SoS1: O1 and R1-Like Reasoning LLMs are Sum-of-Square Solvers
Large Language Models (LLMs) have achieved human-level proficiency across
diverse tasks, but their ability to perform rigorous mathematical problem
solving remains an open challenge. In this work, we investigate a fundamental
yet computationally intractable problem: determining whether a given
multivariate polynomial is nonnegative. This problem, closely related to
Hilbert's Seventeenth Problem, plays a crucial role in global polynomial
optimization and has applications in various fields. First, we introduce
SoS-1K, a meticulously curated dataset of approximately 1,000 polynomials,
along with expert-designed reasoning instructions based on five progressively
challenging criteria. Evaluating multiple state-of-the-art LLMs, we find that
without structured guidance, all models perform only slightly above the random
guess baseline 50%. However, high-quality reasoning instructions significantly
improve accuracy, boosting performance up to 81%. Furthermore, our 7B model,
SoS-7B, fine-tuned on SoS-1K for just 4 hours, outperforms the 671B DeepSeek-V3
and GPT-4o-mini in accuracy while only requiring 1.8% and 5% of the computation
time needed for letters, respectively. Our findings highlight the potential of
LLMs to push the boundaries of mathematical reasoning and tackle NP-hard
problems.
67c51b459d5807d6674b3d3c
Kechen Li
claimed_verified
2025-03-04T08:51:29.578Z
6742deb4d3ad4510c12da658
/avatars/91407d854560ef9a2facd80fa8fab6ec.svg
Kechen Li
user
Kechen-Li
67c51b459d5807d6674b3d3d
Wenqi Zhu
67c51b459d5807d6674b3d3e
Coralia Cartis
67c51b459d5807d6674b3d3f
Tianbo Ji
admin_assigned
2025-03-03T09:35:49.782Z
64bb61e876a6e2efcc728e22
/avatars/b0ed1c9f13fd1f2c99d202155001e39b.svg
Tianbo Ji
user
jitianbo
67c51b459d5807d6674b3d40
Shiwei Liu
claimed_verified
2025-03-03T11:14:45.635Z
65b04d2291e63920a7898c9e
https://cdn-avatars.huggingface.co/v1/production/uploads/65b04d2291e63920a7898c9e/iUHs235G4bqK-KnH_94ti.jpeg
Liu
user
Shiweiliuiiiiiii
SoS1: O1 and R1-Like Reasoning LLMs are Sum-of-Square Solvers
Large Language Models (LLMs) have achieved human-level proficiency across
diverse tasks, but their ability to perform rigorous mathematical problem
solving remains an open challenge. In this work, we investigate a fundamental
yet computationally intractable problem: determining whether a given
multivariate polynomial is nonnegative. This problem, closely related to
Hilbert's Seventeenth Problem, plays a crucial role in global polynomial
optimization and has applications in various fields. First, we introduce
SoS-1K, a meticulously curated dataset of approximately 1,000 polynomials,
along with expert-designed reasoning instructions based on five progressively
challenging criteria. Evaluating multiple state-of-the-art LLMs, we find that
without structured guidance, all models perform only slightly above the random
guess baseline 50%. However, high-quality reasoning instructions significantly
improve accuracy, boosting performance up to 81%. Furthermore, our 7B model,
SoS-7B, fine-tuned on SoS-1K for just 4 hours, outperforms the 671B DeepSeek-V3
and GPT-4o-mini in accuracy while only requiring 1.8% and 5% of the computation
time needed for letters, respectively. Our findings highlight the potential of
LLMs to push the boundaries of mathematical reasoning and tackle NP-hard
problems.
LLMs are investigated for solving nonnegative polynomial problems, showing significant improvement with reasoning instructions, and suggesting potential for tackling NP-hard mathematical challenges.
Large Language Models LLMs nonnegative polynomials Hilbert's Seventeenth Problem global polynomial optimization SoS-1K reasoning instructions SoS-7B DeepSeek-V3 GPT-4o-mini computational intractability NP-hard problems parameter-efficient fine-tuning