cs.CLApr 7, 2026
SaveUnderstanding Performance Gap Between Parallel and Sequential Sampling in Large Reasoning Models
Organizations: Google DeepMind · National University of Singapore
Abstract
Large Reasoning Models (LRMs) have shown remarkable performance on challenging questions, such as math and coding. However, to obtain a high quality solution, one may need to sample more than once. In principal, there are two sampling strategies that can be composed to form more complex processes: sequential sampling and parallel sampling. In this paper, we first compare these two approaches with rigor, and observe, aligned with previous works, that parallel sampling seems to outperform sequential sampling even though the latter should have more representation power. To understand the underline reasons, we make three hypothesis on the reason behind this behavior: (i) parallel sampling outperforms due to the aggregator operator; (ii) sequential sampling is harmed by needing to use longer contexts; (iii) sequential sampling leads to less exploration due to conditioning on previous answers. The empirical evidence on various model families and sizes (Qwen3, DeepSeek-R1 distilled models, Gemini 2.5) and question domains (math and coding) suggests that the aggregation and context length do not seem to be the main culprit behind the performance gap. In contrast, the lack of exploration seems to play a considerably larger role, and we argue that this is one main cause for the performance gap.
Figures & tables
Figure 1: Illustration of sampling approaches considered in this work. In parallel sampling, given a prompt, multiple independent thinking traces and solutions are sampled, and then the solutions are aggregated to the final one. In Markov sequential sampling, solutions are sampled in a sequential chain, for current generation, only last solution is considered as input. While for auto-regressive sequential sampling, the input includes all previous solutions.
Figure 2: Comparisons of AIME2025 performance between parallel and sequential sampling using ( Left ) Qwen3-14B; ( Middle ) DeepSeek-R1-Distill Qwen-14B; ( Right ) Gemini 2.5 Flash. In parallel sampling, majority voting aggregation is adopted. Parallel sampling performs better than sequential sampling on the math competition scenario.
Figure 3: Comparisons of LiveCodeBench v5 performance between parallel and sequential sampling using ( Left ) Gemini 2.5 Flash; ( Middle ) DeepSeek-R1-Distill Qwen-14B; ( Right ) DeepSeek-R1-Distill Qwen-7B. In parallel sampling, best-of-N aggregation is adopted. Parallel sampling performs better than sequential sampling on code generation questions.
Figure 4: Comparisons of AIME2025 performance between parallel and sequential sampling using ( Left ) Qwen3-14B; ( Middle ) DeepSeek-R1-Distill Qwen-14B; ( Right ) Gemini 2.5 Flash. Both sampling approaches use majority vote aggregation.
Figure 5: Comparisons of LiveCodeBench v5 performance of Gemini 2.5 Flash/DeepSeek-R1-Distill Qwen-14B/7B under parallel sampling and sequential sampling using the best-of-N aggregation. ( Top ) Each solution is rewarded based on public and generated tests. ( Bottom ) Each solution is rewarded based on public and private tests.
Figure 6: ( Left ) input context length, ( Middle ) overall context length, ( Right ) thinking traces length when using Gemini 2.5 Flash on LiveCodeBench v5 under parallel and sequential sampling. Even though there is a clear performance gap between Markov sequential sampling and parallel sampling, the input context lengths are similar. In addition, the overall context length of auto-regressive sampling is not significantly accumulated compared to others.
| 0 (baseline) | 1k | 2k | 4k | 8k | 16k | 32k | |
|---|---|---|---|---|---|---|---|
| Acc. | 62.2 2.0 | 62.4 2.8 | 61.2 1.2 | 62.8 1.6 | 63.0 1.6 | 62.7 1.5 | 63.2 2.9 |
| Agg. | 77.84 | 74.85 | 77.84 | 79.64 | 76.65 | 79.64 | 77.84 |
Table 1: Accuracy (Acc.) and aggregation accuracy on 8 solutions (Agg.) of Gemini 2.5 Flash on LiveCodeBench v5 with different irrelevant input context lengths. Best-of-N aggregation is adopted based on rewards using public and private tests. It is observed that input context length does not make significant difference on the performance, including both single-solution accuracy and aggregation accuracy.
Figure 7: An example (with content truncation) of two solutions in an auto-regressive sequential chain using Qwen3-14B on a question from AIME2025. The ground-truth answer is 735, but the LRM keeps making mistakes and becomes “lazy” to explore new alternatives.
| Domain | Model | gemini-embedding-001 | text-embedding-3-small | ||||
|---|---|---|---|---|---|---|---|
| Parallel | Markov | Auto. | Parallel | Markov | Auto. | ||
| Coding | Gemini 2.5 Flash | 0.973 | 0.989 | 0.990 | 0.877 | 0.963 | 0.966 |
| Gemini 2.5 Pro | 0.972 | 0.979 | 0.980 | 0.839 | 0.902 | 0.904 | |
| Distill Qwen-14B | 0.966 | 0.985 | NA | 0.854 | 0.947 | NA | |
| Distill Qwen-7B | 0.955 | 0.968 | NA | 0.813 | 0.878 | NA | |
| Math | Gemini 2.5 Flash | 0.972 | 0.977 | 0.978 | 0.916 | 0.946 | 0.945 |
Table 2: Averaged pair-wise cosine similarities (higher means more similar) among solutions generated by parallel and sequential sampling. We consider two embedding models.
Figure 8: Performance of Gemini 2.5 Flash on LiveCodeBench v5 using parallel sampling with different rounds of previous samples in input context. Here “Round ” refers to previous solutions in a sequence chain. Best-of-N aggregation with both public and private tests for rewarding is applied here.
Figure 9: Comparisons of LiveCodeBench v5 performance of Gemini 2.5 Flash/DeepSeek-R1-Distill Qwen-14B/7B under parallel sampling and sequential sampling using the best-of-N aggregation. Each solution is rewarded based on public and private tests. For sequential sampling, the running errors on public and private tests are provided as high-quality guidance.
Figure 10: Visualization of induction heads in Qwen3-14B using auto-regressive sequential sampling on AIME2025 questions. ( Left ) All 7 solutions are incorrect; ( Middle ) all 7 solutions are correct; ( Right ) the 2rd/3rd solution are incorrect while the others are correct.
Figure 11: Visualization of induction heads in Qwen3-14B using auto-regressive sequential sampling on AIME2025 questions. We consider 1/3/7 previous solutions in input context.
Appendix figures & tables23 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 12: Comparisons of AIME2025 performance between parallel and sequential sampling using Qwen3-14B/8B and DeepSeek-R1-Distill Qwen-14B/7B. In parallel sampling, majority voting aggregation is adopted. In sequential sampling, we apply a feedback prompt of “Please re-answer” in ( Top ) row and “Please review your previous response and find problems with your answer. Based on the problems you found, improve your answer.” (self-refinement feedback) in ( Bottom ) row.
Figure 13: Comparisons of AIME2025 performance between parallel and sequential sampling using Gemini 2.5 Flash. In parallel sampling, majority voting aggregation is adopted. In sequential sampling, we apply a feedback prompt of ( Left ) “Please re-answer” and ( Right ) “Please review your previous response and find problems with your answer. Based on the problems you found, improve your answer.”.
Figure 14: Code generation prompt of Gemini 2.5 for LiveCodeBench.
Figure 15: Code generation prompt of DeepSeek-R1-Distill Qwen models for LiveCodeBench.
Figure 16: Code generation prompt of Gemini 2.5 for LiveCodeBench in Markov sequential sampling.
Figure 17: Feedback prompt design when there are running errors in sequential sampling.
Figure 18: Prompt design for generating test cases based on public tests.
Figure 19: Comparisons of LiveCodeBench v5 performance between parallel and sequential sampling using Gemini 2.5 Flash on ( Left ) all questions; ( Middle ) medium questions; ( Right ) hard questions.
Figure 20: Comparisons of LiveCodeBench v5 performance between parallel and sequential sampling using DeepSeek-R1-Distill Qwen-14B on ( Left ) all questions; ( Middle ) medium questions; ( Right ) hard questions.
Figure 21: Comparisons of LiveCodeBench v5 performance between parallel and sequential sampling using DeepSeek-R1-Distill Qwen-7B on ( Left ) all questions; ( Middle ) medium questions; ( Right ) hard questions.
Figure 22: Comparisons of AIME2025 performance between parallel and sequential sampling using ( Left ) Qwen3-14B; ( Middle ) DeepSeek-R1-Distill Qwen-14B; ( Right ) Gemini 2.5 Flash. Both sampling approaches use the ideal best-of-N aggregation with the assumption of perfect verifier.
Figure 23: Comparisons (with different difficulty levels) of LiveCodeBench v5 performance between parallel and sequential sampling using Gemini 2.5 Flash. Both sampling approaches adopt the same best-of-N aggregation based on rewards on public and private tests. In sequential sampling, we apply feedback prompt of ( Top ) “Please re-answer” and ( Bottom ) self-refinementing.
Figure 24: Comparisons (with different difficulty levels) of LiveCodeBench v5 performance between parallel and sequential sampling using Gemini 2.5 Pro. Both sampling approaches adopt the same best-of-N aggregation based on rewards on public and private tests. In sequential sampling, we apply feedback prompt of ( Top ) “Please re-answer” and ( Bottom ) self-refinementing.
Figure 25: Comparisons (with different difficulty levels) of LiveCodeBench v5 performance between parallel and sequential sampling using DeepSeek-R1-Distill Qwen-14B. Both sampling approaches adopt the same best-of-N aggregation based on rewards on public and private tests. In sequential sampling, we apply feedback prompt of ( Top ) “Please re-answer” and ( Bottom ) self-refinementing.
Figure 26: Comparisons (with different difficulty levels) of LiveCodeBench v5 performance between parallel and sequential sampling using DeepSeek-R1-Distill Qwen-7B. Both sampling approaches adopt the same best-of-N aggregation based on rewards on public and private tests. In sequential sampling, we apply feedback prompt of ( Top ) “Please re-answer” and ( Bottom ) self-refinementing.
Figure 27: ( Left ) input context length, ( Middle ) overall context length, ( Right ) thinking traces length when using Gemini 2.5 Flash on AIME2025 under parallel and sequential sampling.
Figure 28: ( Left ) input context length, ( Middle ) overall context length, ( Right ) thinking traces length when using Qwen3-14B on AIME2025 under parallel and sequential sampling.
Figure 29: ( Left ) input context length, ( Middle ) overall context length, ( Right ) thinking traces length when using DeepSeek-R1-Distill Qwen-14B on AIME2025 under parallel and sequential sampling.
Figure 30: ( Left ) input context length, ( Middle ) overall context length, ( Right ) thinking traces length when using Gemini 2.5 Pro on LiveCodeBench v5 under parallel/sequential sampling.
Figure 31: Comparisons (with different difficulty levels) of LiveCodeBench v5 performance of (from Top to Bottom ) DeepSeek-R1-Distill Qwen-14B/7B, Gemini 2.5 Flash/Pro under parallel sampling and sequential sampling using the best-of-N aggregation. Each solution is rewarded based on public and private tests. For sequential sampling, the running errors on public and private tests are provided as high-quality guidance.
Figure 32: Solution generated by Gemini 2.5 Pro repeatedly appear in the auto-regressive sequential sampling when solving “Accumulating Many Times” from LiveCodeBench.
Figure 33: (Continued) solution generated by Gemini 2.5 Pro repeatedly appear in the auto-regressive sequential sampling when solving “Accumulating Many Times” from LiveCodeBench.
Figure 34: (Continued) solution generated by Gemini 2.5 Pro repeatedly appear in the auto-regressive sequential sampling when solving “Accumulating Many Times” from LiveCodeBench.