An Imperfect Verifier is Good Enough: Learning with Noisy Rewards
cs.LG, cs.AI
Submitted: 2026-04-09
Updated: 2026-09-27
Comments: 31 pages, 11 figures, 9 tables
Code: https://github.com/huggingface/MathVerify
License: http://creativecommons.org/licenses/by-sa/4.0/
The gist: Reinforcement Learning with Verifiable Rewards (RLVR) is widely used for post-training Large Language Models, but practical verifiers can make errors.
Terminology
Abstract
Reinforcement Learning with Verifiable Rewards (RLVR) is widely used for post-training Large Language Models, but practical verifiers can make errors. We study how the rate and structure of reward noise affect RLVR in code generation, with a preliminary scientific-reasoning check. In multi-seed Qwen3 8B experiments on MBPP, mean validation reward over two fixed late evaluations is within 1 percentage point of the clean baseline at the tested resampled group-rollout noise rates through 20%, and within about 2 points at 30%. Confidence intervals allow larger losses; these point estimates do not establish a general tolerance threshold. We also examine full-program pass@k, four controlled noise structures, two model-based verifiers, and policy models from three families spanning 4B-9B parameters. We derive conditional advantage distributions for symmetric and asymmetric group noise, including retained format penalties, and show why clipping limits simple gradient-scaling arguments. The analysis identifies information preserved by whole-group corruption and limits on interpreting our asymmetric sweep as a precision-recall comparison. Overall, the results indicate that imperfect verification can support effective RLVR in the tested settings, while aggregate error rates alone do not characterize the learning signal.
Sources
- Program Synthesis with Large Language Models
- Constitutional AI: Harmlessness from AI Feedback
- Reinforcement Learning with Verifiable yet Noisy Rewards under Imperfect Verifiers
- Exploration vs Exploitation: Rethinking RLVR through Clipping, Entropy, and Spurious Reward
- SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training
- Process Reinforcement through Implicit Rewards
- DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning
- Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data
- Rubrics as Rewards: Reinforcement Learning Beyond Verifiable Domains
- AdvancedIF: Rubric-Based Benchmarking and Reinforcement Learning for Advancing LLM Instruction Following
- Measuring Mathematical Problem Solving With the MATH Dataset
- From Accuracy to Robustness: A Study of Rule- and Model-based Verifiers in Mathematical Reasoning
- Reinforcement Learning via Self-Distillation
- Evaluating Feature Dependent Noise in Preference-based Reinforcement Learning
- Let's Verify Step by Step
- Examining Reasoning LLMs-as-Judges in Non-Verifiable LLM Post-Training
- Understanding R1-Zero-Like Training: A Critical Perspective
- Noise-corrected GRPO: From Noisy Rewards to Unbiased Gradients
- Rate or Fate? RLV$^\varepsilon$R: Reinforcement Learning with Verifiable Noisy Rewards
- GPQA: A Graduate-Level Google-Proof Q&A Benchmark
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