Simon Park

Jui-Hui Chung, Ziyang Cai, Zihao Li et al.

We introduce Goedel-Architect, an agentic framework for formal theorem proving in Lean 4 centered on blueprint generation and refinement. A blueprint is a dependency graph of definitions and lemmas that builds up to the main theorem. First, Goedel-Architect generates a blueprint of formally stated definitions and lemmas, along with declared dependencies. This blueprint is optionally guided by a natural language proof. Then, a tool-equipped Lean prover component closes each open lemma node in parallel using relevant dependencies. Failed lemmas in turn drive refinement of the global blueprint. This strategy contrasts with other mainstream approaches which use recursive lemma decomposition, and can inefficiently loop on dead-end strategies. Using the open-weight DeepSeek-V4-Flash (284B-A13B) as the backbone, Goedel-Architect attains 99.2% pass@1 on MiniF2F-test and 75.6% pass@1 on PutnamBench. With an optional natural-language proof seeding the initial blueprint on the harder problems, we additionally close the remaining two MiniF2F-test problems (reaching 100%), lift PutnamBench to 88.8% (597/672), and solve 4/6 on IMO 2025, 11/12 on Putnam 2025, and 3/6 on USAMO 2026. This represents state-of-the-art performance for an open-source pipeline at a price point up to 500x less than comparable open-source pipelines.

Vedant Shah, Dingli Yu, Kaifeng Lyu et al. · mila, tsinghua

Current LLM training positions mathematical reasoning as a core capability. With publicly available sources fully tapped, there is unmet demand for diverse and challenging math questions. Relying solely on human experts is both time-consuming and costly, while LLM-generated questions often lack the requisite diversity and difficulty. We present a design framework that combines the strengths of LLMs with a human-in-the-loop approach to generate a diverse array of challenging math questions. We leverage LLM metacognition skills [Didolkar et al., 2024] of a strong LLM to extract core "skills" from existing math datasets. These skills serve as the basis for generating novel and difficult questions by prompting the LLM with random pairs of core skills. The use of two different skills within each question makes finding such questions an "out of distribution" task for both LLMs and humans. Our pipeline employs LLMs to iteratively generate and refine questions and solutions through multiturn prompting. Human annotators then verify and further refine the questions, with their efficiency enhanced via further LLM interactions. Applying this pipeline on skills extracted from the MATH dataset [Hendrycks et al., 2021] resulted in MATH$^2$ - a dataset of higher-quality math questions, as evidenced by: (a) Lower performance of all models on MATH$^2$ than on MATH (b) Higher performance on MATH when using MATH$^2$ questions as in-context examples. Although focused on mathematics, our methodology seems applicable to other domains requiring structured reasoning, and potentially as a component of scalable oversight. Also of interest is a striking relationship observed between models' performance on the new dataset: the success rate on MATH$^2$ is the square on MATH, suggesting that successfully solving the question in MATH$^2$ requires a nontrivial combination of two distinct math skills.

Simran Kaur, Simon Park, Anirudh Goyal et al. · mila

We introduce Instruct-SkillMix, an automated approach for creating diverse, high quality SFT data for instruction-following. The pipeline involves two stages, each leveraging an existing powerful LLM: (1) Skill extraction: uses the LLM to extract core "skills" for instruction-following by directly prompting the model. This is inspired by ``LLM metacognition'' of Didolkar et al. (2024); (2) Data generation: uses the powerful LLM to generate (instruction, response) data that exhibit a randomly chosen pair of these skills. Here, the use of random skill combinations promotes diversity and difficulty. The estimated cost of creating the dataset is under $600. Vanilla SFT (i.e., no PPO, DPO, or RL methods) on data generated from Instruct-SkillMix leads to strong gains on instruction following benchmarks such as AlpacaEval 2.0, MT-Bench, and WildBench. With just 4K examples, LLaMA-3-8B-Base achieves 42.76% length-controlled win rate on AlpacaEval 2.0, a level similar to frontier models like Claude 3 Opus and LLaMA-3.1-405B-Instruct. Ablation studies also suggest plausible reasons for why creating open instruction-tuning datasets via naive crowd-sourcing has proved difficult. In our dataset, adding 20% low quality answers (``shirkers'') causes a noticeable degradation in performance. The Instruct-SkillMix pipeline seems flexible and adaptable to other settings.

Simon Park, Simran Kaur, Sanjeev Arora

While reinforcement learning (RL) successfully enhances reasoning in large language models, its role in fostering compositional generalization (the ability to synthesize novel skills from known components) is often conflated with mere length generalization. To this end, we study what RL post-training teaches about skill composition and how the structure of the composition affects the skill transfer. We focus on the Countdown task (given n numbers and a target, form an expression that evaluates to the target) and analyze model solutions as expression trees, where each subtree corresponds to a reusable subtask and thus can be viewed as a ``skill.'' Tracking tree shapes and their success rates over training, we find: (i) out-of-distribution (OOD) generalization to larger n and to unseen tree shapes, indicating compositional reuse of subtasks; (ii) a structure-dependent hierarchy of learnability -- models master shallow balanced trees (workload is balanced between subtasks) before deep unbalanced ones, with persistent fragility on right-heavy structures (even when the composition depth is the same as some left-heavy structures). Our diagnostic reveals what is learned, in what order, and where generalization fails, clarifying how RL-only post-training induces OOD generalization beyond what standard metrics such as pass@k reveal.

Simon Park, Abhishek Panigrahi, Yun Cheng et al. · princeton

Vision Language Models (VLMs) are impressive at visual question answering and image captioning. But they underperform on multi-step visual reasoning -- even compared to LLMs on the same tasks presented in text form -- giving rise to perceptions of modality imbalance or brittleness. Towards a systematic study of such issues, we introduce a synthetic framework for assessing the ability of VLMs to perform algorithmic visual reasoning, comprising three tasks: Table Readout, Grid Navigation, and Visual Analogy. Each has two levels of difficulty, SIMPLE and HARD, and even the SIMPLE versions are difficult for frontier VLMs. We propose strategies for training on the SIMPLE version of tasks that improve performance on the corresponding HARD task, i.e., simple-to-hard (S2H) generalization. This controlled setup, where each task also has an equivalent text-only version, allows a quantification of the modality imbalance and how it is impacted by training strategy. We show that 1) explicit image-to-text conversion is important in promoting S2H generalization on images, by transferring reasoning from text; 2) conversion can be internalized at test time. We also report results of mechanistic study of this phenomenon. We identify measures of gradient alignment that can identify training strategies that promote better S2H generalization. Ablations highlight the importance of chain-of-thought.