Cross-lingual Self-Consistency for Multilingual Reasoning with Language Models

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

As LLMs reduce English-centric bias, a surprising trend emerges: non-English responses sometimes outperform English on reasoning tasks. We hypothesize that language functions as a latent variable that structurally modulates the model's internal inference pathways, rather than merely serving as an output medium. To test this, we conducted a Polyglot Thinking Experiment, in which models were prompted to solve identical problems under language-constrained and language-unconstrained conditions. Results show that non-English responses often achieve higher accuracy, and the best performance frequently occur when language is unconstrained, suggesting that multilinguality broadens the model's latent reasoning space. Based on this insight, we propose polyGRPO (Polyglot Group Relative Policy Optimization), an RL framework that treats language variation as an implicit exploration signal. It generates polyglot preference data online under language-constrained and unconstrained conditions, optimizing the policy with respect to both answer accuracy and reasoning structure. Trained on only 18.1K multilingual math problems without chain-of-thought annotations, polyGRPO improves the base model (Qwen2.5-7B-Instruct) by 6.72% absolute accuracy on four English reasoning testset and 6.89% in their multilingual benchmark. Remarkably, it is the only method that surpasses the base LLM on English commonsense reasoning task (4.9%), despite being trained solely on math data-highlighting its strong cross-task generalization. Further analysis reveals that treating language as a latent variable expands the model's latent reasoning space, yielding consistent and generalizable improvements in reasoning performance.

Reasoning language models (RLMs) achieve strong performance on complex reasoning tasks, but still exhibit substantial multilingual reasoning gaps, largely due to language-understanding failures in non-English inputs. English translation can mitigate these failures by expressing non-English inputs in a form that RLMs can more reliably interpret, yet translating every input is unnecessary when the model can reason reliably from the original query. To address this challenge, we propose Luar, a Language Understanding Boundary-aware Reinforcement Learning framework that trains RLMs to selectively invoke translation when direct understanding is unreliable. Luar trains the model to choose between solving the original input directly and reasoning over its English translation, encouraging translation only when translator-augmented reasoning is expected to substantially outperform direct reasoning. Across multilingual reasoning benchmarks, Luar outperforms standard GRPO and other training-based baselines, with particularly large gains on low-resource languages. Further analysis shows that Luar avoids unnecessary translation in cases where direct reasoning is sufficient, while extending its translator-call behavior to unseen low-resource languages. Together, our work suggests a selective approach to multilingual reasoning: RLMs can learn to invoke translation only when their direct understanding is unreliable. The project will be made publicly available at https://github.com/deokhk/LUAR