Reinforcement Learning with Verifiable Rewards (RLVR) has become a powerful approach for improving the reasoning capabilities of large language models (LLMs). While RLVR is designed for tasks with verifiable ground-truth answers, real-world verifiers (e.g., static code checkers) can introduce errors into the reward signal. Prior analyses have largely treated such errors as random and independent across samples, concluding that errors merely slow training with limited effect on final performance. However, practical verifiers tend to exhibit systematic errors. This introduces a risk of models learning unwanted consistent behavior from a structurally incorrect reward signal. In this work, we study the impact of such systematic verification errors on RLVR. Through controlled experiments on arithmetic tasks, we show that systematic false negatives lead to similar effects as random noise. On the other hand, systematic false positives can cause a wide range of behaviors from sub-optimal plateaus to performance collapse. Crucially, these outcomes are not determined by the overall error rate but by the specific pattern of introduced errors, making pre-hoc mitigation difficult. Our results show that, in contrast to prior conclusions, realistic verification errors can critically shape RLVR outcomes and that verifier quality has to be understood beyond its sample-level error rate.
Figures & tables
Figure 1: Overview of our results. On the left, we categorize systematic errors in RLVR based on their effect on training dynamics, including delayed training, suboptimal plateaus, and training collapse. On the right, we show representative results of these different error types in a controlled experiment using the Reasoning Gym dataset ( Stojanovski et al., 2025 ) .
Figure 2: Main results across two models (left: OLMo, right: Qwen). We observe the following trends: (i) random noise and systematic FNR lead to delayed training but similar final performance, while systematic FPR leads to plateauing or collapsing behaviors, depending on the trigger used. We provide results with Dr. GRPO and SAPO in Figure 12 .
Figure 3: Output examples under different word-based FPs. The clean and Certainly -triggered runs produce correct arithmetic reasoning, while the python -triggered fails by using incorrect code-based output.
Figure 4: Distribution of tri-gram and final performance (OLMo). The trigrams that remained below 50% FPR are highlighted in red.
Figure 5
Figure 7: Alternation of oracle and noisy verifier. FPR is measured only for the steps with the noisy verifier.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Dataset
Dataset size
50,000
min_terms
3
max_terms
6
min_digits
3
max_digits
6
min_decimal_places
3
Appendix
Table 1: Training Configuration
Figure 8: An example query used for training.
Figure 9
Figure 11: Examples for learned behaviors in the trigram-based error patterns. For a question 481.869 + 519.413 + 776.7711 - 734.133 , (ground truth: 1043.9201), we provide a representative output example for positive / negative conditional advantage (defined in Section 4.3 ), and low / high frequency. For each caption, we also provide the conditional advantage (CA), frequency (Freq), and the converged reward (R) of the when the trigram is used as the FP trigger.
Figure 12: Results with DAPO, Dr. GRPO, and SAPO.
Figure 13: Multi-seed runs across two models. Across three random seeds, the qualitative dynamics remain stable: clean training improves the reward consistently, relative-error and Certainly -triggered FP plateau, and python -triggered FP collapses.
Figure 14: Systematic but non-deterministic FP. (i) For error patterns that lead to a suboptimal plateau with 100% FPR (relative-error based FP, “Certainly”-triggered FP), the training dynamics improve as error rate decreases. (ii) For triggers that induce collapse (“python”-triggered FP), when p=80% , training is substantially delayed, but it no longer fully collapses as it does in the main experiment with p=100% . Interestingly, the Qwen run initially follows a collapsing trajectory, but the reward begins to recover after around step 100, eventually reaching a suboptimal level within our timeframe. When p is further reduced to 50% , the model no longer exploits the false-positive pattern, and the training curve is close to ideal, with only a slight delay.
Figure 15: Combined FP/FN error. When combining FP and FN rules (denoted as MIX in the legend), we consider two variants for rollouts that satisfy both the FP and FN conditions: treating such rollouts as (i) FNs or (ii) FPs. Compared with the pure-FP setting, we find that adding format-based FN rule has no strong effect, preserving the same qualitative behavior. In contrast, language-based FN rule with variant (i) results in a mere delay for Qwen because the FN rule forces Qwen to generate its answer in Chinese (as analyzed in Section 4.5 of the main paper), which naturally suppresses the generation of the FP trigger "Certainly". For OLMo, however, the same setting leads to complete collapse.
Figure 16: Results on OLMo3-32B. The qualitative observation is consistent with the main experiments on smaller models, where verification errors result in delay (random noise and systematic FN), plateauing (relative-error-based and "Certainly"-triggered FP), or collapse ("python"-triggered FP). We note that the gap between the ideal run and the delayed runs is larger because, due to resource constraints, we trained the model for 250 steps instead of 500.
Reinforcement learning with verifiable rewards (RLVR) has become a principal recipe for improving the reasoning of large language models (LLMs). It rests on the assumption that the checker scoring each response is itself correct. In practice, rule-based checkers reject correct answers in unexpected formats, unit tests accept incorrect programs, and LLM judges can be led to approve empty or adversarial responses. This work surveys 58 studies that measure, analyze, or mitigate such verifier errors. We ask what errors verifiers make, how they affect training, which mitigations work against which errors, and how the findings extend to self-generated, rubric- and judge-based, and verifier-free rewards. The evidence suggests that the structure of an error matters more than its rate. Random errors mostly slow learning, though under a fixed budget or KL regularization they also lower final accuracy. False positives that the policy can learn to trigger can cause plateaus or collapse, sometimes even with a low initial error rate. Whether a verifier wrongly rejects or accepts depends on what it checks and who built it, not on its family. Its errors shift as the policy improves, and most mitigations have been tested against only one kind of error. We compile the findings into comparison tables, reconcile apparent conflicts in the literature, and propose a reporting protocol to make future results comparable.
Savyasachi Gupta, Aniruddha Srinivas Joshi
Independent Researcher TX, USA · Independent Researcher CA, USA
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.
Reinforcement Learning with Verifiable Rewards (RLVR) replaces costly human labeling with automated verifiers. To reduce verifier hacking, many RLVR systems binarize rewards to {0,1}, but imperfect verifiers inevitably introduce false negatives (rejecting correct answers) and false positives (accepting incorrect ones). We formalize verifier unreliability as a stochastic reward channel with asymmetric noise rates ρ0 and ρ1 -- the FP rate and the FN rate, respectively. From this abstraction we derive two lightweight corrections: (i) a backward correction that yields an unbiased surrogate reward and thus an unbiased policy-gradient estimator in expectation, and (ii) a forward correction that reweights score-function terms so the expected update aligns with the clean gradient direction and requires only the FN rate. We implement both as lightweight hooks in a group relative policy optimization pipeline, both corrections improve RLVR for math reasoning under synthetic and real verifier noise, with the forward variant being more stable under heavier noise. Finally, an appeals mechanism with a lightweight LLM verifier estimates the FN rate online and further improves performance.
Xin-Qiang Cai, Wei Wang, Feng Liu +3
RIKEN AIP, Tokyo, Japan · The University of Tokyo, Tokyo, Japan · The University of Melbourne, Melbourne, Australia