Simple and Provable Scaling Laws for the Test-Time Compute of Large Language Models (2411.19477v4)

Abstract: We propose two simple, principled and practical algorithms that enjoy provable scaling laws for the test-time compute of 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.

• The paper introduces a two-stage algorithm that generates candidate solutions and uses tournament-style knockouts to select the optimal one.
• The methodology demonstrates that failure probability decreases exponentially with increased candidates (N) and comparisons (K) under minimal probability assumptions.
• Empirical results on the MMLU-Pro benchmark confirm significant accuracy improvements, particularly in math and engineering domains.

An Analysis of the Two-Stage Algorithm for Scaling LLM Test-Time Compute

This paper introduces a novel two-stage algorithm aimed at improving the test-time compute efficacy of LLMs by generating candidate solutions and employing a tournament-style knockout process to select the optimal solution. The authors propose a theoretical foundation, supported by empirical evidence, suggesting that the failure probability of this algorithm declines exponentially with increased computational resources, represented by the parameters N and K.

Methodology and Theoretical Contributions

The two-stage algorithm comprises a generation stage and a knockout stage. In the generation stage, N candidate solutions are generated using LLMs. These solutions are subsequently compared in the knockout stage. Each pair of solutions is compared K times, and the solution with more favorable outcomes progresses to the next round. Importantly, the algorithm is designed to function with just a black-box LLM, eliminating the need for external verification or reward models, which simplifies implementation.

The paper establishes a scaling law demonstrating that the algorithm's failure probability decreases exponentially with respect to N and K. This result is contingent upon two conditions:

1. LLMs must possess a non-zero probability of generating a correct solution (p_gen > 0).
2. The likelihood of correctly identifying the superior solution in pairwise comparisons exceeds random chance (p_comp > 0.5).

The presented algorithm is not only efficient and scalable but also supports parallel and distributed computation, which allows for significant improvements in accuracy with an increase in computational resources. This characteristic is validated using the MMLU-Pro benchmark, which provides a robust testbed for evaluating the efficacy of LLMs on various categories of multiple-choice questions.

Empirical Validation

The experiments were conducted across 14 categories of the MMLU-Pro benchmark, using the Llama3.1-70B-Instruct and Qwen2.5-72B-Instruct models. The empirical results confirmed the theoretical expectations, showcasing improved accuracy as the computational resources (N and K) were scaled. Furthermore, the efficacy of the two-stage method varied across different domains, with notable improvements in math and engineering-related problems, whereas more modest gains were observed in knowledge-heavy domains such as psychology.

A particularly compelling aspect of the paper is the distinction between scenarios where LLMs can significantly benefit from pairwise comparisons, especially for questions requiring analytical reasoning over factual recall.

Implications and Future Directions

The research presented in this paper has several practical and theoretical implications. It demonstrates a clear path toward enhancing the reliability of LLMs in high-stakes scenarios by leveraging higher test-time compute. From a theoretical standpoint, the established scaling law provides a framework for future algorithms that seek similar guarantees.

However, the paper also opens avenues for further exploration. The proposed anytime variant of the algorithm could be highly practical, allowing responsive adjustment to available computational resources without predefined parameters. Additionally, the manuscript alludes to the potential for relaxing the assumption of efficient pairwise comparison, which may expand the applicability of this approach across more complex tasks.

Future work could also explore pushing the theoretical optimal limits of computational or sample complexity under varying assumptions. This could lead to the development of algorithms that more effectively balance accuracy and compute efficiency, thereby enhancing the applicability of LLMs in real-world problem-solving contexts.

Conclusion

The two-stage algorithm proposed in this paper is a significant step toward scalable LLM-based problem-solving, with empirical evidence supporting its theoretical foundation. By framing an efficient mechanism that scales with computational investment, this work contributes valuable insights and tools for both current applications of LLMs and future research directions in AI scalability and reliability.