Expected Value Alignment for Generative Reward Modeling in Formal Mathematics Verification
Authors: Shihao Ji, Haotao Tan, Zihui Song, Mingyu Li
Abstract
Large Language Models (LLMs) are increasingly used with formal interactive theorem provers such as Lean 4. Scaling these systems with reinforcement learning or search methods requires process reward models (PRMs) that can evaluate intermediate reasoning steps. Existing reward-model designs expose a practical trade-off. Value-head models provide continuous scores but modify the generative model interface, while generative reward models preserve textual rationales but are poorly matched to continuous floating-point regression because numeric values are split across tokens. We introduce Expected Value Alignment (EVA), a reward-modeling procedure that keeps the surface output discrete while extracting continuous scores from the model's token distribution. The model emits integer scores in a structured JSON format, and EVA computes a continuous score as the expectation over the logits of the corresponding anchor tokens. Training combines the causal language modeling objective with an auxiliary mean squared error loss on these expected values. We instantiate EVA in \textit{Leibniz}, a reward model for Lean 4 formal verification, and evaluate it against zero-shot and reward-modeling baselines. The evaluation demonstrates that continuous logit-based scoring significantly reduces discretization artifacts while retaining the interpretability of generative critiques.
Recent neural theorem provers use reinforcement learning with verifiable rewards (RLVR), where proof assistants provide binary correctness signals. While verifiable rewards are cheap and scalable without reward hacking issues, they suffer from sparse credit assignment: models receive no learning signal from difficult problems where partial progress goes unrewarded. This motivates learned reward models that can evaluate proof quality beyond binary verification. However, comparing reward models is challenging since it typically requires expensive RL training ablations. To address this, we introduce \textbf{FormalRewardBench}, the first benchmark for evaluating reward models in formal theorem proving with Lean 4. Our benchmark consists of 250 preference pairs where correct proofs are paired with incorrect variants generated through five expert curated error injection strategies: forced mistakes, minimal single-point variations, verbose incorrect proofs, natural language justification, and Python code injection. We evaluate frontier LLMs (e.g., Claude Opus 4.5), judge LLMs (e.g., CompassJudger-1-14B), general-purpose LLMs (e.g., Qwen2.5-72B-Instruct), and specialized theorem proving models (e.g., DeepSeek-Prover-V2-7B). Our results reveal that frontier LLMs achieve the highest performance (59.8%) while specialized theorem provers perform the worst (24.4%), suggesting that theorem proving ability does not transfer to proof evaluation. We provide further insights on various error injection mechanisms, highlighting the challenging nature of most injection mechanisms. We release \textbf{FormalRewardBench} publicly to encourage more research on developing reward models in formal mathematics.
Zeynel A. Uluşan, Burak S. Akbudak, Can S. Erer +1
Automated formal verification remains challenging for large language models because data for proof assistants and verification-aware languages is scarce, and correctness depends on satisfying precise machine-checkable specifications rather than producing plausible code. This thesis studies how verifier environments can improve LLM generation of verified programs and proofs through reinforcement learning from verifiable rewards (RLVR) and verifier-guided inference-time search. First, we train open-source models in Dafny with RLVR using Group Relative Policy Optimization (GRPO) and related variants, assembling generated candidates into complete programs and scoring them with compiler and verifier outcomes. Initial experiments on an APPS-derived Dafny dataset increased verified reward from 2.2% to 58.1%, but revealed specification hacking, where models exploit weak formal specifications instead of implementing the intended solutions. After filtering underspecified and vulnerable tasks, multi-turn RLVR on the refined benchmark improves the verified pass rate from 9.7% to 31.1%. Second, we develop a verifier-guided inference scaffold in Lean that treats proof generation as structured search over decomposed subgoals, verifier feedback, diagnostics, and repair. With a fixed base model, the full scaffold with proof reviser improves pass rate on an initial VeriCoding pilot set from 46.2% under direct repair to 69.2%. On the larger VERINA dataset, whole-task decomposition plus proof reviser solves 7 of 42 previously unsolved tasks. We also introduce Dalek-Bench, a repository-scale Lean benchmark derived from the Rust curve25519-dalek verification project; preliminary results remain weak, indicating that stronger progress evaluation and task-specific tool-use policies are still needed.
Neurosymbolic systems rely on mathematical solvers to guarantee reasoning correctness, yet solvers are fundamentally blind to whether a formal translation maintains strict reference-equivalence to a designated formalization. We formalize this vulnerability as Verdict-Preserving-Unfaithfulness (VPU): a failure mode where an incorrect encoding executes successfully and matches the expected verdict. We theoretically prove that structural, verdict-only verification heuristics are mathematically bounded to chance-level detection on these deceptively valid traces. To resolve this, we introduce Generative Verification (GenV), which distills an offline Z3-equivalence oracle into a reference-free, continuous reference-equivalence score by repurposing the language model's native vocabulary space. Mechanistic analysis via decision-projected logit lenses and sparse autoencoders shows this generative readout natively extracts precise spatial error coordinates without explicit localization training. Empirically, our oracle-mined verifier (GenV+HN) achieves 0.961 AUROC in reference-equivalence verification, generalizes zero-shot across unseen translators and divergent formal styles, and yields an 11.3-point downstream accuracy gain in agentic test-time compute allocation.