The Two-Hump Problem: Bridging the Difficulty Gap in Mathematical Reinforcement Learning

• URL: http://arxiv.org/abs/2606.21611v1
• Date: Fri, 19 Jun 2026 17:14:47 GMT
• Title: The Two-Hump Problem: Bridging the Difficulty Gap in Mathematical Reinforcement Learning
• Abstract summary: Andrews-Curtis (AC) conjecture is an illustrative example of such problems.<n>We identify a critical structural barrier in the AC landscape: a "Two-Hump" distribution, where problem instances are either trivially solvable or effectively impossible.<n>We release new comprehensive benchmark datasets including AC-19 (125,192 AC-trivial presentations of varying difficulty with length at most 19) and AC-1M (1,136,154 hard AC-trivial presentations of length at most 30), the first large-scale, publicly available datasets of this kind.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Mathematical search problems present a unique challenge for Reinforcement Learning (RL) due to vast search spaces and sparse rewards. In previous works, the Andrews-Curtis (AC) conjecture was established as an illustrative example of such problems. In this work, we identify a critical structural barrier in the AC landscape: a "Two-Hump" distribution, where problem instances are either trivially solvable or effectively impossible, with a scarcity of intermediate "hard-but-solvable" instances required for effective learning. We tackle this challenge through two primary avenues: novel data generation techniques to populate the difficulty gap, and significant algorithmic enhancements including the introduction of supermoves and Transformer-based architectures. We demonstrate substantial performance improvements over previous baselines, and release new comprehensive benchmark datasets including AC-19 (125,192 AC-trivial presentations of varying difficulty with length at most 19) and AC-1M (1,136,154 hard AC-trivial presentations of length at most 30), the first large-scale, publicly available datasets of this kind.

Related papers

• $S^3$-R1: Learning to Retrieve and Answer Step-by-Step with Synthetic Data [2.003941363902692]
  We introduce S3-R1 (Synthetic data and stabilized Search R1), a framework that couples a datacentric approach with denser learning signals.<n>We first develop a synthetic generation and pipeline that derives diverse, multi-hop questions from existing documents.<n>We then pair this expanded training set with a reward structure that evaluates both intermediate search quality and the correctness final answer.
  arXiv  Detail & Related papers  (2026-05-02T05:01:05Z)
• QueST: Incentivizing LLMs to Generate Difficult Problems [77.75835742350644]
  Large Language Models have achieved strong performance on reasoning tasks, solving competition-level coding and math problems.<n>Existing competitive coding datasets contain only thousands to tens of thousands of problems.<n>We propose QueST, a novel framework which combines difficulty-aware graph sampling and difficulty-aware rejection fine-tuning.
  arXiv  Detail & Related papers  (2025-10-20T16:29:53Z)
• ScaleDiff: Scaling Difficult Problems for Advanced Mathematical Reasoning [51.946959481392064]
  Large Reasoning Models (LRMs) have shown impressive capabilities in complex problem-solving.<n>We propose ScaleDiff, a pipeline designed to scale the creation of difficult problems.<n>We show that our pipeline can effectively transfer advanced reasoning capabilities without relying on larger, more expensive teacher models.
  arXiv  Detail & Related papers  (2025-09-25T12:22:44Z)
• Staying in the Sweet Spot: Responsive Reasoning Evolution via Capability-Adaptive Hint Scaffolding [59.60915947702282]
  Reinforcement learning with verifiable rewards (RLVR) has achieved remarkable success in enhancing the reasoning capabilities of large language models (LLMs)<n>Existing RLVR methods often suffer from exploration inefficiency due to mismatches between the training data's difficulty and the model's capability.<n>We propose SEELE, a novel supervision-aided RLVR framework that dynamically adjusts problem difficulty to stay within the high-efficiency region.
  arXiv  Detail & Related papers  (2025-09-08T17:36:21Z)
• Unlocking the Potential of Difficulty Prior in RL-based Multimodal Reasoning [69.64809103333839]
  We investigate how explicitly modeling problem's difficulty prior information shapes the effectiveness of reinforcement learning based fine-tuning for multimodal reasoning.<n>Our approach demonstrates significant performances across various multi-modal mathematical reasoning benchmarks with only 2K+0.6K two-stage training data.
  arXiv  Detail & Related papers  (2025-05-19T15:43:10Z)
• Neumann Series-based Neural Operator for Solving Inverse Medium Problem [5.980803988596087]
  This study introduces a novel approach by integrating a Neumann series structure within a neural network framework. Experiments demonstrate that our methodology not only accelerates computations but also significantly enhances generalization performance.
  arXiv  Detail & Related papers  (2024-09-14T16:54:30Z)
• OrCo: Towards Better Generalization via Orthogonality and Contrast for Few-Shot Class-Incremental Learning [57.43911113915546]
  Few-Shot Class-Incremental Learning (FSCIL) introduces a paradigm in which the problem space expands with limited data. FSCIL methods inherently face the challenge of catastrophic forgetting as data arrives incrementally. We propose the OrCo framework built on two core principles: features' orthogonality in the representation space, and contrastive learning.
  arXiv  Detail & Related papers  (2024-03-27T13:30:48Z)
• Features for the 0-1 knapsack problem based on inclusionwise maximal solutions [0.7734726150561086]
  We formulate several new computationally challenging problems related to the IMSs of arbitrary 0-1 knapsack problem instances. We derive a set of 14 computationally expensive features, which we calculate for two large datasets on a supercomputer in approximately 540 CPU-hours. We show that the proposed features contain important information related to the empirical hardness of a problem instance.
  arXiv  Detail & Related papers  (2022-11-16T12:48:35Z)
• Few-shot Partial Multi-view Learning [103.33865779721458]
  We propose a new task called few-shot partial multi-view learning. It focuses on overcoming the negative impact of the view-missing issue in the low-data regime. We conduct extensive experiments to evaluate our method.
  arXiv  Detail & Related papers  (2021-05-05T13:34:43Z)
• Graph Minors Meet Machine Learning: the Power of Obstructions [0.90238471756546]
  We show the utility of using obstructions for training neural networks. Experiments show that training with obstructions results in a huge reduction in number of iterations needed for convergence.
  arXiv  Detail & Related papers  (2020-06-08T15:40:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.