We propose two simple, principled and practical algorithms that enjoy provable scaling laws for the test-time compute of large language models (LLMs). The first one is a two-stage knockout-style algorithm: given an input problem, it first generates multiple candidate solutions, and then aggregate them via a knockout tournament for the final output. Assuming that the LLM can generate a correct solution with non-zero probability and do better than a random guess in comparing a pair of correct and incorrect solutions, we prove theoretically that the failure probability of this algorithm decays to zero exponentially or by a power law (depending on the specific way of scaling) as its test-time compute grows. The second one is a two-stage league-style algorithm, where each candidate is evaluated by its average win rate against multiple opponents, rather than eliminated upon loss to a single opponent. Under analogous but more robust assumptions, we prove that its failure probability also decays to zero exponentially with more test-time compute. Both algorithms require a black-box LLM and nothing else (e.g., no verifier or reward model) for a minimalistic implementation, which makes them appealing for practical applications and easy to adapt for different tasks. Through extensive experiments with diverse models and datasets, we validate the proposed theories and demonstrate the outstanding scaling properties of both algorithms.

翻译：我们提出了两种简单、原则性强且实用的算法，它们在大型语言模型（LLMs）的测试时计算方面享有可证明的缩放定律。第一种是两阶段淘汰赛式算法：给定一个输入问题，它首先生成多个候选解决方案，然后通过淘汰赛聚合它们以产生最终输出。假设LLM能以非零概率生成正确解，并且在比较一对正确与错误解时表现优于随机猜测，我们从理论上证明，随着测试时计算的增加，该算法的失败概率以指数方式或幂律方式（取决于具体的缩放方式）衰减至零。第二种是两阶段联赛式算法，其中每个候选解通过其与多个对手的平均胜率进行评估，而非因单次输给对手而被淘汰。在类似但更稳健的假设下，我们证明其失败概率也随着测试时计算的增加以指数方式衰减至零。两种算法仅需一个黑盒LLM即可实现最小化实现（例如，无需验证器或奖励模型），这使它们在实际应用中具有吸引力，并易于适应不同任务。通过使用多样化模型和数据集的广泛实验，我们验证了所提出的理论，并展示了两种算法出色的缩放特性。