Inference-Time Diversity in RL-Trained Lean Theorem Provers: A Diagnostic Study

TL;DR

This study investigates inference-time diversity in RL-trained Lean theorem provers, revealing that strategic prompt variations significantly improve theorem-solving performance and are specific to RL training stages.

Contribution

It introduces a diagnostic approach to understanding inference-time diversity effects, demonstrating that prompt diversity can mitigate mode-collapse in RL-trained provers.

Findings

Doubling sampling budget yields no additional solved theorems.

A fixed schedule of tactic skeletons improves theorem-solving by 45%.

Diversity interventions have RL-specific effects, not seen in SFT-trained models.

Abstract

RL-trained Lean theorem provers mode-collapse at inference time: on miniF2F-test with DeepSeek-Prover-V1.5-RL, doubling the i.i.d.\ sampling budget from $k{=}32$ to $k{=}64$ produces zero additional solved theorems (42/244 in both cases). A fixed schedule of 15 tactic skeletons breaks this plateau and recovers a $+45%$ relative improvement at $k{=}16$ (mean $\Delta = +12.3 \pm 4.2$ theorems across $n{=}3$ seeds, sign preserved in every seed). A controlled diversity ablation rules out the prompt-diversity confound: tactic skeletons help, paraphrases match the baseline, and irrelevant Lean comments actively degrade. A leave-one-out formalization-difficulty stratification reveals a structural-content gradient across the three perturbations. The phenomenon is RL-specific: V1.5-Base proves zero theorems regardless of intervention, identifying RL as the stage that creates the proof capability which subsequently collapses; extending to two additional 7B Lean provers, RL-trained DeepSeek-Prover-V2-7B contributes $+3$ frontier solves no i.i.d.\ baseline can reach despite a flat aggregate, while SFT-trained Goedel-Prover does not ($-10.0 \pm 4.4$ theorems, $n{=}3$, sign preserved every seed). Inference-time structural diversity is a cheap, complementary axis for RL-trained provers, orthogonal to scaling model size or training compute.