6 Discussion and related work

本文为论文《s1：用简单方法实现测试时计算扩展》「论文原文」板块的对应章节，按 arXiv 官方全文逐字保留，层级与「中文全文翻译」板块一一对应。

6.1 Sample-efficient reasoning

Models

There are a number of concurrent efforts to build models that replicate the performance of o1 (OpenAI, 2024). For example, DeepSeek-r1 and k1.5 (DeepSeek-AI et al., 2025; Team et al., 2025) are built with reinforcement learning methods, while others rely on SFT using tens of thousands of distilled examples (Team, 2025; Xu et al., 2025; Labs, 2025). We show that SFT on only 1,000 examples suffices to build a competitive reasoning model matching o1-preview and produces a model that lies on the pareto frontier (Figure 2). Further, we introduce budget forcing which combined with our reasoning model leads to the first reproduction of OpenAI’s test-time scaling curves (OpenAI, 2024). Why does supervised finetuning on just 1,000 samples lead to such performance gains? We hypothesize that the model is already exposed to large amounts of reasoning data during pretraining which spans trillions of tokens. Thus, the ability to perform reasoning is already present in our model. Our sample-efficient finetuning stage just activates it and we scale it further at test time with budget forcing. This is similar to the ”Superficial Alignment Hypothesis” presented in LIMA (Zhou et al., 2023), where the authors find that 1,000 examples can be sufficient to align a model to adhere to user preferences.

Benchmarks and methods

To evaluate and push the limits of these models, increasingly challenging benchmarks have been introduced, such as Olympiad-level science competitions (He et al., 2024a; Jain et al., 2024a; Zhong et al., 2023) and others (Srivastava et al., 2023; Glazer et al., 2024; Su et al., 2024; Kim et al., 2024; Phan et al., 2025). To enhance models’ performance on reasoning-related tasks, researchers have pursued several strategies: Prior works have explored continuing training language models on specialized corpora related to mathematics and science (Azerbayev et al., 2023; Yang et al., 2024), sometimes even synthetically generated data (Yu et al., 2024). Others have developed training methodologies specifically aimed at reasoning performance (Zelikman et al., 2022; Zelikman et al., 2024; Luo et al., 2025; Yuan et al., 2025). Another significant line of work focuses on prompting-based methods to elicit and improve reasoning abilities, including methods like Chain-of-Thought prompting (Wei et al., 2023; Yao et al., 2023a; Yao et al., 2023b; Bi et al., 2023; Fu et al., 2023; Zhang et al., 2024b; Hu et al., 2024). These combined efforts aim to advance the reasoning ability of language models, enabling them to handle more complex and abstract tasks effectively.

6.2 Test-time scaling

Methods

As we introduce in §3, we differentiate two methods to scale test-time compute: parallel and sequential. The former relies on multiple solution attempts generated in parallel and selecting the best outcome via specific criteria. These criteria include choosing the most frequent response for majority voting or the best response based on an external reward for Best-of-N (Brown et al., 2024; Irvine et al., 2023; Snell et al., 2024). Unlike repeated sampling, previous sequential scaling methods let the model generate solution attempts sequentially based on previous attempts, allowing it to refine each attempt based on previous outcomes (Snell et al., 2024; Hou et al., 2025; Lee et al., 2025). Tree-based search methods (Gandhi et al., 2024; Wu et al., 2024a) offer a hybrid approach between sequential and parallel scaling, such as Monte-Carlo Tree Search (MCTS) (Liu et al., 2024; Zhang et al., 2023; Zhou et al., 2024; Choi et al., 2023) and guided beam search (Xie et al., 2023). REBASE (Wu et al., 2024a) employs a process reward model to balance exploitation and pruning during tree search. Empirically, REBASE has been shown to outperform sampling-based methods and MCTS (Wu et al., 2024a). Reward models (Lightman et al., 2023; Wang et al., 2024b; Wang et al., 2024c) play a key role in these methods. They come in two variants: outcome reward models and process reward models. Outcome reward models (Xin et al., 2024; Ankner et al., 2024) assign a score to complete solutions and are particularly useful in Best-of-N selection, while process reward models (Lightman et al., 2023; Wang et al., 2024b; Wu et al., 2024a) assess individual reasoning steps and are effective in guiding tree-based search methods.

Limits to further test-time scaling

We have shown that budget forcing allows extrapolating test-time compute in §4, e.g., improving AIME24 performance from 50% to 57%. However, it has two key limitations when scaling further: it eventually flattens out (Figure 4), and the context window of the underlying language model constrains it. Despite these constraints, our work shows test-time scaling across a wide range of accuracies (Figure 1), partly because scaling down test-time compute behaves predictably and does not suffer from these constraints.

Continuing test-time scaling will require approaches that can further extrapolate test-time compute. How can we get such extrapolation? There may be improvements to budget forcing such as rotating through different strings, not only “Wait”, or combining it with frequency penalties or higher temperature to avoid repetitive loops. An exciting direction for future work is also researching whether applying budget forcing to a reasoning model trained with reinforcement learning yields better extrapolation; or if RL allows for new ways of test-time scaling beyond budget forcing. Our work defines the right metrics (§3.2) – Control, Scaling, and Performance – to enable future research and progress on extrapolating test-time compute.

Figure 7: Scaling further with parallel scaling methods. All metrics averaged over the 30 questions in AIME24. Average thinking tokens for REBASE do not account for the additional compute from the reward model. For sequential scaling, we prompt the model to use up to (from left to right) 32, 64, 256, and 512 steps. For REBASE and majority voting we generate 16 parallel trajectories to aggregate across.

Parallel scaling as a solution

Parallel scaling offers one solution to the limits of sequential scaling, thus we augment our sequentially scaled model with two methods: (I) Majority voting: After generating $k$ solutions, the final solution is the most frequent one across generations; (II) Tree search via REBASE: We use the REBASE process reward model, which is initialized from LLaMA-34B and further finetuned on a synthetic process reward modeling dataset (Wu et al., 2024a). We then aggregate the solutions generated by REBASE via majority voting. As shown in Figure 7, augmenting our model with REBASE scales better than majority voting, and even sequential scaling in this scenario. However, REBASE requires an additional forward pass at each step for the reward model adding some computation overhead. For sequential scaling, when prompted to use up to 512 steps, for 12 out of the 30 evaluation questions the model generates a response that exceeds the context window leading to a large performance drop. Overall, we find that these parallel scaling methods complement sequential scaling thus they offer an avenue for scaling test-time compute even further; beyond fixed context windows.