# TD-Grokking: Learning from Zero-Reward Problems by Training-Time Decomposition

arXiv: https://arxiv.org/abs/2606.09883 · HTML: https://mlsanity.com/paper/2606.09883/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 45 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 26 of 26)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [What Disease does this Patient Have? A Large-scale Open Domain Question Answering Dataset from Medical Exams](https://mlsanity.com/paper/2009.13081/) (2020)
- [PubMedQA: A Dataset for Biomedical Research Question Answering](https://mlsanity.com/paper/1909.06146/) (2019)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Measuring and Narrowing the Compositionality Gap in Language Models](https://mlsanity.com/paper/2210.03350/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Plan-and-Solve Prompting: Improving Zero-Shot Chain-of-Thought Reasoning by Large Language Models](https://mlsanity.com/paper/2305.04091/) (2023)
- [Decomposed Prompting: A Modular Approach for Solving Complex Tasks](https://mlsanity.com/paper/2210.02406/) (2022)
- [MedMCQA : A Large-scale Multi-Subject Multi-Choice Dataset for Medical domain Question Answering](https://mlsanity.com/paper/2203.14371/) (2022)
- [Improve Mathematical Reasoning in Language Models by Automated Process Supervision](https://mlsanity.com/paper/2406.06592/) (2024)
- [A Diverse Corpus for Evaluating and Developing English Math Word Problem Solvers](https://mlsanity.com/paper/2106.15772/) (2021)
- [ProcessBench: Identifying Process Errors in Mathematical Reasoning](https://mlsanity.com/paper/2412.06559/) (2024)
- [Benchmarking Large Language Models on Answering and Explaining Challenging Medical Questions](https://mlsanity.com/paper/2402.18060/) (2024)
- [Scaf-GRPO: Scaffolded Group Relative Policy Optimization for Enhancing LLM Reasoning](https://mlsanity.com/paper/2510.19807/) (2025)
- [QuestA: Expanding Reasoning Capacity in LLMs via Question Augmentation](https://mlsanity.com/paper/2507.13266/) (2025)
- [Nudging the Boundaries of LLM Reasoning](https://mlsanity.com/paper/2509.25666/) (2025)
- [RL Grokking Recipe: How Does RL Unlock and Transfer New Algorithms in LLMs?](https://mlsanity.com/paper/2509.21016/) (2025)
- [Self-Hinting Language Models Enhance Reinforcement Learning](https://mlsanity.com/paper/2602.03143/) (2026)
- [Learning to Hint for Reinforcement Learning](https://mlsanity.com/paper/2604.00698/) (2026)

## Abstract (arXiv)

Large language models (LLMs) have made remarkable progress in reasoning tasks, largely driven by post-training paradigms, especially reinforcement learning with verifiable rewards (RLVR). However, a critical bottleneck persists: RLVR fails on highly challenging zero-reward problems, where all sampled reasoning trajectories yield uniformly failed outcomes, providing no optimization signal to drive model improvement. Prior efforts to address this limitation, such as dense process supervision, partial reward assignment, or prefix-guided exploration, suffer from inherent task constraints or do not fully equip the policy model with the capabilities necessary to solve the original intractable problems. To address this, we propose TD-Grokking, a training-time decomposition framework for zero-reward problems. It recursively decomposes intractable root problems into self-contained, verifiable subproblems, forming hierarchical trees where solvable leaves provide non-zero rewards. Evaluations on mathematical and medical tasks show that TD-Grokking outperforms vanilla GRPO as well as all baseline approaches. Together with detailed analysis, these results confirm that training-time decomposition effectively converts zero-reward examples into usable training signals, enabling consistent performance gains. Our code and datasets are available at https://anonymous.4open.science/r/TD-Grokking-6567/.