RL on binary rewards boosts LLM factual recall by ~27% relative across models by redistributing probability mass to latent correct answers rather than acquiring new knowledge.
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Scaling Relationship on Learning Mathematical Reasoning with Large Language Models
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abstract
Mathematical reasoning is a challenging task for large language models (LLMs), while the scaling relationship of it with respect to LLM capacity is under-explored. In this paper, we investigate how the pre-training loss, supervised data amount, and augmented data amount influence the reasoning performances of a supervised LLM. We find that pre-training loss is a better indicator of the model's performance than the model's parameter count. We apply supervised fine-tuning (SFT) with different amounts of supervised data and empirically find a log-linear relation between data amount and model performance, and we find better models improve less with enlarged supervised datasets. To augment more data samples for improving model performances without any human effort, we propose to apply Rejection sampling Fine-Tuning (RFT). RFT uses supervised models to generate and collect correct reasoning paths as augmented fine-tuning datasets. We find with augmented samples containing more distinct reasoning paths, RFT improves mathematical reasoning performance more for LLMs. We also find RFT brings more improvement for less performant LLMs. Furthermore, we combine rejection samples from multiple models which push LLaMA-7B to an accuracy of 49.3\% on GSM8K which outperforms the supervised fine-tuning (SFT) accuracy of 35.9\% significantly.
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representative citing papers
POPO uses bounded importance sampling on positive rollouts and a siamese policy network to achieve implicit negative gradients and stable optimization, matching or exceeding GRPO on math benchmarks such as 36.67% on AIME 2025.
Reference-sampled weighted SFT with prompt-normalized Boltzmann weights induces the same policy as fixed-reference KL-regularized RLVR, with BOLT as the estimator and a finite one-shot error decomposition separating coverage, variance, and other terms.
Small 7B reasoning models were fine-tuned on synthetic and curated QFT problems using RL and SFT, yielding performance gains, error analysis, and public release of data and traces.
Self-CriTeach lets an LLM generate symbolic domains that supply both chain-of-thought training data and structured rewards, producing a planning-enhanced model with better success rates and generalization.
High-Entropy Sum (HES) selects high-quality reasoning data for LLMs by summing entropy of the top highest-entropy tokens, matching full-dataset performance with top 20% in SFT and outperforming baselines in RFT and RL.
Step Rejection Fine-Tuning masks loss on erroneous steps identified by a critic LLM in unresolved trajectories, raising SWE-bench Verified resolution rate by 3.7% to 32.2% versus 2.4% for trajectory-level rejection.
CauSim turns scarce causal reasoning labels into scalable supervised data by having LLMs incrementally construct complex executable structural causal models.
RL for LLM reasoning acts as sparse policy selection at high-entropy tokens already present in the base model, enabling ReasonMaxxer—an efficient contrastive method that recovers most RL gains at three orders of magnitude lower cost.
S^3-R1 generates synthetic intermediate-difficulty multi-hop questions and applies dense rewards for search quality plus answer correctness, yielding up to 10% better out-of-domain generalization than baselines.
Reinforcement fine-tuning calibration makes LLM distillability adjustable, allowing optimized knowledge transfer or model IP safeguards via a combined task-KL-calibration objective.
An empirical evaluation of 22 agentic frameworks on BBH, GSM8K, and ARC benchmarks shows stable performance in 12 frameworks but highlights orchestration failures and weaker mathematical reasoning.
Lightning OPD is an offline on-policy distillation method that matches standard OPD performance at 4x efficiency by enforcing teacher consistency between SFT and distillation phases.
AIRA₂ improves AI research agents via asynchronous multi-GPU workers, hidden consistent evaluation, and interactive ReAct agents, reaching 81.5-83.1% percentile rank on MLE-bench-30 and exceeding human SOTA on 6 of 20 AIRS-Bench tasks.
SAM 3D reconstructs 3D objects from single images with geometry, texture, and pose using human-model annotated data at scale and synthetic-to-real training, achieving 5:1 human preference wins.
LightReasoner distills supervision signals from SLM-LLM behavioral divergence to improve LLM reasoning on math benchmarks with up to 28.1% accuracy gains and 90-99% reductions in resources.
SIE framework automatically constructs scalable, verifiable reasoning environments from structured data, improving in-domain performance and enabling generalization to out-of-domain math and logic tasks.
ARPO adds entropy-based adaptive rollouts and stepwise advantage attribution to RL for LLM agents, outperforming prior trajectory-level methods on 13 benchmarks with half the tool budget.
VLA-RL applies online RL to pretrained VLAs, yielding a 4.5% gain over strong baselines on 40 LIBERO manipulation tasks and matching commercial models like π₀-FAST.
Entropy minimization on self-generated outputs elicits strong reasoning in pretrained LLMs, matching or exceeding supervised RL methods on benchmarks.
Search-o1 integrates agentic retrieval-augmented generation and a Reason-in-Documents module into large reasoning models to dynamically supply missing knowledge and improve performance on complex science, math, coding, and QA tasks.
Process advantage verifiers trained to predict step-level progress under a distinct prover policy improve LLM reasoning accuracy by over 8% and sample efficiency by 5-6x over outcome reward models.
Step-DPO performs preference optimization on individual reasoning steps rather than complete answers, producing nearly 3% accuracy gains on MATH for 70B+ parameter models with 10K preference pairs.
StarCoder2-15B matches or beats CodeLlama-34B on code tasks despite being smaller, and StarCoder2-3B outperforms prior 15B models, with open weights and exact training data identifiers released.
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