mlsanityData to 7 Oct 2026

Milestones over Outcome: Unlocking Geometric Reasoning with Sub-Goal Verifiable Reward

arXiv 2601.05073 ↗First posted 8 Jan 2026cs.LGcited by 4

On the map: LLM reasoning → Spatial and embodied reasoning

also in 1 more
Abstract (arXiv)

Multimodal Large Language Models (MLLMs) struggle with complex geometric reasoning, largely because "black box" outcome-based supervision fails to distinguish between lucky guesses and rigorous deduction. To address this, we introduce a paradigm shift towards subgoal-level evaluation and learning. We first construct GeoGoal, a benchmark synthesized via a rigorous formal verification data engine, which converts abstract proofs into verifiable numeric subgoals. This structure reveals a critical divergence between reasoning quality and outcome accuracy. Leveraging this, we propose the Sub-Goal Verifiable Reward (SGVR) framework, which replaces sparse signals with dense rewards based on the Skeleton Rate. Experiments demonstrate that SGVR not only enhances geometric performance (+9.7%) but also exhibits strong generalization, transferring gains to general math (+8.0%) and other general reasoning tasks (+2.8%), demonstrating broad applicability across diverse domains.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning. Builds on 38 of its 65 references that are in the map. First posted on arXiv 8 Jan 2026.

Builds on

38 of 65 references.

Closest ancestors · 5

Same direction, newest first

  1. 2025GeoBench: Rethinking Multimodal Geometric Problem-Solving via Hierarchical Evaluation30 Dec
  2. 2025TrustGeoGen: Formal-Verified Data Engine for Trustworthy Multi-modal Geometric Problem Solving22 Apr
  3. 2025GeoSense: Evaluating Identification and Application of Geometric Principles in Multimodal Reasoning17 Apr
  4. 2023GeomVerse: A Systematic Evaluation of Large Models for Geometric Reasoning19 Dec
  5. 2023A Multi-Modal Neural Geometric Solver with Textual Clauses Parsed from Diagram22 Feb

Other references · 33

  1. 2023GPT-4 Technical Reportcited by 18,132
  2. 2017Proximal Policy Optimization Algorithmscited by 14,441
  3. 2023LLaMA: Open and Efficient Foundation Language Modelscited by 13,958
  4. 2023Llama 2: Open Foundation and Fine-Tuned Chat Modelscited by 12,865
  5. 2025DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learningcited by 6,146
  6. 2025Qwen2.5-VL Technical Reportcited by 5,418
  7. 2024GPT-4o System Cardcited by 4,467
  8. 2025Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilitiescited by 3,655
  9. 2023Evaluating Object Hallucination in Large Vision-Language Modelscited by 1,696
  10. 2023MME: A Comprehensive Evaluation Benchmark for Multimodal Large Language Modelscited by 1,624
  11. 2024OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problemscited by 1,099
  12. 2022Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Taskscited by 1,057
21 more
  1. 2022Solving math word problems with process- and outcome-based feedbackcited by 871
  2. 2023Gemini: A Family of Highly Capable Multimodal Modelscited by 810
  3. 2023Multimodal Chain-of-Thought Reasoning in Language Modelscited by 740
  4. 2023HallusionBench: An Advanced Diagnostic Suite for Entangled Language Hallucination and Visual Illusion in Large Vision-Language Modelscited by 572
  5. 2023Textbooks Are All You Needcited by 561
  6. 2023Textbooks Are All You Need II: phi-1.5 technical reportcited by 559
  7. 2024DeepSeek-VL2: Mixture-of-Experts Vision-Language Models for Advanced Multimodal Understandingcited by 559
  8. 2021Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoningcited by 376
  9. 2025Accelerating scientific discovery with Co-Scientistcited by 353
  10. 2025MM-Eureka: Exploring the Frontiers of Multimodal Reasoning with Rule-based Reinforcement Learningcited by 294
  11. 2024LiveBench: A Challenging, Contamination-Limited LLM Benchmarkcited by 242
  12. 2025MME-CoT: Benchmarking Chain-of-Thought in Large Multimodal Models for Reasoning Quality, Robustness, and Efficiencycited by 123
  13. 2024MAVIS: Mathematical Visual Instruction Tuning with an Automatic Data Enginecited by 111
  14. 2025VisuLogic: A Benchmark for Evaluating Visual Reasoning in Multi-modal Large Language Modelscited by 102
  15. 2025Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2cited by 95
  16. 2024OpenRLHF: An Easy-to-use, Scalable and High-performance RLHF Frameworkcited by 70
  17. 2025InternAgent: When Agent Becomes the Scientist -- Building Closed-Loop System from Hypothesis to Verificationcited by 41
  18. 2024DocGenome: An Open Large-scale Scientific Document Benchmark for Training and Testing Multi-modal Large Language Modelscited by 34
  19. 2025DeepTheorem: Advancing LLM Reasoning for Theorem Proving Through Natural Language and Reinforcement Learningcited by 23
  20. 2023StructChart: On the Schema, Metric, and Augmentation for Visual Chart Understandingcited by 18
  21. 2024Newclid: A User-Friendly Replacement for AlphaGeometrycited by 9

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 4.

4 papers.

  1. 2026PAGER: Bridging the Semantic-Execution Gap in Point-Precise Geometric GUI Controlcited by 1
  2. 2026MIRROR: Learning from the Other View for Multi-Modal Reasoningcited by 1
  3. 2026Draw2Think: Harnessing Geometry Reasoning through Constraint Engine Interactioncited by 0
  4. 2026Reactivating Test-Time Scaling for Plane Geometry Problem Solvingcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026GeoChallenge: A Multi-Answer Multiple-Choice Benchmark for Geometric Reasoning with Diagrams
  2. 2026Learning Where Outcomes Change:Credit-Addressable Reasoning for Multimodal Geometry
  3. 2024Beyond Lines and Circles: Unveiling the Geometric Reasoning Gap in Large Language Models
  4. 2025Geoint-R1: Formalizing Multimodal Geometric Reasoning with Dynamic Auxiliary Constructions
  5. 2025Reliable Proof Generation with LLMs via Analogical Retrieval and Symbolic Verification: A Case Study in Euclidean Geometry
  6. 2026When and Why Does Unsupervised RL Succeed in Mathematical Reasoning? A Manifold Envelopment Perspective
  7. 2025Generalizable Geometric Image Caption Synthesis
  8. 2025GeoThought: A Dataset for Enhancing Mathematical Geometry Reasoning in Vision-Language Models