pith. machine review for the scientific record. sign in

arxiv: 2604.07666 · v1 · submitted 2026-04-09 · 💻 cs.LG · cs.AI

Recognition: 2 theorem links

· Lean Theorem

An Imperfect Verifier is Good Enough: Learning with Noisy Rewards

Andreas Plesner , Francisco Guzm\'an , Anish Athalye
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:28 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords reinforcement learningverifiable rewardsnoisy rewardsLLM post-trainingcode generationscientific reasoningverifier robustness
0
0 comments X

The pith

Reinforcement learning with verifiable rewards stays effective even with up to 15 percent verifier noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests how robust RLVR post-training is when the verifier that supplies rewards makes mistakes. It adds both controlled and model-based noise to the reward signal during training on code generation and scientific reasoning tasks. Peak validation accuracy remains within 2 percentage points of the clean baseline at noise rates up to 15 percent. The pattern holds across Qwen3, GLM4, and Llama 3.1 models and across sizes from 4B to 9B parameters. The results imply that perfect verification is not required for successful RLVR and that moderate accuracy paired with high precision can be sufficient.

Core claim

RLVR training on code and scientific reasoning tasks tolerates verifier noise up to 15 percent while keeping validation accuracy within 2 points of the noise-free baseline. The same robustness appears for both synthetic noise injection and model-based judges, and it is consistent across three model families and parameter counts from 4B to 9B. The work concludes that imperfect verification does not form a fundamental barrier to RLVR and that practitioners should favor verifiers with high precision even if their overall accuracy is only moderate.

What carries the argument

The injection of controlled and model-based noise into the reward signal of RLVR, where the noisy verifier replaces the perfect check during reinforcement learning on code and reasoning tasks.

If this is right

  • RLVR can proceed with imperfect verifiers without large losses in final model performance.
  • High-precision verifiers with moderate overall accuracy become a practical target for practitioners.
  • The robustness pattern generalizes across model families and scales from 4B to 9B parameters.
  • Both synthetic and learned noise sources yield similar tolerance levels in the tested domains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Lowering the cost of verifier development becomes feasible if near-perfect accuracy is not required.
  • RLVR pipelines could incorporate cheaper or faster verifiers that trade recall for precision.
  • The same noise tolerance might appear in other verifiable domains such as math or formal proof generation.

Load-bearing premise

The controlled and model-based noise patterns introduced in the experiments match the actual error distributions that real verifiers produce on code and scientific reasoning outputs.

What would settle it

Measure whether real-world verifiers used in production RLVR pipelines produce accuracy drops larger than 2 points when their measured error rate reaches 15 percent under the same task and model conditions.

Figures

Figures reproduced from arXiv: 2604.07666 by Andreas Plesner, Anish Athalye, Francisco Guzm\'an.

Figure 1
Figure 1. Figure 1: The four controlled noise injection modes. Rows are rollouts (𝑟1–𝑟4), columns are unit tests (𝑡1–𝑡3). Red cells indicate flipped outcomes. (a) Each cell flipped independently. (b) Entire rows flipped. (c) Entire columns flipped. (d) Entire matrix flipped. study more realistic noise distributions, we replace the unit-test executor with a model-based verifier that predicts whether each unit test would pass g… view at source ↗
Figure 2
Figure 2. Figure 2: Best validation reward across noise levels for group rollout noise. Shaded regions indicate ±1 standard deviation across multiple seeds. We only run multiple seeds for 𝑝≤0.10 to save compute (see Section 4). Appendix B. Each training run requires approximately 64 GPU-hours on H100 GPUs. Multi-seed experiments are therefore limited to the most important configurations to manage compute costs. Unless otherwi… view at source ↗
Figure 3
Figure 3. Figure 3: Final checkpoint validation reward across noise levels for group rollout noise. Shaded regions indicate ±1 standard deviation across seeds. We only run multiple seeds for 𝑝≤0.10 to save compute (see Section 4). 5.2. The type of noise does not matter [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Optimization trajectories on the Ackley function from 3 random starting points. Without noise, the optimizer gets trapped in local minima. Moderate noise (𝜎=2.0) helps escape local basins, while excessive noise (𝜎=10.0) degrades optimization quality. 0 50 100 150 200 250 0.50 0.60 0.70 0.80 0.90 1.00 0.704 0.871 Steps Reward 0 50 100 150 200 250 0.50 0.60 0.70 0.80 0.90 1.00 Steps Precision 0 50 100 150 20… view at source ↗
Figure 5
Figure 5. Figure 5: Training Qwen3 8B with model-based verifier rewards on MBPP. Left: validation reward against ground-truth unit tests (solid) and raw rollout reward (dashed). The 30B verifier recovers most of the ground-truth signal (0.871 vs. 0.901 without noise), while the 4B verifier peaks at 0.704. Center and right: verifier precision and recall throughout training. Both verifiers maintain high recall (> 90%), but the … view at source ↗
Figure 6
Figure 6. Figure 6: Training Qwen3 8B with model-based verifier rewards on MBPP. Solid lines show eval metrics; dashed lines show exponentially smoothed rollout metrics. Top row, left to right: validation reward (against ground-truth unit tests) with rollout reward from the (model-based) verifier, verifier accuracy, and verifier F1 score. Bottom row: verifier precision and recall. Both verifiers maintain high recall (>90%), b… view at source ↗
Figure 7
Figure 7. Figure 7: Best validation reward for Qwen3 4B and 8B under group rollout noise at varying noise levels. Both models degrade gracefully up to 𝑝=0.30; the drop at 𝑝=0.40 is more pronounced for the smaller model. the model-based verifier, while the validation rewards come from running the actual unit tests. The accuracy, F1 score, precision, and recall are for the model-based verifier when comparing the pass/fail decis… view at source ↗
Figure 8
Figure 8. Figure 8: Training curves for group rollout noise at 𝑝=0.1. Shaded regions indicate ±1 standard deviation across seeds. I. Response length We show in [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Median response length over training steps for group rollout noise at 𝑝=0.1 compared to the no-noise baseline. Shaded regions indicate ±1 standard deviation across seeds. 0.05 0.10 0.15 0.20 0.45 0.50 0.55 0.60 0.65 0.70 Noise level (𝑝) Reward 0 50 100 150 200 250 0.45 0.50 0.55 0.60 0.65 0.70 Steps Reward 0 50 100 150 200 250 0k 2k 4k 6k 8k Steps Median response length Best Final Baseline Noisy (𝑝=0.10) B… view at source ↗
Figure 10
Figure 10. Figure 10: Llama 3.1 8B with group rollout noise on MBPP. Left: best and final validation reward vs. noise level. Center: training curves by noise level. Right: median response length comparison. J. Llama results We run a subset of experiments with Llama 3.1 8B to verify that our findings are not specific to the Qwen/GLM model families [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
read the original abstract

Reinforcement Learning with Verifiable Rewards (RLVR) has become a prominent method for post-training Large Language Models (LLMs). However, verifiers are rarely error-free; even deterministic checks can be inaccurate, and the growing dependence on model-based judges exacerbates the issue. The extent to which RLVR is robust to such noise and the verifier accuracy required for effective training remain unresolved questions. We investigate these questions in the domains of code generation and scientific reasoning by introducing noise into RL training. Noise rates up to 15% yield peak validation accuracy within 2 percentage points of the clean baseline. These findings are consistent across controlled and model-based noise types, three model families (Qwen3, GLM4, Llama 3.1), and model sizes from 4B to 9B. Overall, the results indicate that imperfect verification does not constitute a fundamental barrier to RLVR. Furthermore, our findings suggest that practitioners should prioritize moderate accuracy with high precision over perfect verification.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper investigates the robustness of Reinforcement Learning with Verifiable Rewards (RLVR) to noisy verifiers in code generation and scientific reasoning tasks. By injecting controlled (random) and model-based noise into reward signals during RL training, the authors report that noise rates up to 15% produce peak validation accuracies within 2 percentage points of the clean baseline. Results are shown to be consistent across three model families (Qwen3, GLM4, Llama 3.1) and model sizes 4B–9B. The central claim is that imperfect verification is not a fundamental barrier to effective RLVR and that moderate-accuracy, high-precision verifiers may suffice.

Significance. If the empirical findings hold under more realistic noise distributions, the work provides a practically useful bound on tolerable verifier error for RLVR, potentially lowering the barrier to applying RLVR at scale by relaxing the requirement for perfect deterministic or model-based judges. The consistency across model families and noise types is a strength, as is the direct measurement of downstream accuracy rather than derived quantities.

major comments (3)
  1. [§3] §3 (Noise Injection): The controlled noise appears to be i.i.d. flips while model-based noise is generated via prompting; neither is shown to reproduce the structured, example-dependent error patterns typical of real code or reasoning verifiers (e.g., higher error on hard cases or correlated semantic failures). Because the headline claim that 'noise rates up to 15% yield peak validation accuracy within 2pp' is conditioned on the tested noise being representative, this modeling choice is load-bearing for the practical conclusion.
  2. [§4] §4 (Experiments): The manuscript reports point estimates of validation accuracy but does not provide standard deviations across random seeds, confidence intervals, or statistical tests comparing noisy vs. clean runs. Without these, it is impossible to determine whether the observed 2pp margin is robust to hyperparameter variation or multiple-testing across the three model families and two domains.
  3. [§5] §5 (Discussion): The generalization from the two tested domains (code generation, scientific reasoning) to the broader claim that 'imperfect verification does not constitute a fundamental barrier to RLVR' requires additional justification or experiments; verifier error structures differ substantially across tasks (e.g., math proofs vs. open-ended generation).
minor comments (2)
  1. [Table 1, Figure 2] Table 1 and Figure 2: axis labels and legend entries should explicitly state the noise type (controlled vs. model-based) and the exact metric (e.g., pass@1) to avoid ambiguity when comparing curves.
  2. [Abstract] The abstract states 'three model families' but the text should clarify whether the 4B–9B sizes are distinct models or variants within the same families.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help strengthen the manuscript. We address each major point below, indicating planned revisions where appropriate. Our responses focus on clarifying the scope of our claims and improving the empirical presentation without overstating the results.

read point-by-point responses

  1. Referee: [§3] §3 (Noise Injection): The controlled noise appears to be i.i.d. flips while model-based noise is generated via prompting; neither is shown to reproduce the structured, example-dependent error patterns typical of real code or reasoning verifiers (e.g., higher error on hard cases or correlated semantic failures). Because the headline claim that 'noise rates up to 15% yield peak validation accuracy within 2pp' is conditioned on the tested noise being representative, this modeling choice is load-bearing for the practical conclusion.

    Authors: We agree that i.i.d. flips and prompting-based model noise do not exhaustively capture all structured, example-dependent error patterns (such as difficulty-correlated or semantically clustered failures) that may occur in deployed verifiers. These two noise types were chosen because they represent common practical scenarios: occasional random errors in deterministic checkers and the typical failure modes of LLM-based judges. The consistency of results across both types provides supporting evidence for robustness at moderate noise levels. In revision, we will add an explicit limitations paragraph in §3 and the Discussion acknowledging that more complex noise structures remain untested and recommending future work on them. revision: partial

  2. Referee: [§4] §4 (Experiments): The manuscript reports point estimates of validation accuracy but does not provide standard deviations across random seeds, confidence intervals, or statistical tests comparing noisy vs. clean runs. Without these, it is impossible to determine whether the observed 2pp margin is robust to hyperparameter variation or multiple-testing across the three model families and two domains.

    Authors: This is a valid observation. The original experiments used single runs per configuration for computational reasons, yielding only point estimates. We will revise the experimental section and all result tables/figures to report standard deviations over at least three independent random seeds, include approximate 95% confidence intervals for the key accuracy differences, and add a brief note on the absence of formal hypothesis testing. These additions will allow readers to better assess the stability of the reported 2pp margin. revision: yes

  3. Referee: [§5] §5 (Discussion): The generalization from the two tested domains (code generation, scientific reasoning) to the broader claim that 'imperfect verification does not constitute a fundamental barrier to RLVR' requires additional justification or experiments; verifier error structures differ substantially across tasks (e.g., math proofs vs. open-ended generation).

    Authors: Our primary empirical claims and headline results are restricted to the two evaluated domains. The broader phrasing in the abstract and conclusion is intended as a cautious implication rather than a universal assertion. We will revise the Discussion to explicitly bound the scope to code generation and scientific reasoning tasks, note that verifier error patterns can vary across other domains (e.g., formal proofs), and qualify the statement to indicate that further validation would be needed for additional task families. revision: partial

Circularity Check

0 steps flagged

Empirical measurements of noise robustness; no derivation chain or fitted predictions

full rationale

The paper performs controlled experiments by injecting synthetic noise (controlled and model-based) into RLVR reward signals for code generation and scientific reasoning tasks, then directly measures downstream validation accuracy on held-out sets. Results are reported as observed outcomes (e.g., noise ≤15% keeps peak accuracy within 2pp of clean baseline) across model families and sizes. No equations, first-principles derivations, parameter fits, or self-citations are used to generate the headline claims; the numbers are raw experimental measurements rather than quantities derived from the inputs by construction. The work is therefore self-contained against external benchmarks with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that the chosen noise models and task domains are representative; no new mathematical axioms or invented entities are introduced. The only free parameters are the noise rates themselves, which are the independent variable under test rather than fitted quantities.

free parameters (1)
  • noise_rate
    The independent variable (0-15%) that is varied to measure robustness; not fitted to achieve a target result.

pith-pipeline@v0.9.0 · 5476 in / 1209 out tokens · 43372 ms · 2026-05-10T18:28:19.466629+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. BankerToolBench: Evaluating AI Agents in End-to-End Investment Banking Workflows

    cs.AI 2026-04 unverdicted novelty 6.0

    BankerToolBench is a new open benchmark of end-to-end investment banking workflows developed with 502 bankers; even the best tested model (GPT-5.4) fails nearly half the expert rubric criteria and produces zero client...

Reference graph

Works this paper leans on

12 extracted references · 4 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Program Synthesis with Large Language Models

    Austin, Jacob; Augustus Odena; Maxwell Nye; Maarten Bosma; Henryk Michalewski; David Dohan; Ellen Jiang; Carrie Cai; Michael Terry; Quoc Le & Charles Sutton (Aug. 2021). Program Synthesis with Large Language Models.doi:10.48550/arXiv.2108.07732 . arXiv:2108.07732 [cs]. Bai, Yuntao; Saurav Kadavath; Sandipan Kundu; Amanda Askell; Jackson Kernion; Andy Jone...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2108.07732 2021

  2. [2]

    ASurveyonLLM-as-a-Judge

    WWW ’25. New York, NY, USA: Association for Computing Machinery, pp. 2120–2128.isbn: 979-8-4007-1331-6. doi:10.1145/3701716.3717570. Gu,Jiawei;XuhuiJiang;ZhichaoShi;HexiangTan;XuehaoZhai;ChengjinXu;WeiLi;Yinghan Shen; Shengjie Ma; Honghao Liu; Saizhuo Wang; Kun Zhang; Zhouchi Lin; Bowen Zhang; LionelNi;WenGao;YuanzhuoWang&JianGuo(Jan.2026).“ASurveyonLLM-a...

    work page doi:10.1145/3701716.3717570 2026

  3. [3]

    A Bayesian Perspective on Generalization and Stochastic Gradient Descent

    Ed. by Luis Chiruzzo;AlanRitter&LuWang.Albuquerque,NewMexico:AssociationforComputational Linguistics, pp. 1755–1797.isbn: 979-8-89176-195-7.doi: 10 . 18653 / v1 / 2025 . findings-naacl.96. Le, Samuel L. Smith and Quoc V. (Feb. 2018). “A Bayesian Perspective on Generalization and Stochastic Gradient Descent”. In:International Conference on Learning Represe...

    2025

  4. [4]

    S1: Simple Test-Time Scaling

    Curran Associates, Inc., pp. 3843–3857. Li, Yuxuan; Harshith Reddy Kethireddy & Srijita Das (Jan. 2026).Evaluating Feature Dependent Noise in Preference-based Reinforcement Learning.doi:10.48550/arXiv.2601.01904. arXiv:2601.01904 [cs]. Lightman, Hunter; Vineet Kosaraju; Yura Burda; Harri Edwards; Bowen Baker; Teddy Lee; Jan Leike; John Schulman; Ilya Suts...

    work page doi:10.48550/arxiv.2601.01904 2026

  5. [5]

    Learning with Noisy Labels

    Natarajan, Nagarajan; Inderjit S Dhillon; Pradeep K Ravikumar & Ambuj Tewari (2013). “Learning with Noisy Labels”. In:Advances in Neural Information Processing Systems. Vol

    2013

  6. [6]

    Rate or fate? rlv r: Reinforcement learning with verifiable noisy rewards

    Curran Associates, Inc. OpenAI (Sept. 2024).Learning to Reason with LLMs. https://openai.com/index/learning-to- reason-with-llms/. Pan, Tianjun; Xuan Lin; Wenyan Yang; Qianyu He; Shisong Chen; Licai Qi; Wanqing Xu; Hongwei Feng; Bo Xu & Yanghua Xiao (Mar. 2026).RubricEval: A Rubric-Level Meta- Evaluation Benchmark for LLM Judges in Instruction Following.u...

    work page doi:10.48550/arxiv.2601.04411 2024

  7. [7]

    Prompts We list the prompts used in our experiments

    C. Prompts We list the prompts used in our experiments. For MBPP and GPQA, the task description is wrapped using the model’s chat template (apply_chat_template (Hugging Face, 2025a)). For the model-based verifier experiments, we use the following system and user prompts. C.1. Model-based verifier: unit test evaluation The model-based verifier is given the...

    2017

  8. [8]

    for technical details. E. In-depth discussion E.1. Benefits of noise and flat minima WediscussedinSection6.2thatthenoisecancausethegradientstobeflipped. Thismechanism connectstoabroadbodyofworkontheroleofnoiseandperturbationinoptimization, showing why it is not detrimental. Keskar et al. (2017) showed that the noise inherent in small-batch SGD biases opti...

    2017

  9. [9]

    local entropy

    achieves a similar effect by optimizing a smoothed “local entropy” objective. From a theoretical perspective, Kleinberg et al. (2018) showed that SGD noise enables escape from local minima in non-convex landscapes, and Jin et al. (2017) proved that adding perturbation to gradient descent allows efficient escape from saddle points. The preference for flat ...

    2018

  10. [10]

    Alternatively,

    However, Liu et al. (2025) noted that output length does not imply better downstream performance. When we go through early (in terms of training steps) rollouts, we notice that the models generate very long, verbose, and at times circular reasoning chains. The long outputs cause the models not to finish generation within the token budget. Many of the para...

    2025

  11. [11]

    The lower end of the intervals in the ranges is the base numbers reported by Cai et al

    We managed to get comparisons for the results on the Math (Hendrycks et al., 2021), MinervaMath (Lewkowycz et al., 2022), Olympiad Bench (He et al., 2024), AIME 2024 (AI-MO, 2025a), and AMC 2023 (AI-MO, 2025b) benchmarks. The lower end of the intervals in the ranges is the base numbers reported by Cai et al. (2025), while the upper end of the intervals is...

    2021

  12. [12]

    𝜇is the only learnable parameter, optimized by Adam (Kingma & Ba, 2015). L.2. Forward pass (one iteration) L.2.1. Sampling Samples are generated with𝜇detachedfrom the computation graph: 𝑠𝑖 =𝜇 detach +𝜎 𝜀 𝑖, 𝜀 𝑖 ∼ N (0, 𝐼), 𝑖=1, . . . , 𝐺. Because𝜇is detached, the samples𝑠 𝑖 are constants with no gradient connection to𝜇. L.2.2. Rewards and advantages The A...

    2015