum yields consistent gains across models, without the degradation in pass@k often observed under standard RL training. These results indicate that QUESTA enhances both the quality and diversity of candidate solutions, rather than overfitting to a single best trajectory.
+
+{7}------------------------------------------------
+
+<span id="page-7-1"></span>![](_page_7_Figure_1.jpeg)
+
+Figure 5: Training dynamics of QUESTA-Nemotron-1.5B. The first and second charts show the progression of average response length and average reward across rollout samples during the RL process, both of which steadily increase over time. The third chart presents the average entropy. Interestingly, the entropy increases over time, suggesting that QUESTA does not suffer from entropy collapse and instead encourages diverse and exploratory behavior.
+
+<span id="page-7-0"></span>![](_page_7_Figure_3.jpeg)
+
+Figure 6: Pass Rate Distribution on Training Prompts. We compare the success rate on the 26K training set before and after RL, using the average pass rate over 8 samples per question. Although partial solutions are included during QUESTA training, **no hints** are provided during this evaluation. This setup isolates the true impact of QUESTA by assessing its ability to improve performance on problems without hints. QUESTA significantly reduces the number of unsolved or partially solved problems in the training set, especially for hard ones where initial model solves only 0/8 or 1/8 times.
+
+Generalization at Test Time when Hints are Removed. A natural question arises from our approach: since we add partial solutions during RL training, does this improvement persist when hints are removed at evaluation time? To answer this, Figure 6 compares the pre- and post-RL models on the 26K training prompt set, evaluated without any hints. The distribution clearly shifts away from the 0/8–1/8 bins toward higher pass rates, indicating that the model solves a larger fraction of problems even without access to partial solutions. On the evaluation AIME benchmarks, Table 2 further demonstrates that QUESTA expands coverage at Pass@32: for Nemotron-1.5B, the number of unsolved problems drops from 5 to 2 on AIME24 (newly solved indices 2, 13, 29) and from 6 to 3 on AIME25 (newly solved indices 9, 13, 27). Taken together, these results show that our method generalizes well beyond the training setting and helps solve hard problems that are otherwise inaccessible without partial-solution guidance.
+
+#### 5.2 Further Ablations
+
+**Ablation with Difficulty Curriculum.** We first motivate the choice of a two-stage curriculum: RL on *Partial-50* followed by RL on *Partial-25*. From a modeling standpoint, the most appropriate inference distribution for the model should be the original (no-hint) distribution. Hence, during training we should gradually reduce reliance on hints to align the learned policy with the evaluation distribution. This motivates decreasing the partial ratio over time so that the model transitions from scaffolded reasoning to autonomous reasoning.
+
+{8}------------------------------------------------
+
+<span id="page-8-1"></span>Table 2: Indices of **unsolved** problems at Pass@32 on AIME24 and AIME25 (with indices ranging from 0-29). Our method, QUESTA, consistently improve the model capacity on hard cases where the initial model is unable to solve, improving overall coverage at Pass@32.
+
+| Models               | AIME24 Unsodlved Indices | AIME25 Unsolved Indices |
+|----------------------|--------------------------|-------------------------|
+| Nemotron-1.5B        | 2, 3, 13, 21, 29         | 9, 12, 13, 14, 27, 29   |
+| QUESTA-Nemotron-1.5B | 3, 21                    | 12, 14, 29              |
+
+<span id="page-8-0"></span>Table 3: Ablation Study on the Impact of Curriculum Design. This table demonstrates the importance of curriculum learning in improving model performance. The model QUESTA-Nemotron-1.5B-50 was trained entirely with Partial-50 data for 2000 steps, while QUESTA-Nemotron-1.5B followed a curriculum learning approach, starting with 100 steps of Partial-50 data followed by 1900 steps of Partial-25 data. As seen in the table, the curriculum learning approach (QUESTA-Nemotron-1.5B) outperforms training with only Partial-50 data (QUESTA-Nemotron-1.5B-50). Extension with Partial-50-Partial-25-Partial-0 did not yield significant improvements, and thus, are not included in the table.
+
+| Model                                                      | AIME24       | AIME25       | HMMT FEB 25  | Olympiad Bench | BRUMO25      | Avg          |
+|------------------------------------------------------------|--------------|--------------|--------------|----------------|--------------|--------------|
+| Nemotron-1.5B QUESTA-Nemotron-1.5B-50 QUESTA-Nemotron-1.5B | 61.77        | 49.50        | 31.56        | 64.62          | 58.23        | 53.14        |
+|                                                            | 67.18        | 59.38        | 39.17        | 69.41          | 66.15        | 60.26        |
+|                                                            | <b>72.50</b> | <b>62.29</b> | <b>41.67</b> | <b>70.36</b>   | <b>69.48</b> | <b>63.26</b> |
+
+Empirically, Table 3 shows that, under the same 2000-step budget, the curriculum *Partial-50* Partial-25 learns substantially better than training on *Partial-50* alone. We cap the *Partial-50* stage at 100 steps, after which we switch to *Partial-25*. As shown in Figure 11, entropy for QUESTA-Nemotron-1.5B-50 begins to decline beyond 100 steps, so transitioning at this point prevents overconfidence and sustains training stability. We have also tried extending the curriculum from *Partial-25* to *Partial-0* in our experiments, but observed no gains and no increase in response length (see Figure 12).
+
+<span id="page-8-2"></span>Table 4: Performance comparison (Pass@1, averaged over 32 samples) between Nemotron-1.5B and QUESTA-Nemotron-1.5B (By OpenMathReasoning). The two models achieve comparable results, with the version trained on OpenR1 performing slightly better overall.
+
+| Model                                                                    | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25   Avg |
+|--------------------------------------------------------------------------|--------|--------|-------------|----------------|---------------|
+| Nemotron-1.5B QUESTA-50 (with OpenMathReasoning) QUESTA-50 (with OpenR1) | 61.77  | 49.50  | 31.56       | 64.62          | 58.23   53.14 |
+|                                                                          | 66.46  | 58.54  | 36.35       | 66.06          | 63.13   58.11 |
+|                                                                          | 67.18  | 59.38  | 39.17       | 69.41          | 66.15   60.26 |
+
+Ablation with Different Dataset. We also evaluated QUESTA on OpenMathReasoning Moshkov et al. (2025), selecting the 60K questions with pass\_rate\_72b\_tir of 0 or 1/32. Due to time constraints, we trained only the first stage of QUESTA with 50% partial solutions. Table 4 shows that QUESTA-Nemotron-1.5B-50 achieves similar performance as using the OpenR1 dataset. This indicates that our approach generalizes across datasets.
+
+**Other Ablations.** We also conduct an extensive set of comparative experiments and ablation studies, with detailed results provided in Appendix D. These include an ablation of QUESTA without hints (Appendix D.1), experiments with different model backbones (Appendix D.2), and the full set of pass rates and training curves for additional models (Appendix D.3).
+
+#### 6 CONCLUSIONS
+
+In this work we introduced QUESTA, a lightweight data-centric framework that augments hard prompts with partial-solution hints during RL training. Without altering model architecture or reward design, QUESTA sets new state-of-the-art results for 1.5 B-scale models on AIME24, AIME25 and HMMT25. Further, we theoretically demonstrate how question augmentation can improve sample
+
+{9}------------------------------------------------
+
+efficiency. Our analysis shows that the method can potentially be generalized to other domains such as competitive coding, software engineering or other agentic tasks. Designing proper question augmentation pipelines for theses new tasks can be an important and interesting future direction.
+
+#### 7 ETHICS STATEMENT
+
+We use only public, non-PII datasets—OpenR1-Math-220K (Apache 2.0) and OpenMathReasoning (CC BY 4.0)—in full compliance with their licenses (including attribution and modification notices); no new human-subjects data were collected, no re-identification was attempted, and no IRB review was required. Our augmentation pipeline generates math problems and solutions while avoiding harmful or copyrighted non-math content; outputs may inherit source biases, so we report settings transparently, discourage high-stakes deployment or misuse without safeguards and human oversight, and will release artifacts that respect the original licenses.
+
+#### 8 Reproducibility Statement
+
+To ensure reproducibility, we provide the code, dataset and model in the the supplementary materials and anonymous github https://anonymous.4open.science/r/questa932/README.md. In the README.md file included with the code, we present a step-by-step guide for reproducing our results.
+
+#### 9 ACKNOWLEDGEMENTS AND DISCLOSURE OF FUNDING
+
+J.Z. acknowledges support by the National Key R&D Program of China 2024YFA1015800 and Xiongan AI Institute.
+
+#### REFERENCES
+
+<span id="page-9-5"></span>Brown university math olympiad 2025. February 2025.
+
+<span id="page-9-4"></span>Harvard-mit mathematics tournament 2025. February 2025.
+
+<span id="page-9-0"></span>Chenxin An, Zhihui Xie, Xiaonan Li, Lei Li, Jun Zhang, Shansan Gong, Ming Zhong, Jingjing Xu, Xipeng Qiu, Mingxuan Wang, and Lingpeng Kong. Polaris: A post-training recipe for scaling reinforcement learning on advanced reasoning models, 2025.
+
+<span id="page-9-6"></span>Mislav Balunović, Jasper Dekoninck, Ivo Petrov, Nikola Jovanović, and Martin Vechev. Matharena: Evaluating Ilms on uncontaminated math competitions, February 2025.
+
+<span id="page-9-3"></span>Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, Alex Ray, Raul Puri, Gretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios Chantzis, Elizabeth Barnes, Ariel Herbert-Voss, William Hebgen Guss, Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin, Suchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojciech Zaremba. Evaluating large language models trained on code, 2021.
+
+<span id="page-9-1"></span>Ganqu Cui, Yuchen Zhang, Jiacheng Chen, Lifan Yuan, Zhi Wang, Yuxin Zuo, Haozhan Li, Yuchen Fan, Huayu Chen, Weize Chen, et al. The entropy mechanism of reinforcement learning for reasoning language models. *arXiv preprint arXiv:2505.22617*, 2025.
+
+<span id="page-9-2"></span>Wei Fu, Jiaxuan Gao, Xujie Shen, Chen Zhu, Zhiyu Mei, Chuyi He, Shusheng Xu, Guo Wei, Jun Mei, Jiashu Wang, et al. Areal: A large-scale asynchronous reinforcement learning system for language reasoning. *arXiv preprint arXiv:2505.24298*, 2025.
+
+<span id="page-9-7"></span>Zitian Gao, Lynx Chen, Haoming Luo, Joey Zhou, and Bryan Dai. One-shot entropy minimization, 2025.
+
+{10}------------------------------------------------
+
+- <span id="page-10-3"></span>Gemini Team, Google DeepMind. Gemini 2.5: Pushing the frontier with advanced reasoning, multimodality, long context, and next generation agentic capabilities, 2025. Accessed: July 2025.
+- <span id="page-10-2"></span>Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning. *arXiv preprint arXiv:2501.12948*, 2025.
+- <span id="page-10-13"></span>Chaoqun He, Renjie Luo, Yuzhuo Bai, Shengding Hu, Zhen Leng Thai, Junhao Shen, Jinyi Hu, Xu Han, Yujie Huang, Yuxiang Zhang, Jie Liu, Lei Qi, Zhiyuan Liu, and Maosong Sun. Olympiadbench: A challenging benchmark for promoting agi with olympiad-level bilingual multimodal scientific problems, 2024.
+- <span id="page-10-9"></span>Zhenyu Hou, Ziniu Hu, Yujiang Li, Rui Lu, Jie Tang, and Yuxiao Dong. Treerl: Llm reinforcement learning with on-policy tree search, 2025.
+- <span id="page-10-1"></span>Aaron Jaech, Adam Kalai, Adam Lerer, Adam Richardson, Ahmed El-Kishky, Aiden Low, Alec Helyar, Aleksander Madry, Alex Beutel, Alex Carney, et al. Openai o1 system card. *arXiv preprint arXiv:2412.16720*, 2024.
+- <span id="page-10-15"></span>Naman Jain, King Han, Alex Gu, Wen-Ding Li, Fanjia Yan, Tianjun Zhang, Sida Wang, Armando Solar-Lezama, Koushik Sen, and Ion Stoica. Livecodebench: Holistic and contamination free evaluation of large language models for code, 2024.
+- <span id="page-10-10"></span>Amirhossein Kazemnejad, Milad Aghajohari, Eva Portelance, Alessandro Sordoni, Siva Reddy, Aaron Courville, and Nicolas Le Roux. Vineppo: Refining credit assignment in rl training of llms, 2025.
+- <span id="page-10-5"></span>Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017.
+- <span id="page-10-14"></span>Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal Lago, Thomas Hubert, Peter Choy, Cyprien de Masson d'Autume, Igor Babuschkin, Xinyun Chen, Po-Sen Huang, Johannes Welbl, Sven Gowal, Alexey Cherepanov, James Molloy, Daniel J. Mankowitz, Esme Sutherland Robson, Pushmeet Kohli, Nando de Freitas, Koray Kavukcuoglu, and Oriol Vinyals. Competition-level code generation with alphacode. *Science*, 378(6624):1092–1097, December 2022. ISSN 1095-9203. doi: 10.1126/science.abq1158.
+- <span id="page-10-16"></span>Bill Yuchen Lin, Ronan Le Bras, Kyle Richardson, Ashish Sabharwal, Radha Poovendran, Peter Clark, and Yejin Choi. Zebralogic: On the scaling limits of llms for logical reasoning, 2025.
+- <span id="page-10-0"></span>Mingjie Liu, Shizhe Diao, Ximing Lu, Jian Hu, Xin Dong, Yejin Choi, Jan Kautz, and Yi Dong. Prorl: Prolonged reinforcement learning expands reasoning boundaries in large language models. *arXiv preprint arXiv:2505.24864*, 2025.
+- <span id="page-10-6"></span>Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization, 2019.
+- <span id="page-10-4"></span>Haipeng Luo, Qingfeng Sun, Can Xu, Pu Zhao, Jianguang Lou, Chongyang Tao, Xiubo Geng, Qingwei Lin, Shifeng Chen, Yansong Tang, and Dongmei Zhang. Wizardmath: Empowering mathematical reasoning for large language models via reinforced evol-instruct, 2025a. URL <https://arxiv.org/abs/2308.09583>.
+- <span id="page-10-7"></span>Michael Luo, Sijun Tan, Justin Wong, Xiaoxiang Shi, William Y. Tang, Manan Roongta, Colin Cai, Jeffrey Luo, Li Erran Li, Raluca Ada Popa, and Ion Stoica. Deepscaler: Surpassing o1-preview with a 1.5b model by scaling rl, 2025b. Notion Blog.
+- <span id="page-10-11"></span>MAA. American invitational mathematics examination 2024. February 2024.
+- <span id="page-10-12"></span>MAA. American invitational mathematics examination 2025. February 2025.
+- <span id="page-10-8"></span>Saumya Malik, Valentina Pyatkin, Sander Land, Jacob Morrison, Noah A. Smith, Hannaneh Hajishirzi, and Nathan Lambert. Rewardbench 2: Advancing reward model evaluation, 2025.
+- <span id="page-10-17"></span>Jincheng Mei, Chenjun Xiao, Csaba Szepesvari, and Dale Schuurmans. On the global convergence rates of softmax policy gradient methods, 2022.
+
+{11}------------------------------------------------
+
+- <span id="page-11-5"></span>Ivan Moshkov, Darragh Hanley, Ivan Sorokin, Shubham Toshniwal, Christof Henkel, Benedikt Schifferer, Wei Du, and Igor Gitman. Aimo-2 winning solution: Building state-of-the-art mathematical reasoning models with openmathreasoning dataset. *arXiv preprint arXiv:2504.16891*, 2025.
+- <span id="page-11-1"></span>Open-R1 Team. Openr1-math-220k: A large-scale dataset for mathematical reasoning, 2025.
+- <span id="page-11-13"></span>Shubham Parashar, Shurui Gui, Xiner Li, Hongyi Ling, Sushil Vemuri, Blake Olson, Eric Li, Yu Zhang, James Caverlee, Dileep Kalathil, and Shuiwang Ji. Curriculum reinforcement learning from easy to hard tasks improves llm reasoning. 2025.
+- <span id="page-11-11"></span>Yuxiao Qu, Matthew Y. R. Yang, Amrith Setlur, Lewis Tunstall, Edward Emanuel Beeching, Ruslan Salakhutdinov, and Aviral Kumar. Optimizing test-time compute via meta reinforcement finetuning, 2025.
+- <span id="page-11-15"></span>David Rein, Betty Li Hou, Asa Cooper Stickland, Jackson Petty, Richard Yuanzhe Pang, Julien Dirani, Julian Michael, and Samuel R. Bowman. Gpqa: A graduate-level google-proof q&a benchmark, 2023.
+- <span id="page-11-7"></span>John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms, 2017.
+- <span id="page-11-8"></span>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:2506.10947*, 2025.
+- <span id="page-11-6"></span>Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, YK Li, Y Wu, et al. Deepseekmath: Pushing the limits of mathematical reasoning in open language models. *arXiv preprint arXiv:2402.03300*, 2024.
+- <span id="page-11-9"></span>Peiyi Wang, Lei Li, Zhihong Shao, R. X. Xu, Damai Dai, Yifei Li, Deli Chen, Y. Wu, and Zhifang Sui. Math-shepherd: Verify and reinforce llms step-by-step without human annotations, 2024.
+- <span id="page-11-4"></span>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 llm reasoning. *arXiv preprint arXiv:2506.01939*, 2025a.
+- <span id="page-11-16"></span>Yiping Wang, Qing Yang, Zhiyuan Zeng, Liliang Ren, Liyuan Liu, Baolin Peng, Hao Cheng, Xuehai He, Kuan Wang, Jianfeng Gao, Weizhu Chen, Shuohang Wang, Simon Shaolei Du, and Yelong Shen. Reinforcement learning for reasoning in large language models with one training example, 2025b.
+- <span id="page-11-10"></span>Xumeng Wen, Zihan Liu, Shun Zheng, Zhijian Xu, Shengyu Ye, Zhirong Wu, Xiao Liang, Yang Wang, Junjie Li, Ziming Miao, et al. Reinforcement learning with verifiable rewards implicitly incentivizes correct reasoning in base llms. *arXiv preprint arXiv:2506.14245*, 2025.
+- <span id="page-11-12"></span>Zhiheng Xi, Wenxiang Chen, Boyang Hong, Senjie Jin, Rui Zheng, Wei He, Yiwen Ding, Shichun Liu, Xin Guo, Junzhe Wang, Honglin Guo, Wei Shen, Xiaoran Fan, Yuhao Zhou, Shihan Dou, Xiao Wang, Xinbo Zhang, Peng Sun, Tao Gui, Qi Zhang, and Xuanjing Huang. Training large language models for reasoning through reverse curriculum reinforcement learning, 2024. URL <https://arxiv.org/abs/2402.05808>.
+- <span id="page-11-14"></span>Yuchen Yan, Yongliang Shen, Yang Liu, Jin Jiang, Mengdi Zhang, Jian Shao, and Yueting Zhuang. Inftythink: Breaking the length limits of long-context reasoning in large language models. *arXiv preprint arXiv:2503.06692*, 2025.
+- <span id="page-11-2"></span>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 arXiv:2505.09388*, 2025.
+- <span id="page-11-3"></span>Qiying Yu, Zheng Zhang, Ruofei Zhu, Yufeng Yuan, Xiaochen Zuo, Yu Yue, Weinan Dai, Tiantian Fan, Gaohong Liu, Lingjun Liu, et al. Dapo: An open-source llm reinforcement learning system at scale. *arXiv preprint arXiv:2503.14476*, 2025.
+- <span id="page-11-0"></span>Yang Yue, Zhiqi Chen, Rui Lu, Andrew Zhao, Zhaokai Wang, Shiji Song, and Gao Huang. Does reinforcement learning really incentivize reasoning capacity in llms beyond the base model? *arXiv preprint arXiv:2504.13837*, 2025.
+
+{12}------------------------------------------------
+
+- <span id="page-12-2"></span>Kaiyi Zhang, Ang Lv, Jinpeng Li, Yongbo Wang, Feng Wang, Haoyuan Hu, and Rui Yan. Stephint: Multi-level stepwise hints enhance reinforcement learning to reason. *arXiv preprint arXiv:2507.02841*, 2025. doi: 10.48550/arXiv.2507.02841. Submitted Jul 3, 2025.
+- <span id="page-12-0"></span>Rosie Zhao, Alexandru Meterez, Sham Kakade, Cengiz Pehlevan, Samy Jelassi, and Eran Malach. Echo chamber: Rl post-training amplifies behaviors learned in pretraining, 2025.
+- <span id="page-12-1"></span>Xinyu Zhu, Mengzhou Xia, Zhepei Wei, Wei-Lin Chen, Danqi Chen, and Yu Meng. The surprising effectiveness of negative reinforcement in llm reasoning. *arXiv preprint arXiv:2506.01347*, 2025.
+
+{13}------------------------------------------------
+
+## A THE USE OF LARGE LANGUAGE MODELS (LLMS)
+
+The Large Language Models (LLMs) were exclusively utilized to polish the writing and detect potential typos, with no involvement in other aspects.
+
+#### **B** IMPLEMENTATION DETAILS
+
+#### <span id="page-13-0"></span>**B.1 RLVR ALGORITHMS**
+
+We have employed the GRPO algorithm enhanced with a subset of DAPO techniques. Primarily, we have integrated DAPO's Dynamic Sampling Trick and eliminated the KL divergence term, resulting in an optimization objective that is:
+
+$$\begin{split} \mathcal{J}(\theta) &= & \mathbb{E}_{q \sim \mathcal{D}, \{o_i\}_{i=1}^G \sim \pi_{\theta_{\text{old}}}(\cdot|q)} \\ & \left[ \frac{1}{G} \sum_{i=1}^G \frac{1}{|o_i|} \sum_{t=1}^{|o_i|} \min\left(r_{i,t}(\theta) \hat{A}_{i,t}, \, \text{clip}\Big(r_{i,t}(\theta), 1-\varepsilon, 1+\varepsilon\Big) \hat{A}_{i,t}\Big) \right] \\ & \text{s.t.} \quad 0 < \#\Big\{o_i \mid [o_i \text{ is correct}]\}\Big\} < G, \end{split} \tag{1}$$
+
+where
+
+$$r_{i,t}(\theta) = \frac{\pi_{\theta}(o_{i,t} \mid q, o_{i, < t})}{\pi_{\theta_{\text{old}}}(o_{i,t} \mid q, o_{i, < t})}, \quad \hat{A}_{i,t} = \frac{R_i - \text{mean}(\{R_i\}_{i=1}^G)}{\text{std}(\{R_i\}_{i=1}^G)}.$$
+(2)
+
+Our reward function R mirrors that of DeepScaleR (Luo et al., 2025b), employing an Outcome Reward Model. It returns 1 if and only if both the answer and format are correct; otherwise, it returns 0. In summary, our reward function yields:
+
+$$R = \begin{cases} 1, & \text{if the answer (e.g. passes basic LaTeX/Sympy checks)} \\ & \text{and format (e.g. exists  and ) are both correct,} \\ 0, & \text{otherwise.} \end{cases}$$
+ (3)
+
+#### <span id="page-13-1"></span>B.2 LOW-VARIANCE PASS@K ESTIMATION
+
+Pass@k is a measure of a model's problem - solving ability, indicating the probability that the model can generate at least one correct solution in k attempts. Specifically, for each problem  $x_i$  in the evaluation dataset  $\mathcal{D}$ , we generate n samples (where  $n \geq k$ ) and count the correct ones as  $c_i$ . The direct calculation formula is:
+
+$$\operatorname{pass}@k := \mathbb{E}_{x_i \sim \mathcal{D}} \left[ 1 - \left( 1 - \frac{c_i}{n} \right)^k \right] \tag{4}$$
+
+However, this formula has excessive variance and insufficient accuracy. To solve this problem, we adopt the unbiased estimation method proposed by Chen et al. (Chen et al., 2021), using the unbiased estimator of pass@k over the dataset:
+
+$$\operatorname{pass}@k := \mathbb{E}_{x_i \sim \mathcal{D}} \left[ 1 - \frac{\binom{n - c_i}{k}}{\binom{n}{k}} \right] \tag{5}$$
+
+In our experiments, to ensure sufficient accuracy, we set n such that  $2k \le n$ , which helps further reduce the variance of the estimate.
+
+## B.3 MORE RELATED WORKS
+
+Recent studies show that RL algorithms, such as PPO (Schulman et al., 2017) and GRPO (Guo et al., 2025), can greatly enhance model reasoning capabilities. Building on this, several works have refined this paradigm from different perspectives. One method can be adjusting the reward function. Some studies (Zhu et al., 2025; Shao et al., 2025) directly modify the reward function to improve training efficiency. Other methods introduced intermediate process rewards (Wang et al., 2024; Malik et al., 2025), while Wen et al. (Wen et al., 2025) set up a separate correctness judgment for CoT to obtain rewards.
+
+{14}------------------------------------------------
+
+Another novel perspective aims to improve sample efficiency by measuring certainty. For example, TreeRL (Hou et al., 2025) and VinePPO (Kazemnejad et al., 2025) enhanced sample effects by introducing entropy or confidence. MRT (Qu et al., 2025), on the other hand, reused partial trajectories during testing to boost sample efficiency. R3 (Xi et al., 2024) improves RL sample efficiency by decomposing human solution steps and providing preceding steps to guide the model in completing subsequent ones. Further, some research adopted a multi-stage training or reasoning mode, exploring from different angles such as training length (Luo et al., 2025b), question difficulty (Parashar et al., 2025), and fixed-length summaries during reasoning (Yan et al., 2025).
+
+In addition to designing better algorithms, another line of research (Shao et al., 2024; Yue et al., 2025; Zhao et al., 2025) has investigated how reinforcement learning affects the frontier of model capabilities, observing a decay in pass@k when k becomes large. In response to this phenomenon, some works (Yu et al., 2025; Liu et al., 2025; An et al., 2025) maintained entropy stability by adjusting training entropy through methods such as increasing the clipping upper bound, enlarging the temperature coefficient, extending the training length, and periodically updating the KL reference model. StepHint (Zhang et al., 2025) also preserved entropy stability by leveraging intermediate thinking content of iterative length as a prompting signal.
+
+In contrast to the aforementioned research, our work adopts an orthogonal approach by using part of the ground-truth solution as a hint, without requiring any modifications to the existing reinforcement learning infrastructure. We provide both theoretical justification and empirical evidence that this strategy maintains pass@k without compromising the exploratory capacity of the underlying reinforcement learning algorithm.
+
+#### B.4 BENCHMARKS
+
+We evaluate the models' breadth across various tasks in multiple domains, including mathematics, coding, reasoning, and logical inference. For mathematics, we follow DeepScaleR (Luo et al., 2025b) and Nemotron (Moshkov et al., 2025), and conduct assessments on more challenging mathematical datasets such as AIME2024 (MAA, 2024), AIME2025 (MAA, 2025), Olympiad Bench (He et al., 2024), HMMT FEB 25 (hmm, 2025), and BRUMO25 (bru, 2025). Specifically, HMMT25 Feb and BRUMO25 are both sourced from MathArena (Balunović et al., 2025). In the realm of coding, we utilize commonly employed datasets, including Code Contests (Li et al., 2022), Codeforces<sup>1</sup>, and LCB V5 202410-202502 (Jain et al., 2024). For logical reasoning tasks, we assess our models' capabilities using GPQA Diamond (Rein et al., 2023) <sup>2</sup> and Zebraliogic (Lin et al., 2025). The benchmarks related to coding and logical reasoning are all referenced from AReaL (Fu et al., 2025).
+
+## <span id="page-14-0"></span>B.5 TRAINING DATASET
+
+The dataset employed in our study is OpenR1-Math-220K (Open-R1 Team, 2025). Prior to commencing the training of the Partial Solution, we conducted a preliminary screening of the dataset. Specifically, we utilized the DeepSeek-R1-Distill-1.5B (Guo et al., 2025) model to perform eight inference operations on each of the 220k data entries in the OpenR1 dataset. Subsequently, we compared the annotated answers in the OpenR1 dataset with the results generated from each inference to tally the number of correct instances for each data entry. Ultimately, we selected the data entries with 0 or 1 correct instance as the training samples for our study. The final dataset size is 26K.
+
+For controlled comparisons, we further split this 26K subset by re-sampling **Nemotron-1.5B** eight times per item and counting correct completions. We define *Easy Data* as questions with correct counts in [7,8] and train a model on this split, denoted <code>Easy-Nemotron-1.5B</code>. Similarly, we define *Hard Data* as questions with correct counts in [0,1] and train <code>Hard-Nemotron-1.5B</code> on this split.
+
+Additionally, for the augmented data, we perform eight inference passes using the model currently under training. We then select samples for which the number of correct predictions falls within the range of [0,4]. This criterion is motivated by the finding that samples exhibiting higher variance are more beneficial for training (Gao et al., 2025; Wang et al., 2025b). The range [0,4] is chosen because it includes the point of maximum sample variance, which is achieved with four correct predictions out of eight trials. For convenience, we refer to augmented data with partial ratio p as Partial-p data.
+
+<span id="page-14-2"></span><span id="page-14-1"></span><sup>1</sup>https://codeforces.com/
+
+<sup>&</sup>lt;sup>2</sup>In the GPQA Diamond dataset, multiple-choice questions are presented in the form of options rather than directly providing the answer, requiring the model to output only A, B, C, or D.
+
+{15}------------------------------------------------
+
+## <span id="page-15-0"></span>B.6 THE RATIONALE FOR THE CHOICE OF p
+
+<span id="page-15-2"></span>Table 5: Number of problems vs pass rate under different hint levels on OpenMath-Nemotron-1.5B before training. We evaluated OpenMath-Nemotron-1.5B on the OpenR1 dataset after the first round of filtering, with each problem assessed 8 times. The table illustrates the distribution of correct answers (n) where n ∈ {0, 1, . . . , 8}.
+
+| Hint Levels | 0 / 8 | 1 / 8 | 2 / 8 | 3 / 8 | 4 / 8 | 5 / 8 | 6 / 8 | 7 / 8 | 8 / 8 |
+|-------------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
+| Partial-50  | 143   | 224   | 304   | 472   | 710   | 1013  | 1779  | 3655  | 17741 |
+| Partial-25  | 3155  | 1997  | 1814  | 1785  | 1902  | 2175  | 2614  | 3440  | 7159  |
+| Partial-10  | 3589  | 2090  | 1865  | 1842  | 1905  | 2176  | 2653  | 3415  | 6506  |
+| Partial-0   | 3812  | 2218  | 1854  | 1842  | 2007  | 2136  | 2517  | 3264  | 6391  |
+
+In this study, we evaluated the performance of OpenMath-Nemotron-1.5B on the OpenR1 dataset under various hint levels. The evaluation was performed after the first round of filtering, and each problem was assessed 8 times to capture the predictive distribution. The resulting table (Table [5\)](#page-15-2) shows the distribution of correct answers across different hint levels, where the values represent the number of times the model answered correctly (n ∈ {0, 1, . . . , 8}).
+
+The selection of the hint parameter p was primarily based on these evaluation results. As shown in the table, the performance with a Partial-50 hint significantly reduces task difficulty, as evidenced by the high pass rates across most levels. In contrast, Partial-25 (25% hint) exhibits a performance pattern similar to that of the no-hint scenario (Partial-0), with only marginal differences in task difficulty.
+
+This minimal difference in difficulty between Partial-0 and Partial-25 suggests that training with Partial-25 does not provide substantial gains compared to Partial-0. Consequently, we adopted a stepwise design in which the hint level is first set to p = 50%, followed by p = 25%, to evaluate the model's performance under varying conditions.
+
+## B.7 EVALUATION SETUP
+
+We configured the models to have a maximum generation length of 32,768 tokens. In line with DeepSeek-R1 [\(Guo et al., 2025\)](#page-10-2), we utilized pass@k evaluation [\(Chen et al., 2021\)](#page-9-3), with the formula detailed in [B.2.](#page-13-1) We reported pass@1 using a non-zero temperature. Specifically, we used a sampling temperature of 0.7 and a top-p value of 0.95 to generate k responses per question, typically set at 32, with deviations explicitly noted. *Particular attention should be paid to the fact that, although we incorporated partial Solution during training, it was not included in the evaluation phase.*
+
+## <span id="page-15-1"></span>B.8 DETAIL ON PROMPT TEMPLATE
+
+## DeepScaleR Coding's Inference: <|User|>{input}<|Assistant|><think>
+
+## DeepScaleR Others' Inference:
+
+<|User|>{input}
+
+Please reason step by step, and put your final answer within \boxed{}.<|Assistant|><think>
+
+#### Nemotrion Coding's Inference:
+
+<|im\_start|>user
+
+{input}
+
+<|im\_end|> <|im\_start|>assistant
+
+<think>
+
+## Nemotrion Others' Inference:
+
+<|im\_start|>system
+
+Please reason step by step, and put your final answer within \boxed{}.<|im\_end|> <|im\_start|>user
+
+{16}------------------------------------------------
+
+{input}<lim\_endl> <lim\_startl>assistant
+
+## Training prompt with partial solutions (math RL):
+
+{Problem}
+
+## Hint: {Partial Solution}
+
+Please reason step by step, and put your final answer within \boxed{}.
+
+#### C THEORY
+
+#### C.1 Proofs
+
+**Theorem 4.4** (Lower Bound on RL Learnability under Solution Inaccessibility). Given a probability threshold  $\delta_p > 0$ , if for every question  $q \in \mathcal{Q}$ , the model capacity set  $C(q, \delta_p)$  does not intersect with the solution set  $\mathcal{S}(q)$ , i.e.,
+
+$$C(q, \delta_n) \cap \mathcal{S}(q) = \emptyset, \quad \forall q \in \mathcal{Q},$$
+
+then under Assumption 4.3, when training RL for T steps with B samples per step such that  $TB = \Theta(1/\delta_p)$ , there is a constant probability that the RL algorithm will not update the model.
+
+*Proof.* Let  $p_{\text{sol}} = \sum_{\tau^* \in \mathcal{S}(q)} P_{\mu}(\tau^*|q)$  denote the cumulative generation probability of any solution trajectory. By  $C(q, \delta_p) \cap \mathcal{S}(q) = \emptyset$  and Def 4.2:
+
+$$p_{\text{sol}} = \sum_{\tau^* \in \mathcal{S}(q)} P_{\mu}(\tau^*|q) < \delta_p$$
+
+For N=TB independent samples across T steps with batch size B, the probability of complete failure (no solution sampled) is:
+
+$$\mathbb{P}(\text{failure}) = (1 - p_{\text{sol}})^N > (1 - \delta_p)^N$$
+
+Given  $TB = \Theta(1/\delta_p)$ , we have:
+
+$$(1 - \delta_p)^N > (1 - \delta_p)^{\Theta(1/\delta_p)} = \Theta(1).$$
+
+The last inequality follows from the fact that  $(1-x)^{1/x} > \exp(-1/(1-x))$  for  $x \in (0,1)$ . By Assumption 4.3, if no solution is found, the model weights remain unchanged.
+
+<span id="page-16-0"></span>**Lemma C.1** (Upper Bound on Sampling Budget for Solution Given Hint). Given a question  $q \in \mathcal{Q}$ , if there exists a hint  $h_q$  for the question q (Def. 4.5), then if we perform  $TB = \Theta(1/\delta_p') = \Theta(\delta_p^\epsilon/\sqrt{\delta_p})$  i.i.d sampling over the initial model conditioned on  $(q, h_q)$ , we can find a valid solution with a constant probability.
+
+*Proof.* By Definition 4.5, we know:
+
+1. 
+$$P_{\mu}(h_a|q) \geq \delta'_n$$
+
+2. 
+$$\exists s_q \in \mathcal{S}(q) : P_{\mu}(s_q|(q,h_q)) \geq \delta_p'$$
+
+With  $N = TB \ge 10/\delta'_p$  independent samples conditioned on  $(q, h_q)$ , the probability of not finding the solution  $s_q$  is:
+
+$$\mathbb{P}(\text{no solution}) = (1 - P_{\mu}(s_q|(q, h_q)))^N \le (1 - \delta_p')^{10/\delta_p'} \le \exp(-10) < 0.01.$$
+
+Therefore,  $\mathbb{P}(\text{finding solution}) > 0.99$ .
+
+**Theorem 4.6** (Informal Upper Bound on RL Learnability with Hint). If we have a hint  $h_q$  for every question  $q \in \mathcal{Q}$  (Def. 4.5), then there exists an RL algorithm that can output a policy  $\pi_\theta$  such that  $\mathbb{E}_{q \sim \mathrm{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \pi_\theta(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99$  with  $O(1/\delta_p')$  sampling budget with high probability.
+
+{17}------------------------------------------------
+
+This theorem is a direct corollary of the Theorem 5 regarding the bandit setup in (Mei et al., 2022). Because the setup here is relatively simple, we also present a detailed proof for this special case here. We first formalize our setup as follows:
+
+<span id="page-17-0"></span>**Assumption C.2** (Tabular RL with Hint). We consider the tabular RL setting with softmax policy parameterization. There exists a finite set of possible questions  $\mathcal{Q}$  and a finite set of possible solutions  $\mathcal{S}$ . For each question  $q \in \mathcal{Q}$ , there exists a hint  $h_q$ , which is a subset of solutions  $h_q \subseteq \mathcal{S}$ .
+
+The policy is parameterized by a  $|S| \times |Q|$  matrix  $\theta$  in the following way:
+
+$$\mu_{\theta}(s|q) = \frac{\exp(\theta_{s,q})}{\sum_{s' \in S} \exp(\theta_{s',q})}$$
+
+Here the setup is different than the autoregressive setting in our experiments and simplify the model to a tabular setup for the simplicity of analysis. We now restate the assumption on the existence of hint in this setup.
+
+<span id="page-17-1"></span>**Assumption C.3** (Hint Existence, Formal Version of Definition 4.5). For each question  $q \in \mathcal{Q}$ , there exists a hint  $h_q \subseteq \mathcal{S}$  such that  $\sum_{s \in h_q} P_{\mu}(s|q) \ge \delta'_p$ . Further, there exists a solution  $s_q \in \mathcal{S}$  such that  $P_{\mu}(s_q|q) \ge \delta'_p \sum_{s \in h_q} P_{\mu}(s|q)$ .
+
+**RL Algorithm:** We will first sample  $\Theta(1/\delta'_p)$  action based on the policy  $\mu_\theta$  conditioned on the question q and the hint  $h_q$ . Then we will do a one-step policy gradient update on our policy. Noted that here we can reach high reward within one step because the reward function is deterministic.
+
+**Theorem C.4** (Formal Version of Theorem 4.6). *Under Assumption C.2 and Assumption C.3, running* 1 *steps of policy gradient update with sampling budget*  $\Theta(1/\delta'_{v})$ , *the learned policy achieves:* 
+
+$$\mathbb{E}_{q \sim \text{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \mu_{\theta}(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99$$
+
+with probability 0.99.
+
+*Proof.* First, by Assumption C.3, for any question q, we have:
+
+$$\sum_{s \in h_q} P_{\mu}(s|q) \geq \delta_p \quad \text{and} \quad \exists s_q : P_{\mu}(s_q|q) \geq \delta_p' \sum_{s \in h_q} P_{\mu}(s|q)$$
+
+With sampling budget  $N = \Theta(|\mathcal{Q}|/\delta_p')$ , by Lemma C.1 and the union bound, we will find a solution  $s_q$  for every question q with probability at least 0.99. Suppose the found set of solutions for question q is  $S_q$  and all sampled solutions are  $s^{(1)}, \ldots, s^{(N)}$ . Then because
+
+$$\nabla_{\theta} \log \mu_{\theta}(s|q) = e_s - \sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q)e_{s'}$$
+
+We have the policy gradient being
+
+$$\begin{aligned} \text{PG}_{:,q} &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] \nabla_{\theta} \log \mu_{\theta}(s^{(i)}|q) \\ &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] (e_{s^{(i)}} - \sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q) e_{s'}) \\ &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] e_{s^{(i)}} - \left(\frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q]\right) \left(\sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q) e_{s'}\right). \end{aligned}$$
+
+We can make two simple observations:
+
+- 1. For every  $s \notin S_a$ ,  $PG_{s,a} < 0$ .
+- 2. There exists a  $s^* \in S_q$  such that  $PG_{s^*,q} > 0$ .
+
+Therefore, consider the updated parameters
+
+$$\theta'_{s,q} = \theta_{s,q} + \eta PG_{s,q}$$
+
+If  $\eta$  is large enough, we know that  $\sum_{s \in S_-} P_{\mu_{\theta'}}(s|q) \ge 0.99$ . This completes the proof.
+
+{18}------------------------------------------------
+
+## <span id="page-18-0"></span>D ADDITIONAL EXPERIMENTAL RESULTS
+
+#### <span id="page-18-1"></span>D.1 ABLATION STUDY WITHOUT HINT
+
+<span id="page-18-2"></span>Table 6: Ablation without hint on Nemotron-1.5B: Pass@1 (avg@32) on challenging maths benchmarks. "QUESTA-Nemotron-1.5B w/o hint" trains RL on the same data but removes hints from the prompt, while "w/ hint" uses partial-solution hints during training. With hints, the model improves all benchmarks and achieves a +2.82 average gain over w/o hint (63.26 vs. 60.44), on top of the improvements over the base model. The one using hint requires nearly half the number of steps compared to the one not using hint to achieve the same performance.
+
+| Model                                    | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25 | Avg   |
+|------------------------------------------|--------|--------|-------------|----------------|---------|-------|
+| Nemotron-1.5B                            | 61.77  | 49.50  | 31.56       | 64.62          | 58.23   | 53.14 |
+| QUESTA-Nemotron-1.5B w/o hint (2K step)  | 69.48  | 59.79  | 38.85       | 68.05          | 66.04   | 60.44 |
+| QUESTA-Nemotron-1.5B w/ hint (1.1K step) | 69.27  | 60.00  | 37.92       | 69.72          | 68.33   | 61.05 |
+| QUESTA-Nemotron-1.5B w/ hint (2K step)   | 72.50  | 62.29  | 41.67       | 70.36          | 69.48   | 63.26 |
+
+<span id="page-18-3"></span>![](_page_18_Figure_5.jpeg)
+
+Figure 7: Pass@k comparison on Nemotron-1.5B for RL with vs. without hints. Training with hints consistently dominates across k and avoids the performance drop at larger k seen in standard RL. Hints are used only during training; evaluation uses no hints.
+
+We ablate the role of hints by training RL on Nemotron-1.5B with and without partial-solution hints. Here, QUESTA-Nemotron-1.5B w/o hint denotes RL on the same data and schedule but with the hint removed from the prompt; QUESTA-Nemotron-1.5B w/h hint uses identical settings except that the partial solution is provided as a hint during training. As summarized in Table 6, removing the hint still improves over the base model (average Pass@1:  $53.14 \rightarrow 60.44$ ), but adding the hint yields a
+
+{19}------------------------------------------------
+
+further +2.82 average gain ( $60.44 \rightarrow 63.26$ ), with consistent improvements across all five benchmarks. The one using hint requires nearly half the number of steps compared to the one not using hint to achieve the same performance. Note that hints are used only during training; all evaluations are conducted *without* hints.
+
+Figure 7 compares Pass@k curves. The w/hint model lifts the entire curve across k and avoids the degradation at larger k commonly observed in standard RL, while the w/o hint variant brings smaller gains that taper off as k increases. A possible reason for this phenomenon is that, without hints, extremely difficult problems remain unlearned. Consequently, during reinforcement learning training, the model prioritizes improving performance on problems that have become relatively easier as training progresses. This leads the model to become overly confident, thereby reducing its Pass@k metric. In contrast, when hints are provided, the model still prioritizes learning more difficult problems—this is because such problems can provide effective learning signals.
+
+#### <span id="page-19-0"></span>D.2 ABLATION STUDY WITH DIFFERENT MODELS
+
+<span id="page-19-1"></span>Table 7: Performance comparison on DeepScaleR-1.5B: Pass@1 (avg@32) across maths benchmarks. QUESTA consistently improves all tasks and raises the average by +6.50 points.
+
+| Model                  | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25 | Avg   |
+|------------------------|--------|--------|-------------|----------------|---------|-------|
+| DeepScaleR-1.5B        | 40.42  | 31.35  | 19.27       | 52.97          | 37.40   | 36.28 |
+| QUESTA-DeepScaleR-1.5B | 49.16  | 35.94  | 21.77       | 58.69          | 48.33   | 42.78 |
+
+<span id="page-19-3"></span>Table 8: Performance comparison (Pass@1, averaged over 32 samples) showing the impact of QUESTA across benchmarks in other domains, including general knowledge, logic, and coding tasks. We observe minor cross-domain generalization on all these benchmarks, despite QUESTA being applied exclusively in the maths domain.
+
+| Model                  | GPQA Diamond | Zebralogic | Code Contest All | Codeforces | LCB V5 202410-202502 | Avg   |
+|------------------------|--------------|------------|------------------|------------|----------------------|-------|
+| DeepScaleR-1.5B        | 38.5         | 14.26      | 9.07             | 8.79       | 19.57                | 18.04 |
+| QUESTA-DeepScaleR-1.5B | 39.2         | 14.98      | 10.1             | 8.9        | 20.9                 | 18.82 |
+
+<span id="page-19-2"></span>![](_page_19_Figure_8.jpeg)
+
+Figure 8: Pass@k on DeepScaleR-1.5B: QUESTA raises the entire curve across k and avoids the large-k drop often seen in standard RL. The increasing gap with k indicates improved sample diversity rather than overconfident collapse. No hints are used at evaluation.
+
+We next examine model-family transfer by applying QUESTA to DeepScaleR-1.5B. We train for 750 steps on DeepScaleR-1.5B Stage 2 Luo et al. (2025b) on the QUESTA first stage.
+
+{20}------------------------------------------------
+
+<span id="page-20-1"></span>![](_page_20_Figure_1.jpeg)
+
+Figure 9: We conducted training of the DeepScaleR model employing the QUESTA with a dataset comprising 26,000 questions. The average pass rate was calculated from a sample of 8 instances. The initial graph represents the scenario without the incorporation of partial solutions, while the subsequent graph depicts the situation where partial solutions were included. The application of QUESTA significantly diminishes the incidence of unsolved or partially addressed problems within the training dataset. Concurrently, it has come to our attention that our previous method of data curation was not entirely accurate; in fact, the amount of data providing meaningful training signals is less abundant than anticipated, suggesting the potential for further refinement of the dataset.
+
+As shown in Table 7, QUESTA-DeepScaleR-1.5B improves *every* maths benchmark over the base model, achieved an average improvement of 6%, indicating that the benefits of QUESTA are not tied to a single architecture.
+
+Pass@k behavior mirrors these gains. In Figure 8, QUESTA-DeepScaleR lifts the entire Pass@k curve across k and avoids the degradation at larger k reported in standard RL settings. The widening gap at larger k suggests improved candidate diversity rather than overfitting to a single trajectory, consistent with our general pass@k analysis in Appendix B.2. Complementing this, Figure 9 shows that on the 26K training set (evaluated *without* hints), mass shifts away from the 0/8-1/8 bins toward higher pass rates, reducing unsolved or partially solved cases. Hints are used only during training and are removed at evaluation time.
+
+Beyond maths, Table 8 reports out-of-distribution (OOD) results on general knowledge, logic, and coding. QUESTA-DeepScaleR-1.5B achieves small but consistent gains (Avg: 18.04 $\rightarrow$ 18.82; +0.78), suggesting that the improved reasoning patterns transfer modestly beyond the training domain.
+
+## <span id="page-20-0"></span>D.3 SUPPLEMENTAL EXPERIMENT DETAIL
+
+![](_page_20_Figure_7.jpeg)
+
+Figure 10: Training dynamics of QUESTA-Nemotron-1.5B w/o hint.
+
+{21}------------------------------------------------
+
+<span id="page-21-0"></span>![](_page_21_Figure_1.jpeg)
+
+Figure 11: Training dynamics of QUESTA-Nemotron-1.5B-50.
+
+<span id="page-21-1"></span>![](_page_21_Figure_3.jpeg)
+
+Figure 12: Training dynamics of QUESTA-Nemotron-1.5B-50-25-0.
+
+![](_page_21_Figure_5.jpeg)
+
+Figure 13: Training dynamics of Easy-Nemotron-1.5B.
+
+![](_page_21_Figure_7.jpeg)
+
+Figure 14: Training dynamics of Hard-Nemotron-1.5B.
+
+{22}------------------------------------------------
+
+![](_page_22_Figure_1.jpeg)
+
+Figure 15: The training dynamics of QUESTA-DeepScaleR-1.5B. The first and second charts show the changes in the average reward and average inference length of the rollout samples that include all incorrect/correct ones, respectively. The third chart shows the average entropy excluding all incorrect/correct rollout samples.
+
+![](_page_22_Figure_3.jpeg)
+
+Figure 16: Training dynamics of QUESTA-Nemotron-1.5B on OpenMathReasoning (Moshkov et al., 2025). The first and second charts show the changes in the average reward and average inference length of the rollout samples that include all incorrect/correct ones, respectively. The third chart shows the average entropy excluding all incorrect/correct rollout samples. Dynamics closely mirror those on OpenR1-Math-220K, with no entropy collapse.
+
+{23}------------------------------------------------
+
+![](_page_23_Figure_1.jpeg)
+
+Figure 17: We compare pass@k curves of RLVR-trained models with and without QUESTA.
+
+{24}------------------------------------------------
+
+![](_page_24_Figure_1.jpeg)
+
+Figure 18: Comparison of RL training dynamics: Training with only hard problems (green) makes progress very slowly due to sparse rewards, while our method with partial solutions (orange) accelerates learning and consistently achieves higher accuracy across training steps.

iclr26/3MifB0f7qR/parse_report.json

@@ -0,0 +1,77 @@
+{
+  "paper_id": "3MifB0f7qR",
+  "pipeline": "marker_non_llm_v3",
+  "parser": "marker_single",
+  "formats": [
+    "markdown",
+    "chunks"
+  ],
+  "llm_enabled": false,
+  "pdf_path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/iclr26_20260429_1952_all/raw/3MifB0f7qR.pdf",
+  "pdf_sha256": "6345c44f35aac9e12b38a433d5abfddedda7eb28ca4536cb9c84becf98877dd9",
+  "bytes": 913916,
+  "source": "https://openreview.net/pdf?id=3MifB0f7qR",
+  "page_count": 25,
+  "ok": true,
+  "elapsed_seconds": 207.63,
+  "paper2markdown_v3": {
+    "paper_id": "3MifB0f7qR",
+    "pipeline": "Paper2Markdown-V3",
+    "ok": true,
+    "page_count": 25,
+    "chunk_count": 187,
+    "main_body_chunk_count": 77,
+    "appendix_chunk_count": 100,
+    "reference_chunk_count": 10,
+    "model_text_chars": 36398,
+    "raw_markdown_chars": 76702,
+    "sanitized_chars": 72894,
+    "page_provenance": {
+      "min_page": 1,
+      "max_page": 25,
+      "invalid_count": 0
+    },
+    "marker_block_type_counts": {
+      "Caption": 1,
+      "Equation": 24,
+      "FigureGroup": 17,
+      "Footnote": 6,
+      "ListGroup": 7,
+      "PageFooter": 25,
+      "PageHeader": 25,
+      "Picture": 1,
+      "SectionHeader": 38,
+      "Table": 2,
+      "TableGroup": 6,
+      "Text": 134
+    },
+    "asset_count_raw": 18,
+    "asset_count_model_kept": 17,
+    "asset_count_rejected": 1,
+    "asset_reject_reasons": {
+      "kept": 17,
+      "too_small": 1
+    },
+    "artifact_leak_audit": {
+      "ok": true,
+      "hits": {
+        "Anonymous Authors": [],
+        "ACKNOWLEDGMENT": [],
+        "OpenReview": [],
+        "\"accept_label\"": [],
+        "\"decision\"": [],
+        "\"decision_tier\"": [],
+        "\"source_status\"": [],
+        "Meta-review": [],
+        "Official Review": [],
+        "official_reviews": [],
+        "meta_reviews": [],
+        "suggested_verdict_score": []
+      },
+      "artifact_count": 2
+    },
+    "default_model_input": "model_text_v3.txt",
+    "appendix_input": "appendix_text_v3.txt",
+    "reference_input": "reference_text_v3.txt"
+  }
+}

iclr26/3MifB0f7qR/reference_chunks.jsonl

@@ -0,0 +1,10 @@
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0077", "section": "REFERENCES", "page_start": 10, "page_end": 10, "type": "Text", "text": "Brown university math olympiad 2025. February 2025.", "source": "marker_v2", "marker_block_id": "/page/9/Text/9"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0078", "section": "REFERENCES", "page_start": 10, "page_end": 10, "type": "Text", "text": "Harvard-mit mathematics tournament 2025. February 2025.", "source": "marker_v2", "marker_block_id": "/page/9/Text/10"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0079", "section": "REFERENCES", "page_start": 10, "page_end": 10, "type": "Text", "text": "Chenxin An, Zhihui Xie, Xiaonan Li, Lei Li, Jun Zhang, Shansan Gong, Ming Zhong, Jingjing Xu, Xipeng Qiu, Mingxuan Wang, and Lingpeng Kong. Polaris: A post-training recipe for scaling reinforcement learning on advanced reasoning models, 2025.", "source": "marker_v2", "marker_block_id": "/page/9/Text/11"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0080", "section": "REFERENCES", "page_start": 10, "page_end": 10, "type": "Text", "text": "Mislav Balunović, Jasper Dekoninck, Ivo Petrov, Nikola Jovanović, and Martin Vechev. Matharena: Evaluating Ilms on uncontaminated math competitions, February 2025.", "source": "marker_v2", "marker_block_id": "/page/9/Text/12"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0081", "section": "REFERENCES", "page_start": 10, "page_end": 10, "type": "Text", "text": "Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, Alex Ray, Raul Puri, Gretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios Chantzis, Elizabeth Barnes, Ariel Herbert-Voss, William Hebgen Guss, Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin, Suchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojciech Zaremba. Evaluating large language models trained on code, 2021.", "source": "marker_v2", "marker_block_id": "/page/9/Text/13"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0082", "section": "REFERENCES", "page_start": 10, "page_end": 10, "type": "Text", "text": "Ganqu Cui, Yuchen Zhang, Jiacheng Chen, Lifan Yuan, Zhi Wang, Yuxin Zuo, Haozhan Li, Yuchen Fan, Huayu Chen, Weize Chen, et al. The entropy mechanism of reinforcement learning for reasoning language models. arXiv preprint arXiv:2505.22617 , 2025.", "source": "marker_v2", "marker_block_id": "/page/9/Text/14"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0083", "section": "REFERENCES", "page_start": 10, "page_end": 10, "type": "Text", "text": "Wei Fu, Jiaxuan Gao, Xujie Shen, Chen Zhu, Zhiyu Mei, Chuyi He, Shusheng Xu, Guo Wei, Jun Mei, Jiashu Wang, et al. Areal: A large-scale asynchronous reinforcement learning system for language reasoning. arXiv preprint arXiv:2505.24298 , 2025.", "source": "marker_v2", "marker_block_id": "/page/9/Text/15"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0084", "section": "REFERENCES", "page_start": 10, "page_end": 10, "type": "Text", "text": "Zitian Gao, Lynx Chen, Haoming Luo, Joey Zhou, and Bryan Dai. One-shot entropy minimization, 2025.", "source": "marker_v2", "marker_block_id": "/page/9/Text/16"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0085", "section": "REFERENCES", "page_start": 11, "page_end": 11, "type": "ListGroup", "text": "Gemini Team, Google DeepMind. Gemini 2.5: Pushing the frontier with advanced reasoning, multimodality, long context, and next generation agentic capabilities, 2025. Accessed: July 2025. Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning. arXiv preprint arXiv:2501.12948 , 2025. Chaoqun He, Renjie Luo, Yuzhuo Bai, Shengding Hu, Zhen Leng Thai, Junhao Shen, Jinyi Hu, Xu Han, Yujie Huang, Yuxiang Zhang, Jie Liu, Lei Qi, Zhiyuan Liu, and Maosong Sun. Olympiadbench: A challenging benchmark for promoting agi with olympiad-level bilingual multimodal scientific problems, 2024. Zhenyu Hou, Ziniu Hu, Yujiang Li, Rui Lu, Jie Tang, and Yuxiao Dong. Treerl: Llm reinforcement learning with on-policy tree search, 2025. Aaron Jaech, Adam Kalai, Adam Lerer, Adam Richardson, Ahmed El-Kishky, Aiden Low, Alec Helyar, Aleksander Madry, Alex Beutel, Alex Carney, et al. Openai o1 system card. arXiv preprint arXiv:2412.16720 , 2024. Naman Jain, King Han, Alex Gu, Wen-Ding Li, Fanjia Yan, Tianjun Zhang, Sida Wang, Armando Solar-Lezama, Koushik Sen, and Ion Stoica. Livecodebench: Holistic and contamination free evaluation of large language models for code, 2024. Amirhossein Kazemnejad, Milad Aghajohari, Eva Portelance, Alessandro Sordoni, Siva Reddy, Aaron Courville, and Nicolas Le Roux. Vineppo: Refining credit assignment in rl training of llms, 2025. Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017. Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal Lago, Thomas Hubert, Peter Choy, Cyprien de Masson d'Autume, Igor Babuschkin, Xinyun Chen, Po-Sen Huang, Johannes Welbl, Sven Gowal, Alexey Cherepanov, James Molloy, Daniel J. Mankowitz, Esme Sutherland Robson, Pushmeet Kohli, Nando de Freitas, Koray Kavukcuoglu, and Oriol Vinyals. Competition-level code generation with alphacode. Science , 378(6624):1092–1097, December 2022. ISSN 1095-9203. doi: 10.1126/science.abq1158. Bill Yuchen Lin, Ronan Le Bras, Kyle Richardson, Ashish Sabharwal, Radha Poovendran, Peter Clark, and Yejin Choi. Zebralogic: On the scaling limits of llms for logical reasoning, 2025. Mingjie Liu, Shizhe Diao, Ximing Lu, Jian Hu, Xin Dong, Yejin Choi, Jan Kautz, and Yi Dong. Prorl: Prolonged reinforcement learning expands reasoning boundaries in large language models. arXiv preprint arXiv:2505.24864 , 2025. Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization, 2019. Haipeng Luo, Qingfeng Sun, Can Xu, Pu Zhao, Jianguang Lou, Chongyang Tao, Xiubo Geng, Qingwei Lin, Shifeng Chen, Yansong Tang, and Dongmei Zhang. Wizardmath: Empowering mathematical reasoning for large language models via reinforced evol-instruct, 2025a. URL . Michael Luo, Sijun Tan, Justin Wong, Xiaoxiang Shi, William Y. Tang, Manan Roongta, Colin Cai, Jeffrey Luo, Li Erran Li, Raluca Ada Popa, and Ion Stoica. Deepscaler: Surpassing o1-preview with a 1.5b model by scaling rl, 2025b. Notion Blog. MAA. American invitational mathematics examination 2024. February 2024. MAA. American invitational mathematics examination 2025. February 2025. Saumya Malik, Valentina Pyatkin, Sander Land, Jacob Morrison, Noah A. Smith, Hannaneh Hajishirzi, and Nathan Lambert. Rewardbench 2: Advancing reward model evaluation, 2025. Jincheng Mei, Chenjun Xiao, Csaba Szepesvari, and Dale Schuurmans. On the global convergence rates of softmax policy gradient methods, 2022.", "source": "marker_v2", "marker_block_id": "/page/10/ListGroup/175"}
+{"paper_id": "3MifB0f7qR", "chunk_id": "3MifB0f7qR:0086", "section": "REFERENCES", "page_start": 12, "page_end": 12, "type": "ListGroup", "text": "Ivan Moshkov, Darragh Hanley, Ivan Sorokin, Shubham Toshniwal, Christof Henkel, Benedikt Schifferer, Wei Du, and Igor Gitman. Aimo-2 winning solution: Building state-of-the-art mathematical reasoning models with openmathreasoning dataset. arXiv preprint arXiv:2504.16891 , 2025. Open-R1 Team. Openr1-math-220k: A large-scale dataset for mathematical reasoning, 2025. Shubham Parashar, Shurui Gui, Xiner Li, Hongyi Ling, Sushil Vemuri, Blake Olson, Eric Li, Yu Zhang, James Caverlee, Dileep Kalathil, and Shuiwang Ji. Curriculum reinforcement learning from easy to hard tasks improves llm reasoning. 2025. Yuxiao Qu, Matthew Y. R. Yang, Amrith Setlur, Lewis Tunstall, Edward Emanuel Beeching, Ruslan Salakhutdinov, and Aviral Kumar. Optimizing test-time compute via meta reinforcement finetuning, 2025. David Rein, Betty Li Hou, Asa Cooper Stickland, Jackson Petty, Richard Yuanzhe Pang, Julien Dirani, Julian Michael, and Samuel R. Bowman. Gpqa: A graduate-level google-proof q&a benchmark, 2023. John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms, 2017. 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:2506.10947 , 2025. Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, YK Li, Y Wu, et al. Deepseekmath: Pushing the limits of mathematical reasoning in open language models. arXiv preprint arXiv:2402.03300 , 2024. Peiyi Wang, Lei Li, Zhihong Shao, R. X. Xu, Damai Dai, Yifei Li, Deli Chen, Y. Wu, and Zhifang Sui. Math-shepherd: Verify and reinforce llms step-by-step without human annotations, 2024. 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 llm reasoning. arXiv preprint arXiv:2506.01939 , 2025a. Yiping Wang, Qing Yang, Zhiyuan Zeng, Liliang Ren, Liyuan Liu, Baolin Peng, Hao Cheng, Xuehai He, Kuan Wang, Jianfeng Gao, Weizhu Chen, Shuohang Wang, Simon Shaolei Du, and Yelong Shen. Reinforcement learning for reasoning in large language models with one training example, 2025b. Xumeng Wen, Zihan Liu, Shun Zheng, Zhijian Xu, Shengyu Ye, Zhirong Wu, Xiao Liang, Yang Wang, Junjie Li, Ziming Miao, et al. Reinforcement learning with verifiable rewards implicitly incentivizes correct reasoning in base llms. arXiv preprint arXiv:2506.14245 , 2025. Zhiheng Xi, Wenxiang Chen, Boyang Hong, Senjie Jin, Rui Zheng, Wei He, Yiwen Ding, Shichun Liu, Xin Guo, Junzhe Wang, Honglin Guo, Wei Shen, Xiaoran Fan, Yuhao Zhou, Shihan Dou, Xiao Wang, Xinbo Zhang, Peng Sun, Tao Gui, Qi Zhang, and Xuanjing Huang. Training large language models for reasoning through reverse curriculum reinforcement learning, 2024. URL . Yuchen Yan, Yongliang Shen, Yang Liu, Jin Jiang, Mengdi Zhang, Jian Shao, and Yueting Zhuang. Inftythink: Breaking the length limits of long-context reasoning in large language models. arXiv preprint arXiv:2503.06692 , 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 arXiv:2505.09388 , 2025. Qiying Yu, Zheng Zhang, Ruofei Zhu, Yufeng Yuan, Xiaochen Zuo, Yu Yue, Weinan Dai, Tiantian Fan, Gaohong Liu, Lingjun Liu, et al. Dapo: An open-source llm reinforcement learning system at scale. arXiv preprint arXiv:2503.14476 , 2025. Yang Yue, Zhiqi Chen, Rui Lu, Andrew Zhao, Zhaokai Wang, Shiji Song, and Gao Huang. Does reinforcement learning really incentivize reasoning capacity in llms beyond the base model? arXiv preprint arXiv:2504.13837 , 2025.", "source": "marker_v2", "marker_block_id": "/page/11/ListGroup/200"}

iclr26/3MifB0f7qR/reference_text_v3.txt

@@ -0,0 +1,29 @@
+[p. 10 | section: REFERENCES | type: Text]
+Brown university math olympiad 2025. February 2025.
+
+[p. 10 | section: REFERENCES | type: Text]
+Harvard-mit mathematics tournament 2025. February 2025.
+
+[p. 10 | section: REFERENCES | type: Text]
+Chenxin An, Zhihui Xie, Xiaonan Li, Lei Li, Jun Zhang, Shansan Gong, Ming Zhong, Jingjing Xu, Xipeng Qiu, Mingxuan Wang, and Lingpeng Kong. Polaris: A post-training recipe for scaling reinforcement learning on advanced reasoning models, 2025.
+
+[p. 10 | section: REFERENCES | type: Text]
+Mislav Balunović, Jasper Dekoninck, Ivo Petrov, Nikola Jovanović, and Martin Vechev. Matharena: Evaluating Ilms on uncontaminated math competitions, February 2025.
+
+[p. 10 | section: REFERENCES | type: Text]
+Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, Alex Ray, Raul Puri, Gretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios Chantzis, Elizabeth Barnes, Ariel Herbert-Voss, William Hebgen Guss, Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin, Suchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojciech Zaremba. Evaluating large language models trained on code, 2021.
+
+[p. 10 | section: REFERENCES | type: Text]
+Ganqu Cui, Yuchen Zhang, Jiacheng Chen, Lifan Yuan, Zhi Wang, Yuxin Zuo, Haozhan Li, Yuchen Fan, Huayu Chen, Weize Chen, et al. The entropy mechanism of reinforcement learning for reasoning language models. arXiv preprint arXiv:2505.22617 , 2025.
+
+[p. 10 | section: REFERENCES | type: Text]
+Wei Fu, Jiaxuan Gao, Xujie Shen, Chen Zhu, Zhiyu Mei, Chuyi He, Shusheng Xu, Guo Wei, Jun Mei, Jiashu Wang, et al. Areal: A large-scale asynchronous reinforcement learning system for language reasoning. arXiv preprint arXiv:2505.24298 , 2025.
+
+[p. 10 | section: REFERENCES | type: Text]
+Zitian Gao, Lynx Chen, Haoming Luo, Joey Zhou, and Bryan Dai. One-shot entropy minimization, 2025.
+
+[p. 11 | section: REFERENCES | type: ListGroup]
+Gemini Team, Google DeepMind. Gemini 2.5: Pushing the frontier with advanced reasoning, multimodality, long context, and next generation agentic capabilities, 2025. Accessed: July 2025. Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning. arXiv preprint arXiv:2501.12948 , 2025. Chaoqun He, Renjie Luo, Yuzhuo Bai, Shengding Hu, Zhen Leng Thai, Junhao Shen, Jinyi Hu, Xu Han, Yujie Huang, Yuxiang Zhang, Jie Liu, Lei Qi, Zhiyuan Liu, and Maosong Sun. Olympiadbench: A challenging benchmark for promoting agi with olympiad-level bilingual multimodal scientific problems, 2024. Zhenyu Hou, Ziniu Hu, Yujiang Li, Rui Lu, Jie Tang, and Yuxiao Dong. Treerl: Llm reinforcement learning with on-policy tree search, 2025. Aaron Jaech, Adam Kalai, Adam Lerer, Adam Richardson, Ahmed El-Kishky, Aiden Low, Alec Helyar, Aleksander Madry, Alex Beutel, Alex Carney, et al. Openai o1 system card. arXiv preprint arXiv:2412.16720 , 2024. Naman Jain, King Han, Alex Gu, Wen-Ding Li, Fanjia Yan, Tianjun Zhang, Sida Wang, Armando Solar-Lezama, Koushik Sen, and Ion Stoica. Livecodebench: Holistic and contamination free evaluation of large language models for code, 2024. Amirhossein Kazemnejad, Milad Aghajohari, Eva Portelance, Alessandro Sordoni, Siva Reddy, Aaron Courville, and Nicolas Le Roux. Vineppo: Refining credit assignment in rl training of llms, 2025. Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017. Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal Lago, Thomas Hubert, Peter Choy, Cyprien de Masson d'Autume, Igor Babuschkin, Xinyun Chen, Po-Sen Huang, Johannes Welbl, Sven Gowal, Alexey Cherepanov, James Molloy, Daniel J. Mankowitz, Esme Sutherland Robson, Pushmeet Kohli, Nando de Freitas, Koray Kavukcuoglu, and Oriol Vinyals. Competition-level code generation with alphacode. Science , 378(6624):1092–1097, December 2022. ISSN 1095-9203. doi: 10.1126/science.abq1158. Bill Yuchen Lin, Ronan Le Bras, Kyle Richardson, Ashish Sabharwal, Radha Poovendran, Peter Clark, and Yejin Choi. Zebralogic: On the scaling limits of llms for logical reasoning, 2025. Mingjie Liu, Shizhe Diao, Ximing Lu, Jian Hu, Xin Dong, Yejin Choi, Jan Kautz, and Yi Dong. Prorl: Prolonged reinforcement learning expands reasoning boundaries in large language models. arXiv preprint arXiv:2505.24864 , 2025. Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization, 2019. Haipeng Luo, Qingfeng Sun, Can Xu, Pu Zhao, Jianguang Lou, Chongyang Tao, Xiubo Geng, Qingwei Lin, Shifeng Chen, Yansong Tang, and Dongmei Zhang. Wizardmath: Empowering mathematical reasoning for large language models via reinforced evol-instruct, 2025a. URL . Michael Luo, Sijun Tan, Justin Wong, Xiaoxiang Shi, William Y. Tang, Manan Roongta, Colin Cai, Jeffrey Luo, Li Erran Li, Raluca Ada Popa, and Ion Stoica. Deepscaler: Surpassing o1-preview with a 1.5b model by scaling rl, 2025b. Notion Blog. MAA. American invitational mathematics examination 2024. February 2024. MAA. American invitational mathematics examination 2025. February 2025. Saumya Malik, Valentina Pyatkin, Sander Land, Jacob Morrison, Noah A. Smith, Hannaneh Hajishirzi, and Nathan Lambert. Rewardbench 2: Advancing reward model evaluation, 2025. Jincheng Mei, Chenjun Xiao, Csaba Szepesvari, and Dale Schuurmans. On the global convergence rates of softmax policy gradient methods, 2022.
+
+[p. 12 | section: REFERENCES | type: ListGroup]
+Ivan Moshkov, Darragh Hanley, Ivan Sorokin, Shubham Toshniwal, Christof Henkel, Benedikt Schifferer, Wei Du, and Igor Gitman. Aimo-2 winning solution: Building state-of-the-art mathematical reasoning models with openmathreasoning dataset. arXiv preprint arXiv:2504.16891 , 2025. Open-R1 Team. Openr1-math-220k: A large-scale dataset for mathematical reasoning, 2025. Shubham Parashar, Shurui Gui, Xiner Li, Hongyi Ling, Sushil Vemuri, Blake Olson, Eric Li, Yu Zhang, James Caverlee, Dileep Kalathil, and Shuiwang Ji. Curriculum reinforcement learning from easy to hard tasks improves llm reasoning. 2025. Yuxiao Qu, Matthew Y. R. Yang, Amrith Setlur, Lewis Tunstall, Edward Emanuel Beeching, Ruslan Salakhutdinov, and Aviral Kumar. Optimizing test-time compute via meta reinforcement finetuning, 2025. David Rein, Betty Li Hou, Asa Cooper Stickland, Jackson Petty, Richard Yuanzhe Pang, Julien Dirani, Julian Michael, and Samuel R. Bowman. Gpqa: A graduate-level google-proof q&a benchmark, 2023. John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms, 2017. 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:2506.10947 , 2025. Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, YK Li, Y Wu, et al. Deepseekmath: Pushing the limits of mathematical reasoning in open language models. arXiv preprint arXiv:2402.03300 , 2024. Peiyi Wang, Lei Li, Zhihong Shao, R. X. Xu, Damai Dai, Yifei Li, Deli Chen, Y. Wu, and Zhifang Sui. Math-shepherd: Verify and reinforce llms step-by-step without human annotations, 2024. 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 llm reasoning. arXiv preprint arXiv:2506.01939 , 2025a. Yiping Wang, Qing Yang, Zhiyuan Zeng, Liliang Ren, Liyuan Liu, Baolin Peng, Hao Cheng, Xuehai He, Kuan Wang, Jianfeng Gao, Weizhu Chen, Shuohang Wang, Simon Shaolei Du, and Yelong Shen. Reinforcement learning for reasoning in large language models with one training example, 2025b. Xumeng Wen, Zihan Liu, Shun Zheng, Zhijian Xu, Shengyu Ye, Zhirong Wu, Xiao Liang, Yang Wang, Junjie Li, Ziming Miao, et al. Reinforcement learning with verifiable rewards implicitly incentivizes correct reasoning in base llms. arXiv preprint arXiv:2506.14245 , 2025. Zhiheng Xi, Wenxiang Chen, Boyang Hong, Senjie Jin, Rui Zheng, Wei He, Yiwen Ding, Shichun Liu, Xin Guo, Junzhe Wang, Honglin Guo, Wei Shen, Xiaoran Fan, Yuhao Zhou, Shihan Dou, Xiao Wang, Xinbo Zhang, Peng Sun, Tao Gui, Qi Zhang, and Xuanjing Huang. Training large language models for reasoning through reverse curriculum reinforcement learning, 2024. URL . Yuchen Yan, Yongliang Shen, Yang Liu, Jin Jiang, Mengdi Zhang, Jian Shao, and Yueting Zhuang. Inftythink: Breaking the length limits of long-context reasoning in large language models. arXiv preprint arXiv:2503.06692 , 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 arXiv:2505.09388 , 2025. Qiying Yu, Zheng Zhang, Ruofei Zhu, Yufeng Yuan, Xiaochen Zuo, Yu Yue, Weinan Dai, Tiantian Fan, Gaohong Liu, Lingjun Liu, et al. Dapo: An open-source llm reinforcement learning system at scale. arXiv preprint arXiv:2503.14476 , 2025. Yang Yue, Zhiqi Chen, Rui Lu, Andrew Zhao, Zhaokai Wang, Shiji Song, and Gao Huang. Does reinforcement learning really incentivize reasoning capacity in llms beyond the base model? arXiv preprint arXiv:2504.13837 , 2025.

iclr26/3MifB0f7qR/sanitization_report.json

@@ -0,0 +1,60 @@
+{
+  "paper_id": "3MifB0f7qR",
+  "pipeline": "Paper2Markdown-V3",
+  "ok": true,
+  "page_count": 25,
+  "chunk_count": 187,
+  "main_body_chunk_count": 77,
+  "appendix_chunk_count": 100,
+  "reference_chunk_count": 10,
+  "model_text_chars": 36398,
+  "raw_markdown_chars": 76702,
+  "sanitized_chars": 72894,
+  "page_provenance": {
+    "min_page": 1,
+    "max_page": 25,
+    "invalid_count": 0
+  },
+  "marker_block_type_counts": {
+    "Caption": 1,
+    "Equation": 24,
+    "FigureGroup": 17,
+    "Footnote": 6,
+    "ListGroup": 7,
+    "PageFooter": 25,
+    "PageHeader": 25,
+    "Picture": 1,
+    "SectionHeader": 38,
+    "Table": 2,
+    "TableGroup": 6,
+    "Text": 134
+  },
+  "asset_count_raw": 18,
+  "asset_count_model_kept": 17,
+  "asset_count_rejected": 1,
+  "asset_reject_reasons": {
+    "kept": 17,
+    "too_small": 1
+  },
+  "artifact_leak_audit": {
+    "ok": true,
+    "hits": {
+      "Anonymous Authors": [],
+      "ACKNOWLEDGMENT": [],
+      "OpenReview": [],
+      "\"accept_label\"": [],
+      "\"decision\"": [],
+      "\"decision_tier\"": [],
+      "\"source_status\"": [],
+      "Meta-review": [],
+      "Official Review": [],
+      "official_reviews": [],
+      "meta_reviews": [],
+      "suggested_verdict_score": []
+    },
+    "artifact_count": 2
+  },
+  "default_model_input": "model_text_v3.txt",
+  "appendix_input": "appendix_text_v3.txt",
+  "reference_input": "reference_text_v3.txt"
+}

iclr26/3MifB0f7qR/sanitized_v3.txt

@@ -0,0 +1,350 @@
+{0}
+## **ABSTRACT**
+Reinforcement learning (RL) has emerged as a central paradigm for training large language models (LLMs) in reasoning tasks. Yet recent studies (Yue et al., 2025; Liu et al., 2025) question RL's ability to incentivize reasoning capacity beyond the base model. This raises a key challenge: how can RL be adapted to solve harder reasoning problems more effectively? To address this challenge, we propose a simple yet effective strategy via Question Augmentation: introduce partial solutions during training to reduce problem difficulty and provide more informative learning signals. Our method, QuestA, when applied during RL training on math reasoning tasks, not only improves pass@1 but also pass@k—particularly on problems where standard RL struggles to make progress. This enables continual improvement over strong open-source models such as DEEPSCALER and OPENMATH NEMOTRON, further enhancing their reasoning capabilities. We achieve new state-of-the-art results on math benchmarks using 1.5B-parameter models: 72.50% (+10.73%) on AIME24, 62.29% (+12.79%) on AIME25, and 41.67% (+10.11%) on HMMT25. Code, data and model are available at
+![](_page_0_Figure_6.jpeg)
+Figure 1: QUESTA is a data augmentation method that injects partial solutions to effectively scaffold RL training on hard reasoning problems. We construct 26K high-quality augmented prompts from challenging instances in OpenR1 (Open-R1 Team, 2025), and fine-tune models using 32K-context-length RL. When applied to Nemotron-1.5B, QUESTA delivers substantial performance gains—achieving new state-of-the-art results across all math benchmarks for 1.5B-parameter models.
+#### 1 Introduction
+Frontier large language models (LLMs), including OpenAI-O1, O3 (Jaech et al., 2024), DeepSeek-R1 (Guo et al., 2025), Qwen3 (Yang et al., 2025), and Gemini 2.5 (Gemini Team, Google DeepMind,
+<sup>&</sup>lt;sup>‡</sup>This work is independent of and outside of the work at Amazon.
+{1}------------------------------------------------
+<span id="page-1-0"></span>![](_page_1_Figure_1.jpeg)
+Figure 2: We compare pass@k curves of RLVR-trained models, with and without QUESTA. As a controlled experiment, we perform RL training using either easy or hard prompts. Standard RL on easy prompts (red) shows clear degradation in pass@k as k increases compared to the base model (blue). Training on hard prompts (green) improves pass@k, but comes at the cost of substantially longer training. This motivates our development of QUESTA, which scaffolds hard problems to improve training efficiency and delivers consistently stronger results: the RL+QUESTA model (orange) stays above standard RL (red) across all k, while also preserving or improving performance at larger k relative to RL trained with hard prompts.
+2025), have exhibited exceptional performance on high-complexity reasoning tasks spanning mathematics, programming, and formal logic. Recent advances in the field have increasingly prioritized reinforcement learning paradigms (RL), among which *Reinforcement Learning with Verifiable Rewards* (RLVR) has emerged as a scalable and efficient approach to enhancing reasoning capabilities. Using automatically verifiable signals, RLVR enables alignment between model output and objective correctness, thus addressing a critical limitation of traditional RL for reasoning.
+However, the community remains divided on a fundamental question regarding RLVR: does it expand the model's intrinsic reasoning capacity, or merely exploit pre-existing knowledge encoded in the base model? Recent research (Yue et al., 2025; Liu et al., 2025; Zhao et al., 2025) show that while state-of-the-art RL methods (e.g., GRPO, DAPO) (Guo et al., 2025; Yu et al., 2025; An et al., 2025) can enhance the pass@1 metric by reinforcing high-reward completions, they encounter significant limitations when tackling high-difficulty tasks where the base model performs poorly. This phenomenon differs from that observed in Supervised Fine-Tuning (SFT) Luo et al. (2025a). Within the SFT paradigm, enhancing the diversity of problem difficulty serves as a critical factor, as it can effectively improve the model's performance on downstream tasks. However, in the framework of RLVR, the inclusion of easy prompts tends to undermine the model's inherent reasoning capabilities.
+One insightful explanation (Cui et al., 2025; Wang et al., 2025a) for the drop suggests that model overfits on correct solutions and hence causes entropy collapse, limiting its ability to explore. To validate this, we design a controlled setup that separates prompts into easy and hard groups. When applying RLVR on the Nemotron 1.5B model (Moshkov et al., 2025) with the OpenR1 dataset, we find that training on easy prompts leads to a clear decline in pass@k accuracy (Figure 2).
+Given these findings, we observe that training with hard prompts is more beneficial than with easy ones. Yet, RL training on hard problems tends to be much slower, as sparse reward signals and limited sample efficiency hinder progress. The key challenge, then, is **how to structure the learning process to fully expand reasoning capabilities while mitigating the inefficiency of RL on hard tasks**. To this end, we introduce QUESTA: a parsimonious and efficient strategy that dynamically adjusts problem difficulty during RL training. The core contributions of this work are threefold:
+• We notice that the evolution of model capacity in RLVR critically depends on dataset difficulty, underscoring the importance of training on *hard problems* to expand reasoning ability.
+{2}------------------------------------------------
+<span id="page-2-0"></span>![](_page_2_Figure_1.jpeg)
+Figure 3: Comparison of RL training dynamics: Training with only hard problems (green) makes progress very slowly due to sparse rewards, while our method with partial solutions (orange) accelerates training and consistently achieves higher accuracy across training steps.
+- We introduce QUESTA, an efficient procedure that controls difficulty by augmenting hard problems with partial solutions. This approach provides a smooth curriculum within RL training and makes high-difficulty tasks more tractable. Through our **fully open-sourced** training pipeline, QUESTA consistently improves pass@1 and pass@k, enabling 1.5B-parameter models to reach new state-of-the-art performance—72.5% on AIME24, 62.3% on AIME25, and 41.7% on HMMT25 (Table 1).
+- Our theoretical analysis in Section 4 explains why partial-solution augmentation accelerates RL training: by decomposing problems into intermediate steps, the method yields denser reward signals and improves sample efficiency, while still driving the model to master the hardest problems.
+## <span id="page-2-1"></span>2 Tradeoffs between Reasoning Capacity and Learning Efficiency
+Given the ongoing debate on whether reinforcement learning enhances the reasoning capacity of language models, we design a controlled experiment to study how dataset difficulty changes model performances measured by pass@k accuracy. Specifically, we filter out easy problems and hard problems from the 220K OpenR1 dataset, base on model's success rate, each containing around 4K data. We then run RL with GRPO for one thousand steps. This setup allows us to isolate how the choice of prompt difficulty impacts the model's reasoning capacity. In Figure 2 and Figure 3, we provide pass@k comparison and the learning dynamics, we make two observations.
+**RL** with Easy Prompts Hurts pass@k and Reasoning Capacity. Training on easy or already-solvable problems leads to overfitting on shallow patterns, reinforcing confidence rather than expanding reasoning capacity. While pass@1 may rise, output diversity declines and performance on harder benchmarks deteriorates, with pass@k dropping at larger k (see Figure 2). This suggests that the model exploits familiar solution modes instead of exploring new trajectories. To truly expand capacity, RL training should focus on hard problems, where the policy is forced to explore and acquire novel solution strategies.
+**RL** with Hard Prompts Leads to Slow Learning. Training on hard prompts directly targets the reasoning capacity of the model, but the learning process is much slower (see Figure 3) and less sample-efficient. The difficulty arises because RL rewards on these problems are sparse, providing limited gradient signals for policy improvement. We formallize the underlying reason in Section 4 and in Theorem 4.4.
+In practice, not all questions in the training set Q are equally difficult, and one might hope that training on easier examples could generalize to harder ones. However, empirical evidence suggests that RL-based training exhibits a bi-modal pattern in success rates (An et al., 2025): by the end of training, models tend to either solve a question reliably or fail entirely (see Figure 6). This implies that once a question falls outside the model's capacity set, the RL algorithm is unlikely to recover.
+{3}------------------------------------------------
+Together, these results highlight a tension: *easy prompts dilute reasoning capacity, while hard prompts stall learning altogether.* This motivates the need for strategies that can retain the benefits of hard problems while mitigating the inefficiency caused by sparse rewards. To this end, we introduce partial solutions that break a complex question into smaller, more approachable pieces. Theoretical analysis (Theorem 4.6) suggests that appending part of the solutions as hint can greatly improve RL efficiency.
+Empirically, we simply choose the hint to be a part of the solution of the original question q and observe faster learning in Figure 3. Surprisingly, even if we don't explicitly train the model to generate the hint, the model's capacity without hint still continues to improve and lead to steady improvement in problems out of reach in standard RL training (see Table 3). We elaborate on implementation details in the next section.
+## 3 Questa: Question Augmentation with Partial Solutions
+QUESTA is a modular augmentation framework designed to inject partial solution sketches into prompts during reinforcement learning (RL) training. It adresses scenarios where the base model fails to generate correct completions—conditions that typically result in sparse reward signals. Distinct from approaches that modify reward functions or optimization algorithms, QUESTA operates at the input level: it transforms original training prompts into more tractable variants, thereby exposing intermediate reasoning steps to the model.
+#### <span id="page-3-0"></span>**Original Prompt**
+Let $\mathbb N$ be the set of positive integers. A function $f:\mathbb N\to\mathbb N$ satisfies the equation
+$$f(f(\dots f(n)\dots)) = \frac{n^2}{f(f(n))}$$
+with $f(n)$ applications of $f$ ,
+for all positive integers n. Given this information, determine all possible values of f(1000).
+![](_page_3_Picture_9.jpeg)
+#### **Augmented Prompt**
+Let $\mathbb N$ be the set of positive integers. The function $f:\mathbb N\to\mathbb N$ satisfies the equation
+$$f(f(\dots f(n)\dots)) = \frac{n^2}{f(f(n))} \quad \text{with } f(n) \text{ applications of } f,$$
+for all positive integers n. Given this information, determine all possible values of f(1000).
+#### ## Hint: Partial Solution
+Analysis shows that f must be an involution, meaning f(f(n)) = n for all n, and it fixes all odd positive integers, so f(n) = n for odd n. For even positive integers, f either fixes the number or swaps it with another even positive integer in a 2-cycle.
+Please reason step by step, and put your final answer within \boxed{}.
+Figure 4: QUESTA augments each original question in the dataset by prepending the first p% of the solution sketch. In our experiments, we apply augmentation using the solution block rather than the reasoning chain-of-thought. The hint percentage p is computed as the ratio of tokens used as hints to the total number of tokens in the solution sketch.
+Question Augmentation Mechanism For a given problem x with an n-step solution trajectory $y=(y_1,y_2,\ldots,y_n)$ , QUESTA constructs a set of augmented prompts $\{\tilde{x}^{(p)}\}$ , where each $\tilde{x}^{(p)}$ appends the first p steps of the solution as a prefix to the original question. The parameter p (e.g., p=50% or 25%) quantifies the proportion of the solution revealed, thereby enabling precise control over the difficulty of the augmented prompt.
+In our empirical evaluations, we employed the OpenR1-Math-220K dataset (Open-R1 Team, 2025)—a supervised fine-tuning (SFT) corpus containing solution trajectories generated by DEEPSEEK-R1. Each instance in this dataset comprises a detailed chain-of-thought (CoT) section followed by a final solution block. For augmentation, we extracted the final solution (omitting speculative reasoning
+{4}------------------------------------------------
+within the CoT section). The solution was then truncated at a predefined percentage p and prepended to the original question, yielding the augmented prompt used in RL training, as shwon in Figure 4.
+**Targeting High-Difficulty Problems** QUESTA is applied exclusively to prompts where the base model's pass rate is close to zero. Using the OpenR1-Math-220K dataset, we first employ lightweight heuristic filters to reduce the full 220K problems to 26K of the hardest candidates. These problems are then augmented with partial-solution prefixes where we conduct a second difficulty screening: sample multiple completions from the model for each augmented prompt, and only those instances with consistently low pass rates are retained. This two-stage filtering pipeline yields a final pool of no more than 10K problems, ensuring that augmentation resources are concentrated on the most challenging cases where the base model needs additional guidance and scaffolding.
+**Integrating with RL Pipelines** QUESTA exhibits orthogonality to underlying RL algorithms, enabling seamless integration into existing training pipelines (e.g., GRPO (Shao et al., 2024), DAPO (Yu et al., 2025)) without modifications. Specifically, integration requires only replacing the original rollout dataset with the augmented dataset, while retaining the original reward function and policy update mechanism. To further exploit this input-level flexibility, we extended QUESTA with an iterative curriculum RL paradigm:
+- 1. First, augment the dataset with p = 50%, apply the difficulty filtering with the augmented prompt, and conduct reinforcement learning training until the performance saturates.
+- 2. Second, reduce the augmentation from p = 50% to p = 25%, i.e. provide fewer hints. Again, we apply the difficulty filtering, and conduct reinforcement learning training until convergence.
+Here, the rationale for the choice of *p* is provided in Appendix B.6. By keeping the training signals strong at each stage, the method speeds up convergence on difficult tasks and makes QUESTA a simple, plug-and-play approach for curriculum-based RL.
+## <span id="page-4-0"></span>4 THEORY: VARYING LEARNABILITY ENHANCES RL EFFICIENCY
+In this section, we present a theoretical perspective on how question augmentation improves the efficiency of reinforcement learning. Our central thesis is that the primary bottleneck in RL-based reasoning lies in the difficulty of discovering successful trajectories within a finite sampling budget. Question augmentation addresses this challenge by reshaping the *learnability landscape*—making hard problems more discoverable by increasing the likelihood of encountering correct trajectories.
+Motivated by experiments which quantify model capcity with pass@k accuracy, we introduce the following notions of *solution set* (Definition 4.1) and *model capacity set* (Definition 4.2) for a given question q and model $\mu$ . Let $\mathcal V$ be the vocaboluary set, and let $P_{\mu}(q,\tau)$ denote the probability that a language model $\mu$ generates trajectory $\tau \in \mathcal V^*$ when conditioned on input question $q \in \mathcal V^*$ .
+<span id="page-4-1"></span>**Definition 4.1** (Solution Set). Given a question q and a binary reward function $R: \mathcal{V}^* \times \mathcal{V}^* \to \{0, 1\}$ , the *solution set* is defined as:
+$$S(q) = \{ \tau \in \mathcal{V}^* \mid R(q, \tau) = 1 \}.$$
+<span id="page-4-2"></span>**Definition 4.2** (Model Capacity Set). Given a probability threshold $\delta_p > 0$ , a language model $\mu$ , and a question q, define the *model capacity set* $C(q, \delta_p)$ as the smallest set of trajectories whose total probability mass is at least $1 - \delta_p$ :
+$$C(q, \delta_p) = \arg\min_{S \subseteq \mathcal{V}^*} \left\{ |S| \left| \sum_{\tau \in S} P_{\mu}(q, \tau) \ge 1 - \delta_p \right\}. \right.$$
+The Model Capacity Set $C(q, \delta_p)$ intuitively captures the set of most likely output trajectories that the model $\mu$ can generate for a given input q, up to a small probability threshold $\delta_p$ .
+<span id="page-4-3"></span>This formalization leads to a critical insight: if the model's capacity set fails to intersect with the solution set—meaning the model is unlikely to generate any correct completions—then the RL process cannot make progress. To articulate this more formally, we begin by stating a standard assumption satisfied by many popular RL algorithms, such as DAPO and online GRPO:
+{5}------------------------------------------------
+**Assumption 4.3** (Null Gradient from Zero-Reinforcement). The RL algorithm does not update the model weights if none of the sampled rollouts receives a positive reward (i.e., reward = 1).
+Under this assumption, we easily the following lower bound, which states that if all training questions are unreachable within the model's capacity set, the RL process is likely to stall entirely:
+<span id="page-5-0"></span>**Theorem 4.4** (Lower Bound on RL Learnability under Solution Inaccessibility). Given a probability threshold $\delta_p > 0$ , if for every question $q \in \mathcal{Q}$ , the model capacity set $C(q, \delta_p)$ does not intersect with the solution set $\mathcal{S}(q)$ , i.e.,
+$$C(q, \delta_p) \cap \mathcal{S}(q) = \emptyset, \quad \forall q \in \mathcal{Q},$$
+then under Assumption 4.3, when training RL for T steps with B samples per step such that $TB = \Theta(1/\delta_p)$ , there is a constant probability that the RL algorithm will not update the model.
+To overcome this limitation, our method QUESTA provides a simple yet effective solution: augment each question in $\mathcal Q$ with a partial solution to improve the chances of sampling informative trajectories. Formally, we assume the existence of a hint $h_q$ for every question $q \in \mathcal Q$ that can guide the model toward discovering a valid completion.
+<span id="page-5-2"></span>**Definition 4.5** (Question Augmentation). For every question $q \in \mathcal{Q}$ , hint $h_q \in \mathcal{V}^*$ satisfies that for $\delta'_p = \delta_p^{1/2 - \epsilon}$ for some $\epsilon > 0$ :
+- the hint $h_q$ can be generated with a non-neglible probability: $P_{\mu}(h_q|q) \geq \delta_p'$
+- there exists a solution to the hinted problem $s_q \in \mathcal{S}(q)$ such that $s_q$ can be generated with high probability after $s_q$ , i.e.
+$$P_{\mu}(s_q|(q,h_q)) = \delta'_{p}. \quad R(q,h \oplus s_q) = 1.$$
+The hint $h_q$ can exist for every question even when the model's capacity set $C(q, \delta_p)$ does not intersect with the solution set $\mathcal{S}(q)$ . For instance, if every solution can be decomposed into two steps, and the model can generate each step correctly with probability $\delta_p' = \sqrt{o(\delta_p)}$ , then the possibility of generating two steps correctly at the same time is only $o(\delta_p)$ .
+This implies that the sampling budget needed with a hint is asymptotically almost the square root of the budget required without it $(\Theta(1/\delta_p))$ , as given in Theorem 4.4. We further provide a learnability result where we assume the policy is parameterized by a softmax policy parameterization in a classical tabular RL setup.
+<span id="page-5-1"></span>**Theorem 4.6** (Informal Upper Bound on RL Learnability with Hint). If we have a hint $h_q$ for every question $q \in \mathcal{Q}$ (Def. 4.5), then there exists an RL algorithm that can output a policy $\pi_\theta$ such that $\mathbb{E}_{q \sim \mathrm{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \pi_\theta(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99$ with $O(1/\delta_p')$ sampling budget with high probability.
+Theorem 4.6 provides a theoretical guarantee that the model can reach a high training success rate when partial solution is included. Empirically, we observe the model generalizes well both in-distribution and out-of-distribution to hard questions.
+#### 5 EXPERIMENTS
+**Dataset.** We begin with the OpenR1-Math-220K dataset and use DeepSeek-R1-Distill-1.5B as a weak selection model to filter it down to the 26K hardest items. This set serves as our base prompts. We then use Nemotron-1.5B to sample eight generations per prompt and classify problems into Easy Data (7–8 correct answers) and Hard Data (0–1 correct answers), enabling controlled experiments introduced in Section 2. The exact prompt template is provided in Appendix B.8
+**Data Augmentation (QUESTA).** To improve the tractability of the problems, we apply QUESTA to prepend the prompt with partial solutions, i.e. first p% of the full solution in the SFT data provided in the OpenR1-Math-220K dataset. After augmentation, we use the initial model at RL training, either Nemotron-1.5B or DeepScaleR-1.5B, to sample 8 generations per augmented prompts and select samples with 0–4 correct predictions. Full details are provided in Appendix B.5. These high-variance cases provide stronger learning signals and make the training process more effective.
+{6}------------------------------------------------
+<span id="page-6-0"></span>Table 1: Performance comparison (Pass@1, averaged over 32 samples) across maths benchmarks. The best results among the 1.5B models are highlighted in bold. Larger models are shown in gray as reference points. Reported results for DeepSeek-R1-Distill and Qwen3 are taken from their official documentation [\(Guo et al., 2025;](#page-10-2) [Yang et al., 2025\)](#page-11-2), while the rest are self-evaluated. Our QUESTA-Nemotron-1.5B achieves state-of-the-art performance among 1.5B models and, notably, matches or even exceeds the performance of DeepSeek-R1-Distill-32B across several benchmarks, despite being over 20× smaller in parameter count. This demonstrates the effectiveness of QUESTA in enhancing small model capabilities through targeted training.
+| Model | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25 | Avg |
+|--------------------------|--------|--------|-------------|----------------|---------|-------|
+| DeepSeek-R1-Distill-1.5B | 28.7 | 22.3 | 12.0 | 52.4 | 31.8 | 29.44 |
+| Qwen3-1.7B | 48.3 | 36.8 | 22.19 | 56.13 | 44.06 | 41.50 |
+| DeepSeek-R1-Distill-32B | 72.6 | 51.8 | 33 | 65.0 | 68 | 58.08 |
+| Qwen3-8B | 76.0 | 67.3 | 44.79 | 68.56 | 68.33 | 64.99 |
+| Nemotron-1.5B | 61.77 | 49.50 | 31.56 | 64.62 | 58.23 | 53.14 |
+| QUESTA-Nemotron-1.5B | 72.50 | 62.29 | 41.67 | 70.36 | 69.48 | 63.26 |
+Training Setup. We use AReaL [\(Fu et al., 2025\)](#page-9-2) as our RL training framework, applying the GRPO algorithm [\(Guo et al., 2025\)](#page-10-2) without the Kullback–Leibler (KL) divergence loss. Following DAPO [\(Yu et al., 2025\)](#page-11-3), we also dynamically filter out prompts that are either all correct or all incorrect during rollouts. During training, we sample n = 16 responses per prompt with a maximum prompt length of 8192 tokens and a maximum generation length of 24000 tokens, using a sampling temperature of 1.0 and clipping hyperparameters with εlow = εhigh = 0.2. The batch size is 128 with a mini-batch size of 1, equivalent to 128 gradient updates per rollout step. Optimization is performed with AdamW [\(Kingma & Ba, 2017;](#page-10-5) [Loshchilov & Hutter, 2019\)](#page-10-6) using a constant learning rate of 2 × 10<sup>−</sup><sup>5</sup> . Experiments are conducted on eight NVIDIA H800 (80GB) nodes. Full details of our training method are provided in Appendix [B.1.](#page-13-0)
+Evaluation Setup. For each problem in the evaluation benchmarks, we generate 32 samples and report pass@1 results. Generation uses a sampling temperature of 0.7 and a top-p value of 0.95, with k = 32 responses per question unless otherwise specified. *It is important to note that while partial solutions were incorporated during training, no partial solutions are provided at evaluation time.*
+## 5.1 EXPERIMENTAL RESULTS
+Key Results. Table [1](#page-6-0) reports results on challenging math benchmarks. QUESTA yields substantial gains for Nemotron-1.5B, achieving an average improvement of 10% over its baseline and a particularly strong +13% on AIME25. These improvements are consistent across all benchmarks, highlighting the effectiveness of our approach in enhancing problem-solving robustness.
+Compared to other models, QUESTA-Nemotron-1.5B consistently outperforms peers of similar scale, such as DeepSeek-R1-Distill-1.5B and Qwen3-1.7B, and even surpasses larger models like DeepSeek-R1-Distill-32B across all benchmarks. On AIME25 in particular, it exceeds DeepSeek-R1-Distill-32B by a substantial margin of +11%. Against the stronger Qwen3-8B, QUESTA-Nemotron-1.5B remains competitive despite operating at a fraction of the parameter scale.
+Training Dynamics. Figure [5](#page-7-1) summarizes the training dynamics of QUESTA-Nemotron-1.5B. A positive correlation is observed between average response length and model accuracy, reflecting common trends in RL training. Notably, with QUESTA, the entropy during RL training remains stable and does not exhibit significant collapse.
+Pass@k Analysis. Our evaluation follows the standard pass@k methodology, consistent with DeepSeek-R1 [\(Guo et al., 2025\)](#page-10-2), with further details provided in Appendix [B.2.](#page-13-1) In contrast to recent findings that RL-based training can reduce pass@k at larger k values [\(Yue et al., 2025;](#page-11-0) [Liu et al., 2025\)](#page-10-0), our results show that QUESTA preserves—and in many cases modestly improves—performance across a broad range of k. As shown in Figure [2,](#page-1-0) incorporating partial-solution hints within a two-stage curriculum yields consistent gains across models, without the degradation in pass@k often observed under standard RL training. These results indicate that QUESTA enhances both the quality and diversity of candidate solutions, rather than overfitting to a single best trajectory.
+{7}------------------------------------------------
+<span id="page-7-1"></span>![](_page_7_Figure_1.jpeg)
+Figure 5: Training dynamics of QUESTA-Nemotron-1.5B. The first and second charts show the progression of average response length and average reward across rollout samples during the RL process, both of which steadily increase over time. The third chart presents the average entropy. Interestingly, the entropy increases over time, suggesting that QUESTA does not suffer from entropy collapse and instead encourages diverse and exploratory behavior.
+<span id="page-7-0"></span>![](_page_7_Figure_3.jpeg)
+Figure 6: Pass Rate Distribution on Training Prompts. We compare the success rate on the 26K training set before and after RL, using the average pass rate over 8 samples per question. Although partial solutions are included during QUESTA training, **no hints** are provided during this evaluation. This setup isolates the true impact of QUESTA by assessing its ability to improve performance on problems without hints. QUESTA significantly reduces the number of unsolved or partially solved problems in the training set, especially for hard ones where initial model solves only 0/8 or 1/8 times.
+Generalization at Test Time when Hints are Removed. A natural question arises from our approach: since we add partial solutions during RL training, does this improvement persist when hints are removed at evaluation time? To answer this, Figure 6 compares the pre- and post-RL models on the 26K training prompt set, evaluated without any hints. The distribution clearly shifts away from the 0/8–1/8 bins toward higher pass rates, indicating that the model solves a larger fraction of problems even without access to partial solutions. On the evaluation AIME benchmarks, Table 2 further demonstrates that QUESTA expands coverage at Pass@32: for Nemotron-1.5B, the number of unsolved problems drops from 5 to 2 on AIME24 (newly solved indices 2, 13, 29) and from 6 to 3 on AIME25 (newly solved indices 9, 13, 27). Taken together, these results show that our method generalizes well beyond the training setting and helps solve hard problems that are otherwise inaccessible without partial-solution guidance.
+#### 5.2 Further Ablations
+**Ablation with Difficulty Curriculum.** We first motivate the choice of a two-stage curriculum: RL on *Partial-50* followed by RL on *Partial-25*. From a modeling standpoint, the most appropriate inference distribution for the model should be the original (no-hint) distribution. Hence, during training we should gradually reduce reliance on hints to align the learned policy with the evaluation distribution. This motivates decreasing the partial ratio over time so that the model transitions from scaffolded reasoning to autonomous reasoning.
+{8}------------------------------------------------
+<span id="page-8-1"></span>Table 2: Indices of **unsolved** problems at Pass@32 on AIME24 and AIME25 (with indices ranging from 0-29). Our method, QUESTA, consistently improve the model capacity on hard cases where the initial model is unable to solve, improving overall coverage at Pass@32.
+| Models | AIME24 Unsodlved Indices | AIME25 Unsolved Indices |
+|----------------------|--------------------------|-------------------------|
+| Nemotron-1.5B | 2, 3, 13, 21, 29 | 9, 12, 13, 14, 27, 29 |
+| QUESTA-Nemotron-1.5B | 3, 21 | 12, 14, 29 |
+<span id="page-8-0"></span>Table 3: Ablation Study on the Impact of Curriculum Design. This table demonstrates the importance of curriculum learning in improving model performance. The model QUESTA-Nemotron-1.5B-50 was trained entirely with Partial-50 data for 2000 steps, while QUESTA-Nemotron-1.5B followed a curriculum learning approach, starting with 100 steps of Partial-50 data followed by 1900 steps of Partial-25 data. As seen in the table, the curriculum learning approach (QUESTA-Nemotron-1.5B) outperforms training with only Partial-50 data (QUESTA-Nemotron-1.5B-50). Extension with Partial-50-Partial-25-Partial-0 did not yield significant improvements, and thus, are not included in the table.
+| Model | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25 | Avg |
+|------------------------------------------------------------|--------------|--------------|--------------|----------------|--------------|--------------|
+| Nemotron-1.5B QUESTA-Nemotron-1.5B-50 QUESTA-Nemotron-1.5B | 61.77 | 49.50 | 31.56 | 64.62 | 58.23 | 53.14 |
+| | 67.18 | 59.38 | 39.17 | 69.41 | 66.15 | 60.26 |
+| | <b>72.50</b> | <b>62.29</b> | <b>41.67</b> | <b>70.36</b> | <b>69.48</b> | <b>63.26</b> |
+Empirically, Table 3 shows that, under the same 2000-step budget, the curriculum *Partial-50* Partial-25 learns substantially better than training on *Partial-50* alone. We cap the *Partial-50* stage at 100 steps, after which we switch to *Partial-25*. As shown in Figure 11, entropy for QUESTA-Nemotron-1.5B-50 begins to decline beyond 100 steps, so transitioning at this point prevents overconfidence and sustains training stability. We have also tried extending the curriculum from *Partial-25* to *Partial-0* in our experiments, but observed no gains and no increase in response length (see Figure 12).
+<span id="page-8-2"></span>Table 4: Performance comparison (Pass@1, averaged over 32 samples) between Nemotron-1.5B and QUESTA-Nemotron-1.5B (By OpenMathReasoning). The two models achieve comparable results, with the version trained on OpenR1 performing slightly better overall.
+| Model | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25 Avg |
+|--------------------------------------------------------------------------|--------|--------|-------------|----------------|---------------|
+| Nemotron-1.5B QUESTA-50 (with OpenMathReasoning) QUESTA-50 (with OpenR1) | 61.77 | 49.50 | 31.56 | 64.62 | 58.23 53.14 |
+| | 66.46 | 58.54 | 36.35 | 66.06 | 63.13 58.11 |
+| | 67.18 | 59.38 | 39.17 | 69.41 | 66.15 60.26 |
+Ablation with Different Dataset. We also evaluated QUESTA on OpenMathReasoning Moshkov et al. (2025), selecting the 60K questions with pass\_rate\_72b\_tir of 0 or 1/32. Due to time constraints, we trained only the first stage of QUESTA with 50% partial solutions. Table 4 shows that QUESTA-Nemotron-1.5B-50 achieves similar performance as using the OpenR1 dataset. This indicates that our approach generalizes across datasets.
+**Other Ablations.** We also conduct an extensive set of comparative experiments and ablation studies, with detailed results provided in Appendix D. These include an ablation of QUESTA without hints (Appendix D.1), experiments with different model backbones (Appendix D.2), and the full set of pass rates and training curves for additional models (Appendix D.3).
+#### 6 CONCLUSIONS
+In this work we introduced QUESTA, a lightweight data-centric framework that augments hard prompts with partial-solution hints during RL training. Without altering model architecture or reward design, QUESTA sets new state-of-the-art results for 1.5 B-scale models on AIME24, AIME25 and HMMT25. Further, we theoretically demonstrate how question augmentation can improve sample
+{9}------------------------------------------------
+efficiency. Our analysis shows that the method can potentially be generalized to other domains such as competitive coding, software engineering or other agentic tasks. Designing proper question augmentation pipelines for theses new tasks can be an important and interesting future direction.
+#### REFERENCES
+<span id="page-9-5"></span>Brown university math olympiad 2025. February 2025.
+<span id="page-9-4"></span>Harvard-mit mathematics tournament 2025. February 2025.
+<span id="page-9-0"></span>Chenxin An, Zhihui Xie, Xiaonan Li, Lei Li, Jun Zhang, Shansan Gong, Ming Zhong, Jingjing Xu, Xipeng Qiu, Mingxuan Wang, and Lingpeng Kong. Polaris: A post-training recipe for scaling reinforcement learning on advanced reasoning models, 2025.
+<span id="page-9-6"></span>Mislav Balunović, Jasper Dekoninck, Ivo Petrov, Nikola Jovanović, and Martin Vechev. Matharena: Evaluating Ilms on uncontaminated math competitions, February 2025.
+<span id="page-9-3"></span>Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, Alex Ray, Raul Puri, Gretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios Chantzis, Elizabeth Barnes, Ariel Herbert-Voss, William Hebgen Guss, Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin, Suchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojciech Zaremba. Evaluating large language models trained on code, 2021.
+<span id="page-9-1"></span>Ganqu Cui, Yuchen Zhang, Jiacheng Chen, Lifan Yuan, Zhi Wang, Yuxin Zuo, Haozhan Li, Yuchen Fan, Huayu Chen, Weize Chen, et al. The entropy mechanism of reinforcement learning for reasoning language models. *arXiv preprint arXiv:2505.22617*, 2025.
+<span id="page-9-2"></span>Wei Fu, Jiaxuan Gao, Xujie Shen, Chen Zhu, Zhiyu Mei, Chuyi He, Shusheng Xu, Guo Wei, Jun Mei, Jiashu Wang, et al. Areal: A large-scale asynchronous reinforcement learning system for language reasoning. *arXiv preprint arXiv:2505.24298*, 2025.
+<span id="page-9-7"></span>Zitian Gao, Lynx Chen, Haoming Luo, Joey Zhou, and Bryan Dai. One-shot entropy minimization, 2025.
+{10}------------------------------------------------
+- <span id="page-10-3"></span>Gemini Team, Google DeepMind. Gemini 2.5: Pushing the frontier with advanced reasoning, multimodality, long context, and next generation agentic capabilities, 2025. Accessed: July 2025.
+- <span id="page-10-2"></span>Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning. *arXiv preprint arXiv:2501.12948*, 2025.
+- <span id="page-10-13"></span>Chaoqun He, Renjie Luo, Yuzhuo Bai, Shengding Hu, Zhen Leng Thai, Junhao Shen, Jinyi Hu, Xu Han, Yujie Huang, Yuxiang Zhang, Jie Liu, Lei Qi, Zhiyuan Liu, and Maosong Sun. Olympiadbench: A challenging benchmark for promoting agi with olympiad-level bilingual multimodal scientific problems, 2024.
+- <span id="page-10-9"></span>Zhenyu Hou, Ziniu Hu, Yujiang Li, Rui Lu, Jie Tang, and Yuxiao Dong. Treerl: Llm reinforcement learning with on-policy tree search, 2025.
+- <span id="page-10-1"></span>Aaron Jaech, Adam Kalai, Adam Lerer, Adam Richardson, Ahmed El-Kishky, Aiden Low, Alec Helyar, Aleksander Madry, Alex Beutel, Alex Carney, et al. Openai o1 system card. *arXiv preprint arXiv:2412.16720*, 2024.
+- <span id="page-10-15"></span>Naman Jain, King Han, Alex Gu, Wen-Ding Li, Fanjia Yan, Tianjun Zhang, Sida Wang, Armando Solar-Lezama, Koushik Sen, and Ion Stoica. Livecodebench: Holistic and contamination free evaluation of large language models for code, 2024.
+- <span id="page-10-10"></span>Amirhossein Kazemnejad, Milad Aghajohari, Eva Portelance, Alessandro Sordoni, Siva Reddy, Aaron Courville, and Nicolas Le Roux. Vineppo: Refining credit assignment in rl training of llms, 2025.
+- <span id="page-10-5"></span>Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017.
+- <span id="page-10-14"></span>Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal Lago, Thomas Hubert, Peter Choy, Cyprien de Masson d'Autume, Igor Babuschkin, Xinyun Chen, Po-Sen Huang, Johannes Welbl, Sven Gowal, Alexey Cherepanov, James Molloy, Daniel J. Mankowitz, Esme Sutherland Robson, Pushmeet Kohli, Nando de Freitas, Koray Kavukcuoglu, and Oriol Vinyals. Competition-level code generation with alphacode. *Science*, 378(6624):1092–1097, December 2022. ISSN 1095-9203. doi: 10.1126/science.abq1158.
+- <span id="page-10-16"></span>Bill Yuchen Lin, Ronan Le Bras, Kyle Richardson, Ashish Sabharwal, Radha Poovendran, Peter Clark, and Yejin Choi. Zebralogic: On the scaling limits of llms for logical reasoning, 2025.
+- <span id="page-10-0"></span>Mingjie Liu, Shizhe Diao, Ximing Lu, Jian Hu, Xin Dong, Yejin Choi, Jan Kautz, and Yi Dong. Prorl: Prolonged reinforcement learning expands reasoning boundaries in large language models. *arXiv preprint arXiv:2505.24864*, 2025.
+- <span id="page-10-6"></span>Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization, 2019.
+- <span id="page-10-4"></span>Haipeng Luo, Qingfeng Sun, Can Xu, Pu Zhao, Jianguang Lou, Chongyang Tao, Xiubo Geng, Qingwei Lin, Shifeng Chen, Yansong Tang, and Dongmei Zhang. Wizardmath: Empowering mathematical reasoning for large language models via reinforced evol-instruct, 2025a. URL <
+- <span id="page-10-7"></span>Michael Luo, Sijun Tan, Justin Wong, Xiaoxiang Shi, William Y. Tang, Manan Roongta, Colin Cai, Jeffrey Luo, Li Erran Li, Raluca Ada Popa, and Ion Stoica. Deepscaler: Surpassing o1-preview with a 1.5b model by scaling rl, 2025b. Notion Blog.
+- <span id="page-10-11"></span>MAA. American invitational mathematics examination 2024. February 2024.
+- <span id="page-10-12"></span>MAA. American invitational mathematics examination 2025. February 2025.
+- <span id="page-10-8"></span>Saumya Malik, Valentina Pyatkin, Sander Land, Jacob Morrison, Noah A. Smith, Hannaneh Hajishirzi, and Nathan Lambert. Rewardbench 2: Advancing reward model evaluation, 2025.
+- <span id="page-10-17"></span>Jincheng Mei, Chenjun Xiao, Csaba Szepesvari, and Dale Schuurmans. On the global convergence rates of softmax policy gradient methods, 2022.
+{11}------------------------------------------------
+- <span id="page-11-5"></span>Ivan Moshkov, Darragh Hanley, Ivan Sorokin, Shubham Toshniwal, Christof Henkel, Benedikt Schifferer, Wei Du, and Igor Gitman. Aimo-2 winning solution: Building state-of-the-art mathematical reasoning models with openmathreasoning dataset. *arXiv preprint arXiv:2504.16891*, 2025.
+- <span id="page-11-1"></span>Open-R1 Team. Openr1-math-220k: A large-scale dataset for mathematical reasoning, 2025.
+- <span id="page-11-13"></span>Shubham Parashar, Shurui Gui, Xiner Li, Hongyi Ling, Sushil Vemuri, Blake Olson, Eric Li, Yu Zhang, James Caverlee, Dileep Kalathil, and Shuiwang Ji. Curriculum reinforcement learning from easy to hard tasks improves llm reasoning. 2025.
+- <span id="page-11-11"></span>Yuxiao Qu, Matthew Y. R. Yang, Amrith Setlur, Lewis Tunstall, Edward Emanuel Beeching, Ruslan Salakhutdinov, and Aviral Kumar. Optimizing test-time compute via meta reinforcement finetuning, 2025.
+- <span id="page-11-15"></span>David Rein, Betty Li Hou, Asa Cooper Stickland, Jackson Petty, Richard Yuanzhe Pang, Julien Dirani, Julian Michael, and Samuel R. Bowman. Gpqa: A graduate-level google-proof q&a benchmark, 2023.
+- <span id="page-11-7"></span>John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms, 2017.
+- <span id="page-11-8"></span>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:2506.10947*, 2025.
+- <span id="page-11-6"></span>Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, YK Li, Y Wu, et al. Deepseekmath: Pushing the limits of mathematical reasoning in open language models. *arXiv preprint arXiv:2402.03300*, 2024.
+- <span id="page-11-9"></span>Peiyi Wang, Lei Li, Zhihong Shao, R. X. Xu, Damai Dai, Yifei Li, Deli Chen, Y. Wu, and Zhifang Sui. Math-shepherd: Verify and reinforce llms step-by-step without human annotations, 2024.
+- <span id="page-11-4"></span>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 llm reasoning. *arXiv preprint arXiv:2506.01939*, 2025a.
+- <span id="page-11-16"></span>Yiping Wang, Qing Yang, Zhiyuan Zeng, Liliang Ren, Liyuan Liu, Baolin Peng, Hao Cheng, Xuehai He, Kuan Wang, Jianfeng Gao, Weizhu Chen, Shuohang Wang, Simon Shaolei Du, and Yelong Shen. Reinforcement learning for reasoning in large language models with one training example, 2025b.
+- <span id="page-11-10"></span>Xumeng Wen, Zihan Liu, Shun Zheng, Zhijian Xu, Shengyu Ye, Zhirong Wu, Xiao Liang, Yang Wang, Junjie Li, Ziming Miao, et al. Reinforcement learning with verifiable rewards implicitly incentivizes correct reasoning in base llms. *arXiv preprint arXiv:2506.14245*, 2025.
+- <span id="page-11-12"></span>Zhiheng Xi, Wenxiang Chen, Boyang Hong, Senjie Jin, Rui Zheng, Wei He, Yiwen Ding, Shichun Liu, Xin Guo, Junzhe Wang, Honglin Guo, Wei Shen, Xiaoran Fan, Yuhao Zhou, Shihan Dou, Xiao Wang, Xinbo Zhang, Peng Sun, Tao Gui, Qi Zhang, and Xuanjing Huang. Training large language models for reasoning through reverse curriculum reinforcement learning, 2024. URL <
+- <span id="page-11-14"></span>Yuchen Yan, Yongliang Shen, Yang Liu, Jin Jiang, Mengdi Zhang, Jian Shao, and Yueting Zhuang. Inftythink: Breaking the length limits of long-context reasoning in large language models. *arXiv preprint arXiv:2503.06692*, 2025.
+- <span id="page-11-2"></span>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 arXiv:2505.09388*, 2025.
+- <span id="page-11-3"></span>Qiying Yu, Zheng Zhang, Ruofei Zhu, Yufeng Yuan, Xiaochen Zuo, Yu Yue, Weinan Dai, Tiantian Fan, Gaohong Liu, Lingjun Liu, et al. Dapo: An open-source llm reinforcement learning system at scale. *arXiv preprint arXiv:2503.14476*, 2025.
+- <span id="page-11-0"></span>Yang Yue, Zhiqi Chen, Rui Lu, Andrew Zhao, Zhaokai Wang, Shiji Song, and Gao Huang. Does reinforcement learning really incentivize reasoning capacity in llms beyond the base model? *arXiv preprint arXiv:2504.13837*, 2025.
+{12}------------------------------------------------
+- <span id="page-12-0"></span>Rosie Zhao, Alexandru Meterez, Sham Kakade, Cengiz Pehlevan, Samy Jelassi, and Eran Malach. Echo chamber: Rl post-training amplifies behaviors learned in pretraining, 2025.
+- <span id="page-12-1"></span>Xinyu Zhu, Mengzhou Xia, Zhepei Wei, Wei-Lin Chen, Danqi Chen, and Yu Meng. The surprising effectiveness of negative reinforcement in llm reasoning. *arXiv preprint arXiv:2506.01347*, 2025.
+{13}------------------------------------------------
+## A THE USE OF LARGE LANGUAGE MODELS (LLMS)
+The Large Language Models (LLMs) were exclusively utilized to polish the writing and detect potential typos, with no involvement in other aspects.
+#### **B** IMPLEMENTATION DETAILS
+#### <span id="page-13-0"></span>**B.1 RLVR ALGORITHMS**
+We have employed the GRPO algorithm enhanced with a subset of DAPO techniques. Primarily, we have integrated DAPO's Dynamic Sampling Trick and eliminated the KL divergence term, resulting in an optimization objective that is:
+$$\begin{split} \mathcal{J}(\theta) &= & \mathbb{E}_{q \sim \mathcal{D}, \{o_i\}_{i=1}^G \sim \pi_{\theta_{\text{old}}}(\cdot|q)} \\ & \left[ \frac{1}{G} \sum_{i=1}^G \frac{1}{|o_i|} \sum_{t=1}^{|o_i|} \min\left(r_{i,t}(\theta) \hat{A}_{i,t}, \, \text{clip}\Big(r_{i,t}(\theta), 1-\varepsilon, 1+\varepsilon\Big) \hat{A}_{i,t}\Big) \right] \\ & \text{s.t.} \quad 0 < \#\Big\{o_i \mid [o_i \text{ is correct}]\}\Big\} < G, \end{split} \tag{1}$$
+where
+$$r_{i,t}(\theta) = \frac{\pi_{\theta}(o_{i,t} \mid q, o_{i, < t})}{\pi_{\theta_{\text{old}}}(o_{i,t} \mid q, o_{i, < t})}, \quad \hat{A}_{i,t} = \frac{R_i - \text{mean}(\{R_i\}_{i=1}^G)}{\text{std}(\{R_i\}_{i=1}^G)}.$$
+(2)
+Our reward function R mirrors that of DeepScaleR (Luo et al., 2025b), employing an Outcome Reward Model. It returns 1 if and only if both the answer and format are correct; otherwise, it returns 0. In summary, our reward function yields:
+$$R = \begin{cases} 1, & \text{if the answer (e.g. passes basic LaTeX/Sympy checks)} \\ & \text{and format (e.g. exists and ) are both correct,} \\ 0, & \text{otherwise.} \end{cases}$$
+(3)
+#### <span id="page-13-1"></span>B.2 LOW-VARIANCE PASS@K ESTIMATION
+Pass@k is a measure of a model's problem - solving ability, indicating the probability that the model can generate at least one correct solution in k attempts. Specifically, for each problem $x_i$ in the evaluation dataset $\mathcal{D}$ , we generate n samples (where $n \geq k$ ) and count the correct ones as $c_i$ . The direct calculation formula is:
+$$\operatorname{pass}@k := \mathbb{E}_{x_i \sim \mathcal{D}} \left[ 1 - \left( 1 - \frac{c_i}{n} \right)^k \right] \tag{4}$$
+However, this formula has excessive variance and insufficient accuracy. To solve this problem, we adopt the unbiased estimation method proposed by Chen et al. (Chen et al., 2021), using the unbiased estimator of pass@k over the dataset:
+$$\operatorname{pass}@k := \mathbb{E}_{x_i \sim \mathcal{D}} \left[ 1 - \frac{\binom{n - c_i}{k}}{\binom{n}{k}} \right] \tag{5}$$
+In our experiments, to ensure sufficient accuracy, we set n such that $2k \le n$ , which helps further reduce the variance of the estimate.
+## B.3 MORE RELATED WORKS
+Recent studies show that RL algorithms, such as PPO (Schulman et al., 2017) and GRPO (Guo et al., 2025), can greatly enhance model reasoning capabilities. Building on this, several works have refined this paradigm from different perspectives. One method can be adjusting the reward function. Some studies (Zhu et al., 2025; Shao et al., 2025) directly modify the reward function to improve training efficiency. Other methods introduced intermediate process rewards (Wang et al., 2024; Malik et al., 2025), while Wen et al. (Wen et al., 2025) set up a separate correctness judgment for CoT to obtain rewards.
+{14}------------------------------------------------
+Another novel perspective aims to improve sample efficiency by measuring certainty. For example, TreeRL (Hou et al., 2025) and VinePPO (Kazemnejad et al., 2025) enhanced sample effects by introducing entropy or confidence. MRT (Qu et al., 2025), on the other hand, reused partial trajectories during testing to boost sample efficiency. R3 (Xi et al., 2024) improves RL sample efficiency by decomposing human solution steps and providing preceding steps to guide the model in completing subsequent ones. Further, some research adopted a multi-stage training or reasoning mode, exploring from different angles such as training length (Luo et al., 2025b), question difficulty (Parashar et al., 2025), and fixed-length summaries during reasoning (Yan et al., 2025).
+In addition to designing better algorithms, another line of research (Shao et al., 2024; Yue et al., 2025; Zhao et al., 2025) has investigated how reinforcement learning affects the frontier of model capabilities, observing a decay in pass@k when k becomes large. In response to this phenomenon, some works (Yu et al., 2025; Liu et al., 2025; An et al., 2025) maintained entropy stability by adjusting training entropy through methods such as increasing the clipping upper bound, enlarging the temperature coefficient, extending the training length, and periodically updating the KL reference model. StepHint (Zhang et al., 2025) also preserved entropy stability by leveraging intermediate thinking content of iterative length as a prompting signal.
+In contrast to the aforementioned research, our work adopts an orthogonal approach by using part of the ground-truth solution as a hint, without requiring any modifications to the existing reinforcement learning infrastructure. We provide both theoretical justification and empirical evidence that this strategy maintains pass@k without compromising the exploratory capacity of the underlying reinforcement learning algorithm.
+#### B.4 BENCHMARKS
+We evaluate the models' breadth across various tasks in multiple domains, including mathematics, coding, reasoning, and logical inference. For mathematics, we follow DeepScaleR (Luo et al., 2025b) and Nemotron (Moshkov et al., 2025), and conduct assessments on more challenging mathematical datasets such as AIME2024 (MAA, 2024), AIME2025 (MAA, 2025), Olympiad Bench (He et al., 2024), HMMT FEB 25 (hmm, 2025), and BRUMO25 (bru, 2025). Specifically, HMMT25 Feb and BRUMO25 are both sourced from MathArena (Balunović et al., 2025). In the realm of coding, we utilize commonly employed datasets, including Code Contests (Li et al., 2022), Codeforces<sup>1</sup>, and LCB V5 202410-202502 (Jain et al., 2024). For logical reasoning tasks, we assess our models' capabilities using GPQA Diamond (Rein et al., 2023) <sup>2</sup> and Zebraliogic (Lin et al., 2025). The benchmarks related to coding and logical reasoning are all referenced from AReaL (Fu et al., 2025).
+## <span id="page-14-0"></span>B.5 TRAINING DATASET
+The dataset employed in our study is OpenR1-Math-220K (Open-R1 Team, 2025). Prior to commencing the training of the Partial Solution, we conducted a preliminary screening of the dataset. Specifically, we utilized the DeepSeek-R1-Distill-1.5B (Guo et al., 2025) model to perform eight inference operations on each of the 220k data entries in the OpenR1 dataset. Subsequently, we compared the annotated answers in the OpenR1 dataset with the results generated from each inference to tally the number of correct instances for each data entry. Ultimately, we selected the data entries with 0 or 1 correct instance as the training samples for our study. The final dataset size is 26K.
+For controlled comparisons, we further split this 26K subset by re-sampling **Nemotron-1.5B** eight times per item and counting correct completions. We define *Easy Data* as questions with correct counts in [7,8] and train a model on this split, denoted <code>Easy-Nemotron-1.5B</code>. Similarly, we define *Hard Data* as questions with correct counts in [0,1] and train <code>Hard-Nemotron-1.5B</code> on this split.
+Additionally, for the augmented data, we perform eight inference passes using the model currently under training. We then select samples for which the number of correct predictions falls within the range of [0,4]. This criterion is motivated by the finding that samples exhibiting higher variance are more beneficial for training (Gao et al., 2025; Wang et al., 2025b). The range [0,4] is chosen because it includes the point of maximum sample variance, which is achieved with four correct predictions out of eight trials. For convenience, we refer to augmented data with partial ratio p as Partial-p data.
+<span id="page-14-2"></span><span id="page-14-1"></span><sup>1</sup>
+<sup>&</sup>lt;sup>2</sup>In the GPQA Diamond dataset, multiple-choice questions are presented in the form of options rather than directly providing the answer, requiring the model to output only A, B, C, or D.
+{15}------------------------------------------------
+## <span id="page-15-0"></span>B.6 THE RATIONALE FOR THE CHOICE OF p
+<span id="page-15-2"></span>Table 5: Number of problems vs pass rate under different hint levels on OpenMath-Nemotron-1.5B before training. We evaluated OpenMath-Nemotron-1.5B on the OpenR1 dataset after the first round of filtering, with each problem assessed 8 times. The table illustrates the distribution of correct answers (n) where n ∈ {0, 1, . . . , 8}.
+| Hint Levels | 0 / 8 | 1 / 8 | 2 / 8 | 3 / 8 | 4 / 8 | 5 / 8 | 6 / 8 | 7 / 8 | 8 / 8 |
+|-------------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
+| Partial-50 | 143 | 224 | 304 | 472 | 710 | 1013 | 1779 | 3655 | 17741 |
+| Partial-25 | 3155 | 1997 | 1814 | 1785 | 1902 | 2175 | 2614 | 3440 | 7159 |
+| Partial-10 | 3589 | 2090 | 1865 | 1842 | 1905 | 2176 | 2653 | 3415 | 6506 |
+| Partial-0 | 3812 | 2218 | 1854 | 1842 | 2007 | 2136 | 2517 | 3264 | 6391 |
+In this study, we evaluated the performance of OpenMath-Nemotron-1.5B on the OpenR1 dataset under various hint levels. The evaluation was performed after the first round of filtering, and each problem was assessed 8 times to capture the predictive distribution. The resulting table (Table [5\)](#page-15-2) shows the distribution of correct answers across different hint levels, where the values represent the number of times the model answered correctly (n ∈ {0, 1, . . . , 8}).
+The selection of the hint parameter p was primarily based on these evaluation results. As shown in the table, the performance with a Partial-50 hint significantly reduces task difficulty, as evidenced by the high pass rates across most levels. In contrast, Partial-25 (25% hint) exhibits a performance pattern similar to that of the no-hint scenario (Partial-0), with only marginal differences in task difficulty.
+This minimal difference in difficulty between Partial-0 and Partial-25 suggests that training with Partial-25 does not provide substantial gains compared to Partial-0. Consequently, we adopted a stepwise design in which the hint level is first set to p = 50%, followed by p = 25%, to evaluate the model's performance under varying conditions.
+## B.7 EVALUATION SETUP
+We configured the models to have a maximum generation length of 32,768 tokens. In line with DeepSeek-R1 [\(Guo et al., 2025\)](#page-10-2), we utilized pass@k evaluation [\(Chen et al., 2021\)](#page-9-3), with the formula detailed in [B.2.](#page-13-1) We reported pass@1 using a non-zero temperature. Specifically, we used a sampling temperature of 0.7 and a top-p value of 0.95 to generate k responses per question, typically set at 32, with deviations explicitly noted. *Particular attention should be paid to the fact that, although we incorporated partial Solution during training, it was not included in the evaluation phase.*
+## <span id="page-15-1"></span>B.8 DETAIL ON PROMPT TEMPLATE
+## DeepScaleR Coding's Inference: <|User|>{input}<|Assistant|><think>
+## DeepScaleR Others' Inference:
+<|User|>{input}
+Please reason step by step, and put your final answer within \boxed{}.<|Assistant|><think>
+#### Nemotrion Coding's Inference:
+<|im\_start|>user
+{input}
+<|im\_end|> <|im\_start|>assistant
+<think>
+## Nemotrion Others' Inference:
+<|im\_start|>system
+Please reason step by step, and put your final answer within \boxed{}.<|im\_end|> <|im\_start|>user
+{16}------------------------------------------------
+{input}<lim\_endl> <lim\_startl>assistant
+## Training prompt with partial solutions (math RL):
+{Problem}
+## Hint: {Partial Solution}
+Please reason step by step, and put your final answer within \boxed{}.
+#### C THEORY
+#### C.1 Proofs
+**Theorem 4.4** (Lower Bound on RL Learnability under Solution Inaccessibility). Given a probability threshold $\delta_p > 0$ , if for every question $q \in \mathcal{Q}$ , the model capacity set $C(q, \delta_p)$ does not intersect with the solution set $\mathcal{S}(q)$ , i.e.,
+$$C(q, \delta_n) \cap \mathcal{S}(q) = \emptyset, \quad \forall q \in \mathcal{Q},$$
+then under Assumption 4.3, when training RL for T steps with B samples per step such that $TB = \Theta(1/\delta_p)$ , there is a constant probability that the RL algorithm will not update the model.
+*Proof.* Let $p_{\text{sol}} = \sum_{\tau^* \in \mathcal{S}(q)} P_{\mu}(\tau^*|q)$ denote the cumulative generation probability of any solution trajectory. By $C(q, \delta_p) \cap \mathcal{S}(q) = \emptyset$ and Def 4.2:
+$$p_{\text{sol}} = \sum_{\tau^* \in \mathcal{S}(q)} P_{\mu}(\tau^*|q) < \delta_p$$
+For N=TB independent samples across T steps with batch size B, the probability of complete failure (no solution sampled) is:
+$$\mathbb{P}(\text{failure}) = (1 - p_{\text{sol}})^N > (1 - \delta_p)^N$$
+Given $TB = \Theta(1/\delta_p)$ , we have:
+$$(1 - \delta_p)^N > (1 - \delta_p)^{\Theta(1/\delta_p)} = \Theta(1).$$
+The last inequality follows from the fact that $(1-x)^{1/x} > \exp(-1/(1-x))$ for $x \in (0,1)$ . By Assumption 4.3, if no solution is found, the model weights remain unchanged.
+<span id="page-16-0"></span>**Lemma C.1** (Upper Bound on Sampling Budget for Solution Given Hint). Given a question $q \in \mathcal{Q}$ , if there exists a hint $h_q$ for the question q (Def. 4.5), then if we perform $TB = \Theta(1/\delta_p') = \Theta(\delta_p^\epsilon/\sqrt{\delta_p})$ i.i.d sampling over the initial model conditioned on $(q, h_q)$ , we can find a valid solution with a constant probability.
+*Proof.* By Definition 4.5, we know:
+1.
+$$P_{\mu}(h_a|q) \geq \delta'_n$$
+2.
+$$\exists s_q \in \mathcal{S}(q) : P_{\mu}(s_q|(q,h_q)) \geq \delta_p'$$
+With $N = TB \ge 10/\delta'_p$ independent samples conditioned on $(q, h_q)$ , the probability of not finding the solution $s_q$ is:
+$$\mathbb{P}(\text{no solution}) = (1 - P_{\mu}(s_q|(q, h_q)))^N \le (1 - \delta_p')^{10/\delta_p'} \le \exp(-10) < 0.01.$$
+Therefore, $\mathbb{P}(\text{finding solution}) > 0.99$ .
+**Theorem 4.6** (Informal Upper Bound on RL Learnability with Hint). If we have a hint $h_q$ for every question $q \in \mathcal{Q}$ (Def. 4.5), then there exists an RL algorithm that can output a policy $\pi_\theta$ such that $\mathbb{E}_{q \sim \mathrm{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \pi_\theta(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99$ with $O(1/\delta_p')$ sampling budget with high probability.
+{17}------------------------------------------------
+This theorem is a direct corollary of the Theorem 5 regarding the bandit setup in (Mei et al., 2022). Because the setup here is relatively simple, we also present a detailed proof for this special case here. We first formalize our setup as follows:
+<span id="page-17-0"></span>**Assumption C.2** (Tabular RL with Hint). We consider the tabular RL setting with softmax policy parameterization. There exists a finite set of possible questions $\mathcal{Q}$ and a finite set of possible solutions $\mathcal{S}$ . For each question $q \in \mathcal{Q}$ , there exists a hint $h_q$ , which is a subset of solutions $h_q \subseteq \mathcal{S}$ .
+The policy is parameterized by a $|S| \times |Q|$ matrix $\theta$ in the following way:
+$$\mu_{\theta}(s|q) = \frac{\exp(\theta_{s,q})}{\sum_{s' \in S} \exp(\theta_{s',q})}$$
+Here the setup is different than the autoregressive setting in our experiments and simplify the model to a tabular setup for the simplicity of analysis. We now restate the assumption on the existence of hint in this setup.
+<span id="page-17-1"></span>**Assumption C.3** (Hint Existence, Formal Version of Definition 4.5). For each question $q \in \mathcal{Q}$ , there exists a hint $h_q \subseteq \mathcal{S}$ such that $\sum_{s \in h_q} P_{\mu}(s|q) \ge \delta'_p$ . Further, there exists a solution $s_q \in \mathcal{S}$ such that $P_{\mu}(s_q|q) \ge \delta'_p \sum_{s \in h_q} P_{\mu}(s|q)$ .
+**RL Algorithm:** We will first sample $\Theta(1/\delta'_p)$ action based on the policy $\mu_\theta$ conditioned on the question q and the hint $h_q$ . Then we will do a one-step policy gradient update on our policy. Noted that here we can reach high reward within one step because the reward function is deterministic.
+**Theorem C.4** (Formal Version of Theorem 4.6). *Under Assumption C.2 and Assumption C.3, running* 1 *steps of policy gradient update with sampling budget* $\Theta(1/\delta'_{v})$ , *the learned policy achieves:*
+$$\mathbb{E}_{q \sim \text{Uniform}(\mathcal{Q})}[\mathbb{P}_{\tau \sim \mu_{\theta}(\cdot|q)}(\tau \in \mathcal{S}(q))] \geq 0.99$$
+with probability 0.99.
+*Proof.* First, by Assumption C.3, for any question q, we have:
+$$\sum_{s \in h_q} P_{\mu}(s|q) \geq \delta_p \quad \text{and} \quad \exists s_q : P_{\mu}(s_q|q) \geq \delta_p' \sum_{s \in h_q} P_{\mu}(s|q)$$
+With sampling budget $N = \Theta(|\mathcal{Q}|/\delta_p')$ , by Lemma C.1 and the union bound, we will find a solution $s_q$ for every question q with probability at least 0.99. Suppose the found set of solutions for question q is $S_q$ and all sampled solutions are $s^{(1)}, \ldots, s^{(N)}$ . Then because
+$$\nabla_{\theta} \log \mu_{\theta}(s|q) = e_s - \sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q)e_{s'}$$
+We have the policy gradient being
+$$\begin{aligned} \text{PG}_{:,q} &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] \nabla_{\theta} \log \mu_{\theta}(s^{(i)}|q) \\ &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] (e_{s^{(i)}} - \sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q) e_{s'}) \\ &= \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q] e_{s^{(i)}} - \left(\frac{1}{N} \sum_{i=1}^{N} \mathbf{1}[s^{(i)} \in S_q]\right) \left(\sum_{s' \in \mathcal{S}} \mu_{\theta}(s'|q) e_{s'}\right). \end{aligned}$$
+We can make two simple observations:
+- 1. For every $s \notin S_a$ , $PG_{s,a} < 0$ .
+- 2. There exists a $s^* \in S_q$ such that $PG_{s^*,q} > 0$ .
+Therefore, consider the updated parameters
+$$\theta'_{s,q} = \theta_{s,q} + \eta PG_{s,q}$$
+If $\eta$ is large enough, we know that $\sum_{s \in S_-} P_{\mu_{\theta'}}(s|q) \ge 0.99$ . This completes the proof.
+{18}------------------------------------------------
+## <span id="page-18-0"></span>D ADDITIONAL EXPERIMENTAL RESULTS
+#### <span id="page-18-1"></span>D.1 ABLATION STUDY WITHOUT HINT
+<span id="page-18-2"></span>Table 6: Ablation without hint on Nemotron-1.5B: Pass@1 (avg@32) on challenging maths benchmarks. "QUESTA-Nemotron-1.5B w/o hint" trains RL on the same data but removes hints from the prompt, while "w/ hint" uses partial-solution hints during training. With hints, the model improves all benchmarks and achieves a +2.82 average gain over w/o hint (63.26 vs. 60.44), on top of the improvements over the base model. The one using hint requires nearly half the number of steps compared to the one not using hint to achieve the same performance.
+| Model | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25 | Avg |
+|------------------------------------------|--------|--------|-------------|----------------|---------|-------|
+| Nemotron-1.5B | 61.77 | 49.50 | 31.56 | 64.62 | 58.23 | 53.14 |
+| QUESTA-Nemotron-1.5B w/o hint (2K step) | 69.48 | 59.79 | 38.85 | 68.05 | 66.04 | 60.44 |
+| QUESTA-Nemotron-1.5B w/ hint (1.1K step) | 69.27 | 60.00 | 37.92 | 69.72 | 68.33 | 61.05 |
+| QUESTA-Nemotron-1.5B w/ hint (2K step) | 72.50 | 62.29 | 41.67 | 70.36 | 69.48 | 63.26 |
+<span id="page-18-3"></span>![](_page_18_Figure_5.jpeg)
+Figure 7: Pass@k comparison on Nemotron-1.5B for RL with vs. without hints. Training with hints consistently dominates across k and avoids the performance drop at larger k seen in standard RL. Hints are used only during training; evaluation uses no hints.
+We ablate the role of hints by training RL on Nemotron-1.5B with and without partial-solution hints. Here, QUESTA-Nemotron-1.5B w/o hint denotes RL on the same data and schedule but with the hint removed from the prompt; QUESTA-Nemotron-1.5B w/h hint uses identical settings except that the partial solution is provided as a hint during training. As summarized in Table 6, removing the hint still improves over the base model (average Pass@1: $53.14 \rightarrow 60.44$ ), but adding the hint yields a
+{19}------------------------------------------------
+further +2.82 average gain ( $60.44 \rightarrow 63.26$ ), with consistent improvements across all five benchmarks. The one using hint requires nearly half the number of steps compared to the one not using hint to achieve the same performance. Note that hints are used only during training; all evaluations are conducted *without* hints.
+Figure 7 compares Pass@k curves. The w/hint model lifts the entire curve across k and avoids the degradation at larger k commonly observed in standard RL, while the w/o hint variant brings smaller gains that taper off as k increases. A possible reason for this phenomenon is that, without hints, extremely difficult problems remain unlearned. Consequently, during reinforcement learning training, the model prioritizes improving performance on problems that have become relatively easier as training progresses. This leads the model to become overly confident, thereby reducing its Pass@k metric. In contrast, when hints are provided, the model still prioritizes learning more difficult problems—this is because such problems can provide effective learning signals.
+#### <span id="page-19-0"></span>D.2 ABLATION STUDY WITH DIFFERENT MODELS
+<span id="page-19-1"></span>Table 7: Performance comparison on DeepScaleR-1.5B: Pass@1 (avg@32) across maths benchmarks. QUESTA consistently improves all tasks and raises the average by +6.50 points.
+| Model | AIME24 | AIME25 | HMMT FEB 25 | Olympiad Bench | BRUMO25 | Avg |
+|------------------------|--------|--------|-------------|----------------|---------|-------|
+| DeepScaleR-1.5B | 40.42 | 31.35 | 19.27 | 52.97 | 37.40 | 36.28 |
+| QUESTA-DeepScaleR-1.5B | 49.16 | 35.94 | 21.77 | 58.69 | 48.33 | 42.78 |
+<span id="page-19-3"></span>Table 8: Performance comparison (Pass@1, averaged over 32 samples) showing the impact of QUESTA across benchmarks in other domains, including general knowledge, logic, and coding tasks. We observe minor cross-domain generalization on all these benchmarks, despite QUESTA being applied exclusively in the maths domain.
+| Model | GPQA Diamond | Zebralogic | Code Contest All | Codeforces | LCB V5 202410-202502 | Avg |
+|------------------------|--------------|------------|------------------|------------|----------------------|-------|
+| DeepScaleR-1.5B | 38.5 | 14.26 | 9.07 | 8.79 | 19.57 | 18.04 |
+| QUESTA-DeepScaleR-1.5B | 39.2 | 14.98 | 10.1 | 8.9 | 20.9 | 18.82 |
+<span id="page-19-2"></span>![](_page_19_Figure_8.jpeg)
+Figure 8: Pass@k on DeepScaleR-1.5B: QUESTA raises the entire curve across k and avoids the large-k drop often seen in standard RL. The increasing gap with k indicates improved sample diversity rather than overconfident collapse. No hints are used at evaluation.
+We next examine model-family transfer by applying QUESTA to DeepScaleR-1.5B. We train for 750 steps on DeepScaleR-1.5B Stage 2 Luo et al. (2025b) on the QUESTA first stage.
+{20}------------------------------------------------
+<span id="page-20-1"></span>![](_page_20_Figure_1.jpeg)
+Figure 9: We conducted training of the DeepScaleR model employing the QUESTA with a dataset comprising 26,000 questions. The average pass rate was calculated from a sample of 8 instances. The initial graph represents the scenario without the incorporation of partial solutions, while the subsequent graph depicts the situation where partial solutions were included. The application of QUESTA significantly diminishes the incidence of unsolved or partially addressed problems within the training dataset. Concurrently, it has come to our attention that our previous method of data curation was not entirely accurate; in fact, the amount of data providing meaningful training signals is less abundant than anticipated, suggesting the potential for further refinement of the dataset.
+As shown in Table 7, QUESTA-DeepScaleR-1.5B improves *every* maths benchmark over the base model, achieved an average improvement of 6%, indicating that the benefits of QUESTA are not tied to a single architecture.
+Pass@k behavior mirrors these gains. In Figure 8, QUESTA-DeepScaleR lifts the entire Pass@k curve across k and avoids the degradation at larger k reported in standard RL settings. The widening gap at larger k suggests improved candidate diversity rather than overfitting to a single trajectory, consistent with our general pass@k analysis in Appendix B.2. Complementing this, Figure 9 shows that on the 26K training set (evaluated *without* hints), mass shifts away from the 0/8-1/8 bins toward higher pass rates, reducing unsolved or partially solved cases. Hints are used only during training and are removed at evaluation time.
+Beyond maths, Table 8 reports out-of-distribution (OOD) results on general knowledge, logic, and coding. QUESTA-DeepScaleR-1.5B achieves small but consistent gains (Avg: 18.04 $\rightarrow$ 18.82; +0.78), suggesting that the improved reasoning patterns transfer modestly beyond the training domain.
+## <span id="page-20-0"></span>D.3 SUPPLEMENTAL EXPERIMENT DETAIL
+![](_page_20_Figure_7.jpeg)
+Figure 10: Training dynamics of QUESTA-Nemotron-1.5B w/o hint.
+{21}------------------------------------------------
+<span id="page-21-0"></span>![](_page_21_Figure_1.jpeg)
+Figure 11: Training dynamics of QUESTA-Nemotron-1.5B-50.
+<span id="page-21-1"></span>![](_page_21_Figure_3.jpeg)
+Figure 12: Training dynamics of QUESTA-Nemotron-1.5B-50-25-0.
+![](_page_21_Figure_5.jpeg)
+Figure 13: Training dynamics of Easy-Nemotron-1.5B.
+![](_page_21_Figure_7.jpeg)
+Figure 14: Training dynamics of Hard-Nemotron-1.5B.
+{22}------------------------------------------------
+![](_page_22_Figure_1.jpeg)
+Figure 15: The training dynamics of QUESTA-DeepScaleR-1.5B. The first and second charts show the changes in the average reward and average inference length of the rollout samples that include all incorrect/correct ones, respectively. The third chart shows the average entropy excluding all incorrect/correct rollout samples.
+![](_page_22_Figure_3.jpeg)
+Figure 16: Training dynamics of QUESTA-Nemotron-1.5B on OpenMathReasoning (Moshkov et al., 2025). The first and second charts show the changes in the average reward and average inference length of the rollout samples that include all incorrect/correct ones, respectively. The third chart shows the average entropy excluding all incorrect/correct rollout samples. Dynamics closely mirror those on OpenR1-Math-220K, with no entropy collapse.
+{23}------------------------------------------------
+![](_page_23_Figure_1.jpeg)
+Figure 17: We compare pass@k curves of RLVR-trained models with and without QUESTA.
+{24}------------------------------------------------
+![](_page_24_Figure_1.jpeg)
+Figure 18: Comparison of RL training dynamics: Training with only hard problems (green) makes progress very slowly due to sparse rewards, while our method with partial solutions (orange) accelerates learning and consistently achieves higher accuracy across training steps.