Organizations: School of Artificial Intelligence, University of Chinese Academy of Sciences · Institute of Automation, Chinese Academy of Sciences · Wuhan AI Research
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.
Figures & tables
Figure 1: Cross-lingual transfer across open-source reasoning models. The left panel reports average MTI across four benchmarks and eleven languages. The right panel shows average transfer behavior on multilingual MATH500, comparing SFT-tuned and RL-tuned models.
Model
Acc
Off-target
MTI
Qwen2.5-7B-Base
12.00
11.41
-
↬ GRPO on En Data
22.45
3.12
1.95
Qwen2.5-7B-Instruct
22.45
1.43
-
↬ GRPO on En Data
23.51
0.94
1.23
Qwen2.5-Math-7B
12.25
22.59
-
↬ GRPO on En Data
19.37
9.50
2.12
Table 1: The Impact of Initial Model Type. Average Accuracy (%), Off-target rate (%), and MTI across eleven languages on four benchmarks.
Figure 2: Effect of model family under controlled post-training. Multilingual reasoning performance on MATH500. Base denotes the initial model, and +GRPO the model after English-only GRPO.
Figure 3: Effect of model size under controlled English-only post-training. Δ Performance denotes the average accuracy gain over the initial model, averaged across English and unseen languages.
Figure 4: Transfer improves with parallel multilingual supervision. Each panel shows performance as a function of the number of training languages. The red star denotes the English-only baseline ( X=1 ), shown for reference but excluded from the fit. Blue points denote multilingual configurations ( X≥2 ). The solid red curve shows a power-law-like descriptive fit over X≥2 , with the shaded region indicating the 95% confidence interval. The fit summarizes the bilingual-to-multilingual trend; Appendix E.3 provides the marginal-gain analysis.
Budget
Type
Acc
Off-target
MTI
1×
1× En
54.24
0.49
1.16
2×
2× En
55.82
0.58
1.92
2×
1× En + 1× Ru
57.87
0.20
2.50
3×
3× En
57.13
0.42
2.38
3×
1× En + 1× Ru + 1× Fr
58.38
0.24
2.65
Table 2: The Fixed-Budget ablation experiment based on Qwen2.5-7B-Instruct.
Figure 5: Scaling training data based on Qwen2.5-7B-Instruct.
Evaluation
Training
Acc
Off-target
No language control
English-only
57.82
75.76
No language control
English+Ru
58.43
0.36
Language-controlled
English-only
54.20
0.50
Language-controlled
English+Ru
57.90
0.20
Table 3: Prompt ablation. Average accuracy and off-target rate on multilingual MATH500 under two evaluation protocols.
Training Data
Acc
Off-target
MTI
Zh
55.00
0.38
0.86
Zh + Ru
58.13 (+3.13)
0.26 (-0.12)
1.69 (+0.83)
Zh + Ru + Fr
58.76 (+3.76)
0.23 (-0.15)
1.87 (+1.01)
Table 4: Chinese-centric parallel training results on multilingual MATH500.
Figure 6: Parallel vs. non-parallel multilingual supervision. Accuracy differences under parallel and non-parallel data training based on Qwen2.5-7B-Instruct.
Figure 7: The impact of the selected parallel language. Multilingual reasoning performance across different single-language parallel training settings based on Qwen2.5-7B-Instruct.
Appendix figures & tables24 assets
Supplementary material from the paper’s appendix.
Appendix
Model
Initial Model
Size
Training Paradigm
DeepSeek-R1-Distill-Qwen-7B
Qwen2.5-Math-7B-Base
7B
SFT
Open-Reasoner-Zero-7B
Qwen2.5-7B-Base
7B
RL
OpenThinker2-7B
Qwen2.5-7B-Instruct
7B
SFT
OpenThinker3-7B
Qwen2.5-7B-Instruct
7B
SFT
Qwen-2.5-1.5B-SimpleRL-Zoo
Qwen2.5-1.5B-Base
1.5B
RL
Qwen-2.5-7B-SimpleRL-Zoo
Qwen2.5-7B-Base
7B
RL
Appendix
Table 5: The Overview of the Open-source LLMs Used in Observational Study , including their initial model, parameter size, and training paradigm.
Figure 8: Distribution of question difficulty. (a) The 1,000 English questions were utilized in the controlled comparative study and for parallel training. (b) The 1,000 Russian questions for non-parallel training, comprising a separate and non-overlapping set from the questions in (a).
Method
En
Avg
Non-En
Low-res.
Off-tgt.
Base
74.80
50.91
48.52
27.40
0.18
SFT
76.80
50.80
48.20
22.13
4.40
GRPO
79.20
54.24
51.74
29.53
0.49
Appendix
Table 6: Matched SFT-vs.-GRPO comparison on multilingual MATH500. All values are percentages. All models use Qwen2.5-7B-Instruct as the backbone. SFT and GRPO are trained on the same 1K English MATH examples and evaluated under the same multilingual MATH500 protocol. Low-res. denotes average accuracy over bn , sw , and te ; Off-tgt. denotes the off-target language rate.
Setting
En
Non-En
Gap
Low-res.
MTI
English-only
79.20
51.74
27.46
29.53
1.163
+1 parallel
78.40
55.82
22.58
36.27
2.496
+2 parallels
79.00
56.32
22.68
34.67
2.650
+3 parallels
77.80
56.44
21.36
34.73
3.002
+4 parallels
77.20
56.76
20.44
37.13
3.282
+5 parallels
77.40
57.18
20.22
39.33
3.475
Appendix
Table 7: Absolute-transfer metrics for Qwen2.5-7B-Instruct under parallel training. All accuracy and gap values are percentages. Non-En denotes average accuracy over non-English languages; Gap denotes the En–Non-En accuracy gap; Low-res. denotes average accuracy over bn , sw , and te . MTI is reported as an auxiliary relative transfer-efficiency metric.
Fit
R2
MSE
AIC
BIC
Qwen MTI
Linear
0.951
0.0178
-29.523
-29.631
Log
0.974
0.0121
-33.950
-34.058
Power-law
0.977
0.0082
-34.653
-34.762
Saturation
0.982
0.0894
-34.368
-34.530
Llama MTI
Appendix
Table 8: Curve-fitting comparison for MTI in the multilingual regime. Fits are evaluated on configurations with X=2,…,8 . MSE denotes the leave-one-out cross-validation (LOOCV) MSE.
X
Qwen MTI
Qwen Acc.
Llama MTI
Llama Acc.
1
2.005
4.503
3.684
3.341
2
0.414
0.512
0.495
0.862
3
0.015
0.099
0.067
0.004
4
0.019
0.001
0.005
0.015
5
0.016
0.007
0.009
0.000
6
0.009
0.009
0.008
0.000
Appendix
Table 9: Cook’s distance under all-point power-law fits. The power-law curve is fit to all observed points, X=1,…,8 .
Transition
Avg Acc
Non-En
Low-res.
MTI
1→2
+3.70
+4.08
+6.73
+1.333
2→3
+0.50
+0.50
-1.60
+0.154
3→4
+0.00
+0.12
+0.07
+0.352
4→5
+0.20
+0.32
+2.40
+0.280
5→6
+0.40
+0.42
+2.20
+0.193
6→7
+0.40
+0.54
+1.20
+0.059
Appendix
Table 10: Marginal-gain decomposition over consecutive training-language counts. Each row reports the gain from increasing the number of training languages from X to X+1 in the Qwen2.5-7B-Instruct parallel-training experiments.
Metric
1→2
1→8
Fraction
Avg Acc.
+3.70
+5.30
69.8%
Non-En Acc.
+4.08
+6.07
67.2%
Low-res. Acc.
+6.73
+10.93
61.6%
MTI
+1.333
+2.468
54.0%
Appendix
Table 11: Fraction of total gain explained by the first added parallel language. The table compares the gain from English-only to bilingual training ( 1→2 ) with the total gain from English-only to eight-language training ( 1→8 ). Avg Acc. denotes average accuracy over all languages; Non-En Acc. denotes average accuracy over non-English languages; Low-res. Acc. denotes average accuracy over bn , sw , and te .
Group
n
Mean Adv.
W/T/L
Overall
44
+0.995
28 / 8 / 8
Non-English
40
+1.101
28 / 5 / 7
Unseen Non-En
36
+1.213
27 / 5 / 4
Unseen low-res.
12
+1.632
9 / 3 / 0
Added lang. (ru)
4
+0.093
1 / 0 / 3
English
4
-0.063
0 / 3 / 1
Appendix
Table 12: Parallel vs. non-parallel multilingual supervision. The advantage is computed as Accparallel−Accnon-parallel over benchmark-language pairs. W/T/L denotes wins, ties, and losses.
Scope
Pearson r
Spearman ρ
Overall
-0.956
-0.882
MATH500
-0.971
-0.946
AIME24
-0.901
-0.946
AIME25
-0.948
-0.936
GPQA-Diamond
-0.777
-0.727
Appendix
Table 13: Correlation between mean accuracy and relative variance across languages. For each benchmark and language, mean accuracy and CV are computed across open-source models. Negative correlations indicate that lower-accuracy languages tend to have higher relative variance.
Setting
MTI
NGR
RGR
En-only
1.163
0.396
0.732
+1 parallel
2.496
0.963
1.480
+2 parallels
2.650
1.061
1.601
Appendix
Table 14: Robustness of transfer metrics on the main parallel-training comparison. We compare the original MTI with normalized-gain ratio (NGR) and raw-gain ratio (RGR). All three metrics show a large gain from English-only to one-parallel training, followed by a smaller additional gain.
Setting
All
Accbase>20
Accbase>30
Relative-gain MTI
En-only
1.16
1.16
1.23
+1 parallel
2.50
2.42
1.88
+2 parallels
2.65
2.62
2.13
+3 parallels
3.00
3.16
2.58
+4 parallels
3.28
3.25
2.34
Appendix
Table 15: Sensitivity of transfer scores to low-base-accuracy cases. Scores are recomputed on multilingual MATH500 after excluding unseen-language cases with base accuracy below the specified thresholds.
Settings
Accuracy per language
Average
en
es
ru
de
fr
bn
th
sw
zh
ja
te
Acc
Off-target
Qwen2.5-7B-Base
Qwen-Math Template
49.2
31.8
25.2
28.2
30.0
5.8
23.0
2.4
27.0
15.2
1.4
21.7
16.6
Qwen-Instruct Template
50.6
38.0
30.0
33.2
38.4
10.4
26.8
2.4
30.0
27.8
4.4
26.5
15.7
No Template
44.4
38.2
28.2
28.4
35.6
6.0
17.0
0.2
29.2
19.8
1.2
22.6
18.5
Qwen2.5-Math-7B-Base
Appendix
Table 16: The Performance of Base Models with Different Template Settings. Accuracy (%) and Off-target rate (%) across languages for different template settings on multilingual MATH500 benchmark.
Settings
Accuracy per language
Average
en
es
ru
de
fr
bn
th
sw
zh
ja
te
Acc
Off-target
Multilingual MATH500
Qwen2.5-1.5B
19.60
10.60
7.80
1.00
9.80
1.00
3.00
0.00
8.60
2.20
0.00
5.78
21.02
Qwen2.5-Math-7B
56.60
46.40
11.20
33.40
36.80
31.40
28.20
4.20
44.40
25.20
3.20
29.18
17.96
Qwen2.5-7B-Instruct
74.80
69.00
59.60
56.60
62.60
37.60
49.80
18.00
53.00
52.40
26.60
50.91
0.18
Qwen2.5-7B
50.60
38.00
30.00
33.20
38.40
10.40
26.80
2.40
30.00
27.80
4.40
26.55
15.69
Appendix
Table 17: The Performance of Initial Models. Accuracy (%) and Off-target rate (%) across languages for different Initial models.
Setting
(λ1,λ2,λ3)
Accuracy per Language
Avg.
Off-tgt.
MTI
en
es
ru
de
fr
bn
th
sw
zh
ja
te
C1 (Balanced)
(0.8, 0.1, 0.1)
78.4
73.4
66.0
65.6
67.2
48.8
57.4
26.2
57.8
62.0
33.8
57.87
0.20
2.50
C2 (w/o LCR)
(0.9, 0.1, 0.0)
77.8
71.4
66.2
63.4
68.0
44.2
56.4
26.2
59.6
57.8
31.8
56.62
1.24
2.10
C3 (Strong LCR)
(0.7, 0.1, 0.2)
77.2
70.2
65.8
65.8
67.2
43.6
55.4
25.6
60.2
56.2
32.6
56.35
0.38
2.25
C4 (w/o FR)
(0.9, 0.0, 0.1)
75.2
60.8
62.2
56.0
60.2
36.2
43.8
18.0
54.4
31.8
24.8
47.58
0.31
-3.45
C5 (Strong FR)
(0.7, 0.2, 0.1)
78.2
71.6
65.2
62.6
67.8
45.0
60.0
24.8
60.0
60.8
29.2
56.84
0.35
2.23
Appendix
Table 18: Sensitivity analysis of reward hyperparameters. Models are trained on English data with parallel Russian data. (λ1,λ2,λ3) denotes the weights of the three reward components.
Setting
M500
A24
A25
GPQA
Avg.
Off-tgt.
MTI
Base
26.55
1.23
0.42
19.79
12.00
11.41
–
↬ GRPO
52.16
7.10
3.35
27.18
22.45
3.12
1.95
Instruct
50.91
5.93
3.28
29.71
22.45
1.43
–
↬ GRPO
54.24
7.41
3.92
28.47
23.51
0.94
1.23
Math
29.18
4.11
1.88
13.82
12.25
22.59
–
↬ GRPO
45.73
8.84
3.96
18.96
19.37
9.50
2.12
Appendix
Table 19: Impact of model initialization on cross-lingual transfer. All models are initialized from Qwen2.5-7B variants and trained with GRPO on the same English data. We report average accuracy (%) across languages on MATH500 (M500), AIME24 (A24), AIME25 (A25), and GPQA-Diamond (GPQA), together with overall average accuracy, off-target rate (%), and MTI.
Figure 9: The Impact of Different Model Families in Controlled Comparative Study. Multilingual reasoning performance across languages, comparing the influence of model family using Qwen2.5-7B-Instruct and Llama3.1-8B-Instruct as initial models.
Model
bo
my
zh
Qwen2.5-7B-Instruct
10.40
9.20
53.00
+ GRPO (En)
13.60
11.60
58.60
Δ
(+3.20)
(+2.40)
(+5.60)
Llama-3.1-8B-Instruct
0.80
2.40
28.40
+ GRPO (En)
15.00
8.40
33.80
Δ
(+14.20)
(+6.00)
(+5.40)
Appendix
Table 20: Cross-lingual transfer to Sino-Tibetan languages. We report accuracy (%) before and after English-only RPT with GRPO. bo , my , and zh denote Tibetan, Myanmar, and Chinese, respectively. Δ denotes the absolute accuracy gain after GRPO.
Settings
Δ Performance
Average across languages
en
es
ru
de
fr
bn
th
sw
zh
ja
te
Training
Untraining
Qwen2.5-1.5B-Instruct with GRPO on En Data
MATH500
20.40
27.20
17.40
14.40
17.40
6.20
7.80
4.40
16.80
14.20
7.40
20.40
13.32
AIME24
2.08
0.42
-0.21
1.25
0.42
0.21
0.21
0.00
-0.21
0.21
0.21
2.08
0.25
AIME25
1.04
0.21
0.00
0.21
0.00
0.00
0.00
-0.21
0.42
0.21
0.21
1.04
0.10
GPQA-Diamond
9.09
20.71
-0.51
5.05
12.12
1.52
-2.53
2.02
8.59
3.54
0.51
9.09
5.10
Appendix
Table 21: The Impact of Model Size in Interventional Study. Δ Performance on various benchmarks across Qwen2.5-1.5B-Instruct and Qwen2.5-7B-Instruct .
Settings
Accuracy per language
Average
en
es
ru
de
fr
bn
th
sw
zh
ja
te
Acc
Off-target
MTI
Multilingual MATH500
Only English
79.2
70.0
61.0
62.2
68.2
41.8
55.4
19.4
53.4
58.6
27.4
54.2
0.5
1.163
w. One parallel
78.4
73.4
66.0
65.6
67.2
48.8
57.4
26.2
57.8
62.0
33.8
57.9
0.2
2.496
w. Two parallel
79.0
73.4
64.4
67.4
69.2
45.2
60.2
26.2
63.0
61.6
32.6
58.4
0.2
2.650
w. Three parallel
77.8
73.6
64.6
68.4
69.8
46.0
60.8
24.0
62.2
60.8
34.2
58.4
0.4
3.002
Appendix
Table 22: The Detailed Results in Parallel Scaling Law.
Settings
Training Parallel Languages
Only English
en
w. One parallel
en, ru
w. Two parallel
en, ru, fr
w. Three parallel
en, ru, fr, es
w. Four parallel
en, ru, fr, es, de
w. Five parallel
en, ru, fr, es, de, bn
Appendix
Table 23: The Language Settings in Parallel Scaling Law.
Figure 10: Analysis across Selected Parallel Languages. The accuracy and relative gain across various benchmarks with different parallel languages. “Only en” denotes only fine-tuned on English data. “en& LANGUAGE ” indicates the model was fine-tuned on English and a parallel language, with LANGUAGE representing ru, bn, de, zh , respectively.
Lang.
Base
Zh
Zh+Ru
Zh+Ru+Fr
en
74.8
76.8
78.8
78.0
es
69.0
70.4
72.0
72.2
ru
59.6
61.8
64.4
63.6
de
56.6
62.2
65.0
66.4
fr
62.6
66.8
70.0
69.8
bn
37.6
43.0
44.4
45.4
Appendix
Table 24: Parallel scaling with Chinese as the source language. All experiments use Qwen2.5-7B-Instruct and are evaluated on multilingual MATH500. Adding the first parallel language yields the largest gain, while adding a second language provides a smaller additional improvement.
Settings
Accuracy per language
Average
en
es
ru
de
fr
bn
th
sw
zh
ja
te
Acc
Off-target
Multilingual MATH500
DeepSeek-R1-Distill-Qwen-7B
86.20
76.20
67.00
66.40
69.80
40.20
53.80
18.40
70.00
53.20
17.60
56.25
2.69
Open-Reasoner-Zero-7B
81.60
76.00
70.40
69.80
71.20
45.20
63.20
15.00
62.80
61.80
17.60
57.69
5.20
OpenThinker2-7B
86.00
74.80
64.80
63.00
73.00
34.40
58.80
12.60
65.80
70.00
8.40
55.60
36.13
OpenThinker3-7B
85.80
81.80
75.00
69.20
76.40
17.00
48.80
17.40
69.60
65.00
10.40
56.04
60.75
Appendix
Table 25: The Performance of Various Open-source Models. Accuracy (%) and Off-target rate (%) across languages for various open-source models.
Large language models (LLMs) exhibit substantial cross-lingual variation in mathematical reasoning performance, but it remains unclear whether these differences reflect language-specific parameters or a shared mechanism that manifests differently by language. We present a cross-lingual mechanistic analysis of mathematical reasoning in LLMs, enabling us to localize and compare model parameters that support mathematical reasoning across languages. We find that the extracted math-associated parameters exhibit partial cross-lingual overlap, with the strongest overlap concentrated in intermediate model layers. We further observe that English consistently produces the largest set of math-relevant parameters, whereas lower-resource languages reveal smaller sets of relevant parameters. These results suggest that math-related behavior in multilingual LLMs is neither fully language-invariant nor fully language-specific, but instead exhibits partial cross-lingual parameter overlap with systematic language-dependent differences.
Behzad Shomali, Luisa Victor, Tim Selbach +5
Lamarr Institute · University of Bonn · Fraunhofer IAIS
Large Language Models (LLMs) have demonstrated remarkable proficiency in English mathematical reasoning, yet a significant performance disparity persists in multilingual contexts, largely attributed to deficiencies in language understanding. To bridge this gap, we introduce Translation-Augmented Policy Optimization (TAPO), a novel reinforcement learning framework built upon GRPO. TAPO enforces an explicit alignment strategy where the model leverages English as a pivot and follows an understand-then-reason paradigm. Crucially, we employ a step-level relative advantage mechanism that decouples understanding from reasoning, allowing the integration of translation quality rewards without introducing optimization conflicts. Extensive experiments reveal that TAPO effectively synergizes language understanding with reasoning capabilities and is compatible with various models. It outperforms baseline methods in both multilingual mathematical reasoning and translation tasks, while generalizing well to unseen languages and out-of-domain tasks.
Xu Huang, Zhejian Lai, Zixian Huang +2
National Key Laboratory for Novel Software Technology, Nanjing University · Shanghai Artificial Intelligence Laboratory
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
Yuchun Fan, Bei Li, Peiguang Li +9
NLP Lab, School of Computer Science and Engineering, Northeastern University, Shenyang, China · Meituan Inc. 3NiuTrans Research, Shenyang, China