pith. machine review for the scientific record. sign in

arxiv: 2605.11865 · v1 · submitted 2026-05-12 · 📊 stat.ML · cs.LG

Recognition: 2 theorem links

· Lean Theorem

Variance-aware Reward Modeling with Anchor Guidance

Fan Zhou, Liangyu Zhang, Ruijian Han, Shuxing Fang

Pith reviewed 2026-05-13 05:05 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords variance-aware reward modelinganchor guidancenon-identifiabilityBradley-Terry modelpluralistic preferencesRLHFGaussian reward modelsconvergence rate
0
0 comments X

The pith

Two coarse response-level anchors suffice to identify both mean and variance in Gaussian reward models from pairwise preferences.

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

Standard Bradley-Terry models compress disagreement among pluralistic preferences into narrower reward margins. Gaussian reward models that output both a mean and a variance can retain that disagreement information, yet pairwise preferences alone leave the mean and variance unidentifiable. The paper augments the data with two coarse response-level anchor labels and proves that these two labels are enough to restore identifiability. It supplies a joint training objective and derives non-asymptotic convergence rates for the estimated mean and variance functions. On both simulated data and four real-world datasets with diverging preferences, the resulting models improve reward accuracy and downstream RLHF performance in PPO training and best-of-N selection.

Core claim

Anchor-guided Variance-aware Reward Modeling augments standard pairwise preference data with two coarse response-level anchor labels. This augmentation renders the joint mean-and-variance estimation problem identifiable, supports a joint training objective, and yields non-asymptotic convergence guarantees for both the mean and variance estimators. The framework is shown to outperform standard Bradley-Terry and unanchored Gaussian baselines on simulation studies and on four real-world diverging-preference datasets, with corresponding gains in PPO and best-of-N RLHF pipelines.

What carries the argument

Anchor-guided augmentation that supplies two coarse response-level labels to break the non-identifiability of Gaussian mean-variance reward models trained on pairwise preferences.

If this is right

  • Reward models can now preserve disagreement information instead of shrinking margins.
  • Joint mean-variance estimation becomes statistically consistent with explicit convergence rates.
  • Downstream RLHF procedures such as PPO and best-of-N selection receive higher-quality reward signals on pluralistic data.
  • The same two-anchor construction applies directly to any dataset already equipped with coarse response-level annotations.

Where Pith is reading between the lines

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

  • The identification argument may extend to other non-identifiable preference-learning problems once a small number of cheap auxiliary labels are introduced.
  • Data-collection protocols could be redesigned to request only two coarse anchors per response rather than dense preference annotations.
  • The convergence rates supply a concrete sample-complexity target for scaling variance-aware models to larger preference corpora.

Load-bearing premise

Two coarse response-level anchor labels can be obtained reliably and contain enough information to separate mean from variance without adding new biases.

What would settle it

A controlled simulation in which the true reward variance is known shows that the estimated variance function remains unidentifiable or fails to converge at the stated rate even after the two-anchor labels are added.

Figures

Figures reproduced from arXiv: 2605.11865 by Fan Zhou, Liangyu Zhang, Ruijian Han, Shuxing Fang.

Figure 1
Figure 1. Figure 1: Uncertainty representation. (a) BT vs BT(Hard) reward margin. (b) Estimated rˆ versus sˆ for Gaussian trained on HelpSteer2. (c) Estimated rˆ versus sˆ for Two-anchor-DeepSeek-Flash trained on HelpSteer2. (d) Pearson correlation between estimated rˆ and sˆ. Results are all based on the Llama-3.2-3B backbone. 0 20 40 60 80 Accuracy (%) 67.0 58.658.655.6 48.144.7 68.0 60.658.656.1 48.144.7 66.064.5 58.656.3 … view at source ↗
Figure 2
Figure 2. Figure 2: Anchor data efficiency on RewardBench Accuracy. Results are all based on the Llama￾3.2-1B backbone. this requirement, we obtain four diverging preference datasets with empirical soft labels: MultiPref￾8K [Miranda et al., 2025], HelpSteer2-9K [Wang et al., 2025a], HelpSteer3-20K [Wang et al., 2025b], PersonalLLM-40K [Zollo et al., 2025]. These datasets vary in both size and soft-label distributions, coverin… view at source ↗
Figure 3
Figure 3. Figure 3: Mean gold score of BoN and PPO. (a) BoN performance VS baselines. (b) BoN performance of Two-anchor under different reward distribution quantiles. (c) PPO performance VS baselines. Proxy reward models are all trained on the MultiPref. Uncertainty representation [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Test performance of Two-anchor versus baselines. (a) Anchor data efficiency under different [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Additional simulation results for Two-anchor versus baselines. (a) Anchor data efficiency [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reward mean and variance recovery. prompts x ∈ R 10 and responses y ∈ R 10 are drawn i.i.d. from N (0, I). The true reward is defined as r ∗ (x, y) = 2   x ⊤W y | {z } bilinear + 1 3 X 3 k=1 tanh x ⊤ak √ d  tanh y ⊤bk √ d  | {z } nonlinear   , (9) where W ∈ R d×d and {ak, bk} 3 k=1 ⊂ R d are fixed parameters drawn once from N (0, 1/d2 ) and N (0, I) respectively. The true standard deviation… view at source ↗
Figure 7
Figure 7. Figure 7: Soft-label distributions across four preference datasets. [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Anchor data efficiency on RewardBench Brier score. Results are all based on the Llama-3.2-1B backbone and across all four training datasets. 0 1 CE loss 0.6260.6730.6810.7260.729 1.130 0.6220.6690.6720.7260.729 1.130 0.6320.6400.6720.7260.729 1.130 0.6520.6570.662 0.7260.729 1.130 (a) Trained on MultiPref 0.5820.5900.5960.6020.608 0.720 0.5780.5820.5960.6140.621 0.720 0.5820.5900.5950.5960.596 0.720 0.5820… view at source ↗
Figure 9
Figure 9. Figure 9: Anchor data efficiency on RewardBench Cross-entropy. Results are all based on the Llama-3.2-1B backbone and across all four training datasets. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗
read the original abstract

Standard Bradley--Terry (BT) reward models are limited when human preferences are pluralistic. Although soft preference labels preserve disagreement information, BT can only express it by shrinking reward margins. Gaussian reward models provide an alternative by jointly predicting a reward mean and a reward variance, but suffer from a fundamental non-identifiability from pairwise preferences alone. We propose Anchor-guided Variance-aware Reward Modeling, a framework that resolves this non-identifiability by augmenting preference data with two coarse response-level anchor labels. Building on this, we prove that two anchors are sufficient for identification, develop a joint training objective and establish a non-asymptotic convergence rate for both the estimated reward mean and variance functions. Across simulation studies and four real-world diverging-preference datasets, our method consistently improves reward modeling performance and downstream RLHF, including PPO training and best-of-$N$ selection.

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

0 major / 3 minor

Summary. The manuscript proposes Anchor-guided Variance-aware Reward Modeling (AVRM) to address non-identifiability in Gaussian reward models trained solely on pairwise preferences. By augmenting data with two coarse response-level anchor labels, the authors prove that two anchors suffice for identification of both reward mean and variance, introduce a joint training objective, derive non-asymptotic convergence rates for the estimators, and report improved performance on synthetic simulations plus four real-world diverging-preference datasets for reward modeling and downstream RLHF tasks (PPO training and best-of-N selection).

Significance. If the identification result and convergence rates hold under the stated assumptions, the work supplies a minimal, practical augmentation that resolves a known limitation of variance-aware models without requiring strong parametric assumptions on preferences. The non-asymptotic rates and joint objective provide theoretical grounding, while the empirical gains on pluralistic datasets suggest utility for more robust RLHF. This could encourage wider adoption of uncertainty-aware reward models when human disagreement is present.

minor comments (3)
  1. [Abstract] Abstract: the four real-world datasets are not named, which reduces immediate clarity for readers scanning the contribution.
  2. [§4] §4 (Experiments): the simulation setup for verifying non-asymptotic rates would benefit from explicit parameter values and seed details to support reproducibility of the reported convergence behavior.
  3. [Related Work] Related Work: a brief comparison to prior approaches that mitigate non-identifiability via multiple annotators or richer preference data is absent, which would better situate the two-anchor contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript on Anchor-guided Variance-aware Reward Modeling (AVRM) and for recommending minor revision. The referee's description accurately captures the identification result, joint objective, convergence rates, and empirical improvements on pluralistic preference datasets. Since the report lists no major comments, we have no specific points to address point-by-point at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation proceeds from a standard Gaussian reward model with pairwise preferences (known to be non-identifiable), augments it with two external coarse anchor labels, proves identifiability from that augmented data, constructs a joint objective, and derives non-asymptotic rates under stated assumptions. None of these steps reduce by construction to fitted parameters, self-referential definitions, or load-bearing self-citations; the identification argument and rates are presented as consequences of the augmented model rather than renamings or ansatzes imported from prior author work. The central claims therefore retain independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard statistical assumptions for convergence rates and the availability of two coarse anchor labels; no free parameters or invented entities are explicitly introduced in the provided text.

axioms (1)
  • standard math Standard regularity conditions for non-asymptotic convergence rates in statistical estimation
    Invoked to establish the convergence rate for mean and variance estimators.

pith-pipeline@v0.9.0 · 5444 in / 1349 out tokens · 73266 ms · 2026-05-13T05:05:44.814788+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.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · 8 internal anchors

  1. [1]

    the method of paired comparisons , author=

    Rank analysis of incomplete block designs: I. the method of paired comparisons , author=. Biometrika , volume=. 1952 , publisher=

    work page 1952

  2. [2]

    , author=

    The method of paired comparisons for social values. , author=. The Journal of Abnormal and Social Psychology , volume=. 1927 , publisher=

    work page 1927

  3. [3]

    The Annals of Statistics , pages=

    Convergence rate of sieve estimates , author=. The Annals of Statistics , pages=. 1994 , publisher=

    work page 1994

  4. [4]

    Handbook of econometrics , volume=

    Large sample sieve estimation of semi-nonparametric models , author=. Handbook of econometrics , volume=. 2007 , publisher=

    work page 2007

  5. [5]

    Journal of econometrics , volume=

    Convergence rates and asymptotic normality for series estimators , author=. Journal of econometrics , volume=. 1997 , publisher=

    work page 1997

  6. [6]

    2000 , publisher=

    Empirical Processes in M-Estimation , author=. 2000 , publisher=

    work page 2000

  7. [7]

    1996 , publisher=

    Weak Convergence and Empirical Processes , author=. 1996 , publisher=

    work page 1996

  8. [8]

    2013 , publisher=

    Matrix Analysis , author=. 2013 , publisher=

    work page 2013

  9. [9]

    1989 , publisher=

    Generalized Linear Models , author=. 1989 , publisher=

    work page 1989

  10. [10]

    Annals of Statistics , volume=

    Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models , author=. Annals of Statistics , volume=

    work page

  11. [11]

    1998 , publisher=

    Asymptotic Statistics , author=. 1998 , publisher=

    work page 1998

  12. [12]

    and Tikhomirov, Vladimir M

    Kolmogorov, Andrei N. and Tikhomirov, Vladimir M. , journal=

    work page

  13. [13]

    The annals of statistics , pages=

    Optimal global rates of convergence for nonparametric regression , author=. The annals of statistics , pages=. 1982 , publisher=

    work page 1982

  14. [14]

    International Conference on Machine Learning , pages=

    Principled reinforcement learning with human feedback from pairwise or k-wise comparisons , author=. International Conference on Machine Learning , pages=. 2023 , organization=

    work page 2023

  15. [15]

    International Conference on Artificial Intelligence and Statistics , pages=

    Faithful heteroscedastic regression with neural networks , author=. International Conference on Artificial Intelligence and Statistics , pages=. 2023 , organization=

    work page 2023

  16. [16]

    arXiv preprint arXiv:2203.09168 , year=

    On the pitfalls of heteroscedastic uncertainty estimation with probabilistic neural networks , author=. arXiv preprint arXiv:2203.09168 , year=

    work page arXiv

  17. [17]

    Advances in Neural Information Processing Systems , volume=

    Reliable training and estimation of variance networks , author=. Advances in Neural Information Processing Systems , volume=

    work page

  18. [18]

    The Thirty-ninth Annual Conference on Neural Information Processing Systems , year=

    Pairwise Calibrated Rewards for Pluralistic Alignment , author=. The Thirty-ninth Annual Conference on Neural Information Processing Systems , year=

    work page

  19. [19]

    Forty-first International Conference on Machine Learning , year=

    Position: A Roadmap to Pluralistic Alignment , author=. Forty-first International Conference on Machine Learning , year=

    work page

  20. [20]

    AI feedback , author=

    Hybrid preferences: Learning to route instances for human vs. AI feedback , author=. Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) , pages=

    work page

  21. [21]

    The Thirteenth International Conference on Learning Representations , year=

    HelpSteer2-Preference: Complementing Ratings with Preferences , author=. The Thirteenth International Conference on Learning Representations , year=

    work page

  22. [22]

    2025 , eprint=

    HelpSteer3-Preference: Open Human-Annotated Preference Data across Diverse Tasks and Languages , author=. 2025 , eprint=

    work page 2025

  23. [23]

    Personal

    Thomas P Zollo and Andrew Wei Tung Siah and Naimeng Ye and Ang Li and Hongseok Namkoong , booktitle=. Personal. 2025 , url=

    work page 2025

  24. [24]

    Gonzalez and Ion Stoica , booktitle=

    Evan Frick and Tianle Li and Connor Chen and Wei-Lin Chiang and Anastasios Nikolas Angelopoulos and Jiantao Jiao and Banghua Zhu and Joseph E. Gonzalez and Ion Stoica , booktitle=. How to Evaluate Reward Models for. 2025 , url=

    work page 2025

  25. [25]

    Findings of the Association for Computational Linguistics: NAACL 2025 , pages=

    Rewardbench: Evaluating reward models for language modeling , author=. Findings of the Association for Computational Linguistics: NAACL 2025 , pages=

    work page 2025

  26. [26]

    The Fourteenth International Conference on Learning Representations , year=

    Bradley-Terry and Multi-Objective Reward Modeling Are Complementary , author=. The Fourteenth International Conference on Learning Representations , year=

    work page

  27. [27]

    Advances in Neural Information Processing Systems , volume=

    Regularizing hidden states enables learning generalizable reward model for llms , author=. Advances in Neural Information Processing Systems , volume=

    work page

  28. [28]

    International Conference on Machine Learning , pages=

    Scaling laws for reward model overoptimization , author=. International Conference on Machine Learning , pages=. 2023 , organization=

    work page 2023

  29. [29]

    Reward model ensembles help mitigate overoptimization.arXiv preprint arXiv:2310.02743, 2023

    Reward model ensembles help mitigate overoptimization , author=. arXiv preprint arXiv:2310.02743 , year=

    work page arXiv

  30. [30]

    Nature , volume=

    Large language models encode clinical knowledge , author=. Nature , volume=. 2023 , publisher=

    work page 2023

  31. [31]

    Learning and individual differences , volume=

    ChatGPT for good? On opportunities and challenges of large language models for education , author=. Learning and individual differences , volume=. 2023 , publisher=

    work page 2023

  32. [32]

    Advances in neural information processing systems , volume=

    Language models are few-shot learners , author=. Advances in neural information processing systems , volume=

    work page

  33. [33]

    Journal of machine learning research , volume=

    Palm: Scaling language modeling with pathways , author=. Journal of machine learning research , volume=

    work page

  34. [34]

    LLaMA: Open and Efficient Foundation Language Models

    Llama: Open and efficient foundation language models , author=. arXiv preprint arXiv:2302.13971 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  35. [35]

    A Survey of Large Language Models

    A survey of large language models , author=. arXiv preprint arXiv:2303.18223 , volume=

    work page internal anchor Pith review Pith/arXiv arXiv

  36. [36]

    Large Language Models: A Survey

    Large language models: A survey , author=. arXiv preprint arXiv:2402.06196 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  37. [37]

    Tulu 3: Pushing Frontiers in Open Language Model Post-Training

    Tulu 3: Pushing frontiers in open language model post-training , author=. arXiv preprint arXiv:2411.15124 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  38. [38]

    Open- rlhf: An easy-to-use, scalable and high-performance rlhf framework.arXiv preprint arXiv:2405.11143,

    Openrlhf: An easy-to-use, scalable and high-performance rlhf framework , author=. arXiv preprint arXiv:2405.11143 , volume=

    work page arXiv

  39. [39]

    Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback

    Training a helpful and harmless assistant with reinforcement learning from human feedback , author=. arXiv preprint arXiv:2204.05862 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  40. [40]

    Advances in neural information processing systems , volume=

    Training language models to follow instructions with human feedback , author=. Advances in neural information processing systems , volume=

    work page

  41. [41]

    Proximal Policy Optimization Algorithms

    Proximal policy optimization algorithms , author=. arXiv preprint arXiv:1707.06347 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  42. [42]

    DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models

    Deepseekmath: Pushing the limits of mathematical reasoning in open language models , author=. arXiv preprint arXiv:2402.03300 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  43. [43]

    Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems , pages=

    Jury learning: Integrating dissenting voices into machine learning models , author=. Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems , pages=

    work page 2022

  44. [44]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Value kaleidoscope: Engaging ai with pluralistic human values, rights, and duties , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    work page

  45. [45]

    The Twelfth International Conference on Learning Representations , year=

    Distributional Preference Learning: Understanding and Accounting for Hidden Context in RLHF , author=. The Twelfth International Conference on Learning Representations , year=

    work page

  46. [46]

    Findings of the Association for Computational Linguistics: ACL 2023 , pages=

    Which examples should be multiply annotated? active learning when annotators may disagree , author=. Findings of the Association for Computational Linguistics: ACL 2023 , pages=

    work page 2023

  47. [47]

    arXiv preprint arXiv:2310.16048 , year=

    Ai alignment and social choice: Fundamental limitations and policy implications , author=. arXiv preprint arXiv:2310.16048 , year=

    work page arXiv

  48. [48]

    International Conference on Machine Learning , pages=

    Diverging Preferences: When do Annotators Disagree and do Models Know? , author=. International Conference on Machine Learning , pages=. 2025 , organization=

    work page 2025

  49. [49]

    arXiv preprint arXiv:2502.18770 , year=

    Reward shaping to mitigate reward hacking in rlhf , author=. arXiv preprint arXiv:2502.18770 , year=

    work page arXiv

  50. [50]

    arXiv preprint arXiv:2503.22480 , year=

    Probabilistic uncertain reward model , author=. arXiv preprint arXiv:2503.22480 , year=

    work page arXiv

  51. [51]

    Uncertainty- aware reward model: Teaching reward models to know what is unknown, 2025

    Uncertainty-aware reward model: Teaching reward models to know what is unknown , author=. arXiv preprint arXiv:2410.00847 , year=

    work page arXiv

  52. [52]

    2024 , eprint=

    HelpSteer2: Open-source dataset for training top-performing reward models , author=. 2024 , eprint=

    work page 2024

  53. [53]

    Transactions on Machine Learning Research , issn=

    Open Problems and Fundamental Limitations of Reinforcement Learning from Human Feedback , author=. Transactions on Machine Learning Research , issn=. 2023 , url=

    work page 2023

  54. [54]

    Forty-second International Conference on Machine Learning , year=

    Reward Modeling with Ordinal Feedback: Wisdom of the Crowd , author=. Forty-second International Conference on Machine Learning , year=

    work page

  55. [55]

    2025 , eprint=

    Uncertainty Quantification for Large Language Model Reward Learning under Heterogeneous Human Feedback , author=. 2025 , eprint=

    work page 2025

  56. [56]

    The Fourteenth International Conference on Learning Representations , year=

    Learning Ordinal Probabilistic Reward from Preferences , author=. The Fourteenth International Conference on Learning Representations , year=

    work page

  57. [57]

    2025 , school=

    Reward Modeling for Human Preferences , author=. 2025 , school=

    work page 2025

  58. [58]

    2018 , publisher=

    High-Dimensional Probability: An Introduction with Applications in Data Science , author=. 2018 , publisher=

    work page 2018

  59. [59]

    arXiv preprint arXiv:2406.08469 , year=

    Pal: Pluralistic alignment framework for learning from heterogeneous preferences , author=. arXiv preprint arXiv:2406.08469 , year=

    work page arXiv

  60. [60]

    International Conference on Machine Learning , pages=

    MaxMin-RLHF: Alignment with Diverse Human Preferences , author=. International Conference on Machine Learning , pages=. 2024 , organization=

    work page 2024

  61. [61]

    The Thirteenth International Conference on Learning Representations , year=

    Pal: Sample-efficient personalized reward modeling for pluralistic alignment , author=. The Thirteenth International Conference on Learning Representations , year=

    work page

  62. [62]

    The Fourteenth International Conference on Learning Representations , year=

    Swap-guided Preference Learning for Personalized Reinforcement Learning from Human Feedback , author=. The Fourteenth International Conference on Learning Representations , year=

    work page

  63. [63]

    The Thirty-eighth Annual Conference on Neural Information Processing Systems , year=

    Personalizing Reinforcement Learning from Human Feedback with Variational Preference Learning , author=. The Thirty-eighth Annual Conference on Neural Information Processing Systems , year=

    work page

  64. [64]

    2026 , url=

    Distill Not Only Data but Also Rewards: Can Smaller Language Models Surpass Larger Ones? , author=. 2026 , url=

    work page 2026

  65. [65]

    The American Statistician , number=

    An overview of large language models for statisticians , author=. The American Statistician , number=. 2026 , publisher=

    work page 2026

  66. [66]

    Reinforcement Learning from Human Feedback: A Statistical Perspective

    Reinforcement learning from human feedback: A statistical perspective , author=. arXiv preprint arXiv:2604.02507 , year=

    work page internal anchor Pith review Pith/arXiv arXiv