Ivo Petrov, Jasper Dekoninck, Dimitar I. Dimitrov, Martin Vechev
原文摘要
Large language models (LLMs) have become capable mathematical problem-solvers, often producing correct proofs for challenging problems. However, correctness alone is not sufficient: mathematical proofs should also be clear, insightful, contain transferable knowledge, and be generally useful to the reader. While proof quality is subjective and depends on the reader and context, many of its components can be made concrete. In this work, we identify such components and introduce ProofRank, a benchmark curated from challenging competitions that measures several scalable proxies of proof quality: (i) conciseness - quantifying whether proofs avoid unnecessary steps; (ii) computational ease - the extent to which a proof relies on tedious calculations; (iii) cognitive simplicity - how accessible the used proof techniques are; (iv) diversity - how varied a model's proofs for a single problem are; and (v) adaptivity - how well a model can follow a specified technique. We find proof quality varies substantially across models along dimensions that correctness largely fails to capture. Every proof-level proxy can be steered through prompting, but often at a moderate correctness cost. ProofRank thus captures important aspects of mathematical proofs beyond correctness, and indicates the need for models that produce proofs which are not only correct but also valuable.
Jasper Dekoninck, Maximilian Baader, Martin Vechev
原文摘要
The availability of a wide range of large language models (LLMs) embedded in various agentic systems has significantly increased the potential of model selection strategies to improve the cost-performance tradeoff. Existing strategies involve either routing, where a single model is chosen per query, or cascading, which sequentially runs increasingly larger models until a satisfactory answer is found. However, current approaches face three key limitations: they (1) lack formal proofs of optimality, (2) fail to identify the conditions under which these strategies are most effective to improve the cost-performance tradeoff, and (3) are unable to combine both paradigms for further improvements. To address these issues, we first derive a novel optimal strategy for cascading and prove the optimality of an existing routing strategy. Further, we propose cascade routing, a unified framework that integrates routing and cascading into a theoretically optimal strategy. Through our analysis, we identify good quality estimators as the critical factor for the success of model selection paradigms. Finally, in our experiments, we show that cascade routing consistently outperforms the individual approaches by a large margin and we analyze quality estimators to determine when routing and/or cascading are useful paradigms for model selection.