Learning What Reinforcement Learning Can't: Interleaved Online Fine-Tuning for Hardest Questions

Recent advances in large language model (LLM) reasoning have shown that sophisticated behaviors such as planning and self-reflection can emerge through reinforcement learning (RL). However, despite these successes, RL in its current form remains insufficient to induce capabilities that exceed the limitations of the base model, as it is primarily optimized based on existing knowledge of the model rather than facilitating the acquisition of new information. To address this limitation, we employ supervised fine-tuning (SFT) to learn what RL cannot, which enables the incorporation of new knowledge and reasoning patterns by leveraging high-quality demonstration data. We analyze the training dynamics of RL and SFT for LLM reasoning and find that RL excels at maintaining and improving performance on questions within the model's original capabilities, while SFT is more effective at enabling progress on questions beyond the current scope of the model. Motivated by the complementary strengths of RL and SFT, we introduce a novel training approach, \textbf{ReLIFT} (\textbf{Re}inforcement \textbf{L}earning \textbf{I}nterleaved with Online \textbf{F}ine-\textbf{T}uning). In ReLIFT, the model is primarily trained using RL, but when it encounters challenging questions, high-quality solutions are collected for fine-tuning, and the training process alternates between RL and fine-tuning to enhance the model's reasoning abilities. ReLIFT achieves an average improvement of over +5.2 points across five competition-level benchmarks and one out-of-distribution benchmark compared to other zero-RL models. Furthermore, we demonstrate that ReLIFT outperforms both RL and SFT while using only 13\% of the detailed demonstration data, highlighting its scalability. These results provide compelling evidence that ReLIFT overcomes the fundamental limitations of RL and underscores the significant potential.

LogicPuzzleRL: Cultivating Robust Mathematical Reasoning in LLMs via Reinforcement Learning

Large language models (LLMs) excel at many supervised tasks but often struggle with structured reasoning in unfamiliar settings. This discrepancy suggests that standard fine-tuning pipelines may instill narrow, domain-specific heuristics rather than fostering general-purpose thinking strategies. In this work, we propose a "play to learn" framework that fine-tunes LLMs through reinforcement learning on a suite of seven custom logic puzzles, each designed to cultivate distinct reasoning skills such as constraint propagation, spatial consistency, and symbolic deduction. Using a reinforcement learning setup with verifiable rewards, models receive binary feedback based on puzzle correctness, encouraging iterative, hypothesis-driven problem solving. We demonstrate that this training approach significantly improves out-of-distribution performance on a range of mathematical benchmarks, especially for mid-difficulty problems that require multi-step reasoning. Analyses across problem categories and difficulty levels reveal that puzzle training promotes transferable reasoning routines, strengthening algebraic manipulation, geometric inference, and combinatorial logic, while offering limited gains on rote or highly specialized tasks. These findings show that reinforcement learning over logic puzzles reshapes the internal reasoning of LLMs, enabling more robust and compositional generalization without relying on task-specific symbolic tools.

Let's Verify Math Questions Step by Step

Large Language Models (LLMs) have recently achieved remarkable progress in mathematical reasoning. To enable such capabilities, many existing works distill strong reasoning models into long chains of thought or design algorithms to construct high-quality math QA data for training. However, these efforts primarily focus on generating correct reasoning paths and answers, while largely overlooking the validity of the questions themselves. In this work, we propose Math Question Verification (MathQ-Verify), a novel five-stage pipeline designed to rigorously filter ill-posed or under-specified math problems. MathQ-Verify first performs format-level validation to remove redundant instructions and ensure that each question is syntactically well-formed. It then formalizes each question, decomposes it into atomic conditions, and verifies them against mathematical definitions. Next, it detects logical contradictions among these conditions, followed by a goal-oriented completeness check to ensure the question provides sufficient information for solving. To evaluate this task, we use existing benchmarks along with an additional dataset we construct, containing 2,147 math questions with diverse error types, each manually double-validated. Experiments show that MathQ-Verify achieves state-of-the-art performance across multiple benchmarks, improving the F1 score by up to 25 percentage points over the direct verification baseline. It further attains approximately 90% precision and 63% recall through a lightweight model voting scheme. MathQ-Verify offers a scalable and accurate solution for curating reliable mathematical datasets, reducing label noise and avoiding unnecessary computation on invalid questions. Our code and data are available at https://github.com/scuuy/MathQ-Verify.