TAPO: Translation Augmented Policy Optimization for Multilingual Mathematical Reasoning

Recent advances in Reinforcement Post-Training (RPT) have substantially improved the mathematical reasoning capabilities of large language models (LLMs), but it remains unclear how well such gains transfer across languages. In this work, we study cross-lingual mathematical reasoning transfer in English-centric LLMs under controlled multilingual evaluation. We first evaluate a range of English-centric reasoning models on multilingual mathematical reasoning benchmarks and find that transfer is highly variable across initial model choice, target language, and training paradigm. Through controlled comparative experiments, we further find that stronger English-centric initialization does not necessarily lead to stronger relative transfer efficiency, even when it yields higher multilingual reasoning accuracy. Finally, we show that the largest improvement comes from moving beyond English-only supervision: adding a single parallel language yields a substantial gain, while further gains from additional languages are smaller but consistent. Overall, our results highlight the need to evaluate multilingual mathematical reasoning directly, using both absolute performance and relative transfer-efficiency metrics.

While Large Reasoning Models (LRMs) show strong performance in English, they often fail to reason in the language of the query, a phenomenon known as language collapse. Existing RL-based fixes typically add a binary language fidelity reward to the accuracy objective, yet still incur trade-off in accuracy, mid-trace code-switching, and excessive token usage. In this work, we propose AdaMame, a two-stage training recipe for multilingual mathematical reasoning that addresses these limitations by adaptively aligning the reasoning language to the query language without compromising accuracy. The first SFT stage fine-tunes on non-MT reasoning traces across five languages to establish multilingual reasoning capability. In the subsequent RL stage, we introduce AdaMame-GRPO, an adaptation of Group Relative Policy Optimization (GRPO) in which a query-conditioned alignment factor grows progressively during training, guiding the model to first explore diverse reasoning languages before exploiting reasoning in the query language. Evaluated across two benchmarks, two LRMs, and 12 languages, AdaMame-GRPO achieves Pareto-optimal performance across reasoning accuracy, language fidelity, and token efficiency over all baselines, with the strongest gains on out-of-domain, lower-resource languages.

Reinforcement learning has proven effective for enhancing multi-step reasoning in large language models (LLMs), yet its benefits have not fully translated to multilingual contexts. Existing methods struggle with a fundamental trade-off: prioritizing input-language consistency severely hampers reasoning quality, while prioritizing reasoning often leads to unintended language drift toward English. We address this challenge with LANG, a novel framework that leverages language-conditioned hints to guide exploration in non-English reasoning tasks. Our method incorporates two key mechanisms to prevent dependency on these hints: a progressive decay schedule that gradually withdraws scaffolding, and a language-adaptive switch that tailors learning horizons to specific language difficulties. Empirical results on challenging multilingual mathematical benchmarks reveal that LANG substantially enhances reasoning performance without compromising language consistency. Moreover, we show that our framework generalizes beyond mathematics, fostering more consistent language alignment across model layers