# IRIS: Interleaved Reinforcement with Incremental Staged Curriculum for Cross-Lingual Mathematical Reasoning

arXiv: https://arxiv.org/abs/2604.24114 · HTML: https://mlsanity.com/paper/2604.24114/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 22 of its 41 references that are in the map.
- First posted on arXiv 27 Apr 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 22 of 22)

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- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [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)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Multimodal Chain-of-Thought Reasoning in Language Models](https://mlsanity.com/paper/2302.00923/) (2023)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [Generative Verifiers: Reward Modeling as Next-Token Prediction](https://mlsanity.com/paper/2408.15240/) (2024)
- [rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking](https://mlsanity.com/paper/2501.04519/) (2025)
- [InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning](https://mlsanity.com/paper/2402.06332/) (2024)
- [M3Exam: A Multilingual, Multimodal, Multilevel Benchmark for Examining Large Language Models](https://mlsanity.com/paper/2306.05179/) (2023)
- [Multilingual Instruction Tuning With Just a Pinch of Multilinguality](https://mlsanity.com/paper/2401.01854/) (2024)
- [Self-rewarding correction for mathematical reasoning](https://mlsanity.com/paper/2502.19613/) (2025)
- [SOS-1K: A Fine-grained Suicide Risk Classification Dataset for Chinese Social Media Analysis](https://mlsanity.com/paper/2404.12659/) (2024)
- [Logic Contrastive Reasoning with Lightweight Large Language Model for Math Word Problems](https://mlsanity.com/paper/2409.00131/) (2024)
- [XL-Suite: Cross-Lingual Synthetic Training and Evaluation Data for Open-Ended Generation](https://mlsanity.com/paper/2503.22973/) (2025)

## Abstract (arXiv)

Curriculum learning helps language models tackle complex reasoning by gradually increasing task difficulty. However, it often fails to generate consistent step-by-step reasoning, especially in multilingual and low-resource settings where cross-lingual transfer from English to Indian languages remains limited. We propose IRIS: Interleaved Reinforcement with Incremental Staged Curriculum, a two-axis framework that combines Supervised Fine-Tuning on progressively harder problems (vertical axis) with Reverse Curriculum Reinforcement Learning to reduce reliance on step-by-step guidance (horizontal axis). We design a composite reward combining correctness, step-wise alignment, continuity, and numeric incentives, optimized via Group Relative Policy Optimization (GRPO). We release CL-Math, a dataset of 29k problems with step-level annotations in English, Hindi, and Marathi. Across standard benchmarks and curated multilingual test sets, IRIS consistently improves performance, with strong results on math reasoning tasks and substantial gains in low-resource and bilingual settings, alongside modest improvements in high-resource languages.