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arxiv: 2606.03962 · v1 · pith:Q6B4BT7Gnew · submitted 2026-06-02 · 💻 cs.LG · cs.AI

Using Reward Uncertainty to Induce Diverse Behaviour in Reinforcement Learning

Anthony GX-Chen , Ankit Anand , Gheorghe Comanici , Zaheer Abbas , Eser Ayg\"un , David Smalling , Shibl Mourad , Doina Precup
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Andr\'e Barreto Mark Rowland
This is my paper

Pith reviewed 2026-06-28 10:57 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords reinforcement learningreward uncertaintybehavioral diversitycontextual banditspolicy gradientdiverse policiesuncertain rewards
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The pith

Modeling rewards as distributions over functions yields diverse policies as the rational response to uncertainty.

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

The paper claims that classical RL maximizes expected scalar reward and therefore tends toward deterministic policies, but many applications require behavioral diversity. It argues that when the reward function is uncertain, committing to one action can be sub-optimal, so diversity should arise naturally as the rational response. The proposed reformulation replaces the scalar reward with a distribution over reward functions and applies a non-linear objective over sets of actions. This produces calibrated diversity that is controllable via the reward distribution and does not require sacrificing expected reward. The work derives a gradient estimator for the contextual bandit case and shows that the formulation generalizes both vanilla policy gradient and recent action-set methods.

Core claim

By replacing the scalar reward with a distribution over reward functions and applying a non-linear objective over sets of actions, calibrated behavioural diversity emerges naturally, remains controllable through the reward function distribution, and is obtained without sacrificing expected reward.

What carries the argument

The reformulation of the RL objective that replaces a scalar reward with a distribution over reward functions and uses a non-linear objective over sets of actions.

If this is right

  • Calibrated behavioural diversity emerges naturally from the uncertainty model without explicit regularization.
  • The level of diversity remains directly controllable by the choice of reward-function distribution.
  • Expected reward performance is preserved while diversity is achieved.
  • The formulation generalizes both vanilla policy gradient and action-set approaches.
  • It supplies a robust alternative for tasks where standard RL fails to produce sufficient breadth of behavior.

Where Pith is reading between the lines

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

  • The same reformulation could be applied to full Markov decision processes to test whether diversity benefits persist in sequential settings.
  • Links to Bayesian RL may clarify how posterior reward distributions translate into specific diversity patterns.
  • Evaluation on tasks with genuinely ambiguous human preferences could check whether the induced behaviors match desired variety.
  • The non-linear objective over action sets may provide a new route for multi-objective RL without manual weighting.

Load-bearing premise

Diversity is best understood as the rational response to uncertainty in the reward function.

What would settle it

An experiment in which the proposed objective produces the same or less diversity than entropy-regularized RL at identical expected reward, or in which changing the reward-function distribution fails to alter observed behavioral diversity in the predicted direction.

read the original abstract

Classical reinforcement learning (RL) typically seeks a deterministic policy that maximizes the expected sum of a scalar reward. Yet, modern applications such as language model fine-tuning or scientific discovery demand diversity. Existing remedies such as entropy regularization or diversity bonuses often require fragile trade-offs that sacrifice performance for stochasticity or rely on heuristic metrics that can misalign policy rankings. We argue that diversity is more naturally understood as the rational response to uncertainty in the reward. When the reward function is not perfectly known--as is the case with ambiguous preferences or imperfect reward models--committing to a single action can be sub-optimal. Building on this, we propose a fundamental reformulation of the RL objective by replacing the scalar reward with a distribution over reward functions, and applying a non-linear objective over sets of actions. The result is a framework in which calibrated behavioural diversity emerges naturally, remains controllable through the reward function distribution, and is obtained without sacrificing expected reward. Focusing on the contextual bandit setting, we derive a principled gradient estimator for this objective and prove that our formulation naturally generalizes both vanilla policy gradient and more recently developed action-set approaches. Our empirical results demonstrate that this framework offers a robust and theoretically grounded alternative for complex RL tasks where the traditional formulation of the problem fails to induce the desired breadth of agent behaviour.

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

1 major / 0 minor

Summary. The paper argues that behavioral diversity in RL arises naturally from uncertainty over the reward function. It reformulates the objective as a non-linear function over sets of actions drawn from a distribution of reward functions, derives a gradient estimator and generalization proof in the contextual bandit setting (showing it recovers vanilla policy gradient and action-set methods), and presents empirical results claiming robust diversity on complex RL tasks without sacrificing expected reward.

Significance. If the bandit derivation and its claimed generalization extend rigorously to sequential MDPs, the framework would offer a principled, controllable alternative to entropy regularization or heuristic diversity bonuses, particularly for applications like language-model fine-tuning. The explicit proof that the new objective recovers existing methods is a strength, as is the parameter-free character of the core reformulation.

major comments (1)
  1. [Abstract] Abstract: The derivation and proof are explicitly restricted to the contextual bandit setting ('Focusing on the contextual bandit setting, we derive a principled gradient estimator...'), yet the central claims and empirical demonstration target 'complex RL tasks' and general sequential decision making. No reduction or extension is indicated for multi-step MDPs, value functions, or propagation of reward uncertainty over time; this gap is load-bearing for the claim that calibrated diversity 'emerges naturally... without sacrificing expected reward' in general RL.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the scope mismatch between the theoretical analysis and the broader claims. We address the major comment below and commit to revisions that clarify limitations without overstating the current results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The derivation and proof are explicitly restricted to the contextual bandit setting ('Focusing on the contextual bandit setting, we derive a principled gradient estimator...'), yet the central claims and empirical demonstration target 'complex RL tasks' and general sequential decision making. No reduction or extension is indicated for multi-step MDPs, value functions, or propagation of reward uncertainty over time; this gap is load-bearing for the claim that calibrated diversity 'emerges naturally... without sacrificing expected reward' in general RL.

    Authors: We agree that the formal derivation, gradient estimator, and generalization proof are developed and stated only for the contextual bandit setting. The empirical demonstrations on complex tasks illustrate practical behavior but do not constitute a rigorous reduction or extension to multi-step MDPs. In the revised manuscript we will (i) revise the abstract and introduction to explicitly qualify the scope of the theoretical claims, (ii) add a new subsection in the discussion that outlines the additional technical steps required to propagate reward uncertainty through value functions and trajectories, and (iii) note that such an extension remains future work. These changes will remove any implication that the current proofs already cover general RL. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained in bandit setting with explicit proofs

full rationale

The paper's core move is an argumentative reformulation of the RL objective (distribution over rewards plus non-linear set objective) motivated by the external premise that diversity responds to reward uncertainty. It then restricts the formal derivation and gradient estimator to the contextual bandit case, proves generalization to vanilla policy gradient and action-set methods, and reports empirical results on complex tasks. No equation reduces to a prior fit or self-definition, no load-bearing self-citation chain appears, and the bandit-to-RL extension is presented as an empirical claim rather than a deductive reduction. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Since only the abstract is available, the ledger is based on stated elements in the summary; the central claim relies on unverified derivations and the domain assumption about rational response to uncertainty.

axioms (1)
  • domain assumption Diversity is the rational response to uncertainty in the reward function.
    Explicitly argued in the abstract as the foundation for the new objective.
invented entities (1)
  • Distribution over reward functions no independent evidence
    purpose: To represent uncertainty in the reward
    Introduced as the replacement for scalar reward in the reformulation.

pith-pipeline@v0.9.1-grok · 5795 in / 1168 out tokens · 39494 ms · 2026-06-28T10:57:06.648613+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

84 extracted references · 9 canonical work pages · 4 internal anchors

  1. [1]

    Mastering the game of

    Silver, David and Huang, Aja and Maddison, Chris J and Guez, Arthur and Sifre, Laurent and Van Den Driessche, George and Schrittwieser, Julian and Antonoglou, Ioannis and Panneershelvam, Veda and Lanctot, Marc and others , journal=. Mastering the game of. 2016 , publisher=

    2016

  2. [2]

    Olympiad-level formal mathematical reasoning with reinforcement learning , journal =

    Hubert, Thomas and Mehta, Rishi and Sartran, Laurent and Horv. Olympiad-level formal mathematical reasoning with reinforcement learning , journal =. 2026 , pages =

    2026

  3. [3]

    Pawan and Dupont, Emilien and Ruiz, Francisco J

    Romera-Paredes, Bernardino and Barekatain, Mohammadamin and Novikov, Alexander and Balog, Matej and Kumar, M. Pawan and Dupont, Emilien and Ruiz, Francisco J. R. and Ellenberg, Jordan S. and Wang, Pengming and Fawzi, Omar and Kohli, Pushmeet and Fawzi, Alhussein , title =. Nature , year =

  4. [4]

    Proceedings of the International Conference on Learning Representations , year=

    Self-improvement in language models: The sharpening mechanism , author=. Proceedings of the International Conference on Learning Representations , year=

  5. [5]

    Proceedings of the International Conference on Learning Representations , year=

    Does writing with language models reduce content diversity? , author=. Proceedings of the International Conference on Learning Representations , year=

  6. [6]

    McLean and Peter Norgaard and Zahra Shamsi and David Smalling and James Thompson and Subhashini Venugopalan and Brian P

    Eser Aygün and Anastasiya Belyaeva and Gheorghe Comanici and Marc Coram and Hao Cui and Jake Garrison and Renee Johnston and Anton Kast and Cory Y. McLean and Peter Norgaard and Zahra Shamsi and David Smalling and James Thompson and Subhashini Venugopalan and Brian P. Williams and Chujun He and Sarah Martinson and Martyna Plomecka and Lai Wei and Yuchen Z...

  7. [7]

    Proceedings of the International Conference on Learning Representations , year=

    Polychromic Objectives for Reinforcement Learning , author=. Proceedings of the International Conference on Learning Representations , year=

  8. [8]

    Proceedings of the International Conference on Machine Learning , year=

    Optimizing language models for inference time objectives using reinforcement learning , author=. Proceedings of the International Conference on Machine Learning , year=

  9. [9]

    Proceedings of the International Conference on Machine Learning , year=

    Theoretical guarantees on the best-of- n alignment policy , author=. Proceedings of the International Conference on Machine Learning , year=

  10. [10]

    Advances in Neural Information Processing Systems , year=

    Bonbon alignment for large language models and the sweetness of best-of- n sampling , author=. Advances in Neural Information Processing Systems , year=

  11. [11]

    Proceedings of the International Conference on Learning Representations , year=

    Inference-aware fine-tuning for best-of- M sampling in large language models , author=. Proceedings of the International Conference on Learning Representations , year=

  12. [12]

    arXiv preprint arXiv:2509.02534 , year=

    Jointly reinforcing diversity and quality in language model generations , author=. arXiv preprint arXiv:2509.02534 , year=

    work page arXiv

  13. [13]

    Proceedings of the International Conference on Machine Learning , year=

    Asynchronous methods for deep reinforcement learning , author=. Proceedings of the International Conference on Machine Learning , year=

  14. [14]

    2025 , eprint=

    Diversity or Precision? A Deep Dive into Next Token Prediction , author=. 2025 , eprint=

    2025

  15. [15]

    Advances in Neural Information Processing Systems , year=

    The epoch-greedy algorithm for multi-armed bandits with side information , author=. Advances in Neural Information Processing Systems , year=

  16. [16]

    and Barto, Andrew G

    Sutton, Richard S. and Barto, Andrew G. , edition =. Reinforcement Learning: An Introduction , year =

  17. [17]

    Machine Learning , volume=

    Simple statistical gradient-following algorithms for connectionist reinforcement learning , author=. Machine Learning , volume=. 1992 , publisher=

    1992

  18. [18]

    Advances in Neural Information Processing Systems , year=

    Learning to summarize with human feedback , author=. Advances in Neural Information Processing Systems , year=

  19. [19]

    Proceedings of the International Conference on Learning Representations , year=

    Sessa, Pier Giuseppe and Dadashi, Robert and Hussenot, L. Proceedings of the International Conference on Learning Representations , year=

  20. [20]

    2016 , publisher=

    Stochastic dominance: Investment decision making under uncertainty , author=. 2016 , publisher=

    2016

  21. [21]

    Machine learning , volume=

    Empirical evaluation methods for multiobjective reinforcement learning algorithms , author=. Machine learning , volume=. 2011 , publisher=

    2011

  22. [22]

    GX-Chen, Anthony and Prakash, Jatin and Guo, Jeff and Fergus, Rob and Ranganath, Rajesh , booktitle=

  23. [23]

    Autonomous Agents and Multi-Agent Systems , volume=

    A practical guide to multi-objective reinforcement learning and planning , author=. Autonomous Agents and Multi-Agent Systems , volume=. 2022 , publisher=

    2022

  24. [24]

    Orney, Ifdita Hasan and Hamid, Jubayer Ibn and Ramanujam, Shreya S and Wu, Shirley and Hu, Hengyuan and Goodman, Noah and Sadigh, Dorsa and Finn, Chelsea , journal=. Poly-

  25. [25]

    Proceedings of the International Conference on Learning Representations , year=

    Reward model ensembles help mitigate overoptimization , author=. Proceedings of the International Conference on Learning Representations , year=

  26. [26]

    Linearly-solvable

    Todorov, Emanuel , booktitle=. Linearly-solvable

  27. [27]

    , author=

    Maximum entropy inverse reinforcement learning. , author=. Proceedings of the AAAI Conference on Artificial Intelligence , year=

  28. [28]

    Proceedings of the International Conference on Machine Learning , year=

    Reinforcement learning with deep energy-based policies , author=. Proceedings of the International Conference on Machine Learning , year=

  29. [29]

    Journal of Artificial Intelligence Research , volume=

    A survey of multi-objective sequential decision-making , author=. Journal of Artificial Intelligence Research , volume=

  30. [30]

    Constrained

    Altman, Eitan , year=. Constrained

  31. [31]

    , author=

    Multi-criteria reinforcement learning. , author=. Proceedings of the International Conference on Machine Learning , year=

  32. [32]

    Advances in Neural Information Processing Systems , year=

    The steering approach for multi-criteria reinforcement learning , author=. Advances in Neural Information Processing Systems , year=

  33. [33]

    Deep reinforcement learning with attention for slate

    Sunehag, Peter and Evans, Richard and Dulac-Arnold, Gabriel and Zwols, Yori and Visentin, Daniel and Coppin, Ben , journal=. Deep reinforcement learning with attention for slate

  34. [34]

    Non-deterministic policies in

    Fard, M Milani and Pineau, Joelle , journal=. Non-deterministic policies in

  35. [35]

    Advances in Neural Information Processing Systems , year=

    Non-stochastic bandit slate problems , author=. Advances in Neural Information Processing Systems , year=

  36. [36]

    Advances in Neural Information Processing Systems , year=

    Linear submodular bandits and their application to diversified retrieval , author=. Advances in Neural Information Processing Systems , year=

  37. [37]

    Proceedings of the International Conference on Machine Learning , year=

    Learning diverse rankings with multi-armed bandits , author=. Proceedings of the International Conference on Machine Learning , year=

  38. [38]

    , author=

    Marginal Posterior Sampling for Slate Bandits. , author=. Proceedings of the International Joint Conference on Artificial Intelligence , year=

  39. [39]

    Journal of Artificial Intelligence Research , volume=

    Joint optimization of concave scalarized multi-objective reinforcement learning with policy gradient based algorithm , author=. Journal of Artificial Intelligence Research , volume=

  40. [40]

    Breaking the bias barrier in concave multi-objective reinforcement learning.arXiv preprint arXiv:2603.08518, 2026

    Breaking the Bias Barrier in Concave Multi-Objective Reinforcement Learning , author=. arXiv preprint arXiv:2603.08518 , year=

    work page arXiv

  41. [41]

    Puterman , title =

    Martin L. Puterman , title =

  42. [42]

    Understanding the effects of

    Kirk, Robert and Mediratta, Ishita and Nalmpantis, Christoforos and Luketina, Jelena and Hambro, Eric and Grefenstette, Edward and Raileanu, Roberta , booktitle=. Understanding the effects of

  43. [43]

    Proceedings of the Conference on Language Modeling , year=

    Modifying Large Language Model Post-Training for Diverse Creative Writing , author=. Proceedings of the Conference on Language Modeling , year=

  44. [44]

    Findings of the Association for Computational Linguistics , year=

    Creative preference optimization , author=. Findings of the Association for Computational Linguistics , year=

  45. [45]

    The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models

    The entropy mechanism of reinforcement learning for reasoning language models , author=. arXiv preprint arXiv:2505.22617 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  46. [46]

    arXiv:2412.01951 , year=

    Self-improvement in language models: The sharpening mechanism , author=. arXiv preprint arXiv:2412.01951 , year=

    work page arXiv

  47. [47]

    Yang, Chenghao and Holtzman, Ari , journal=

  48. [48]

    The Price of Format: Diversity Collapse in

    Yun, Longfei and An, Chenyang and Wang, Zilong and Peng, Letian and Shang, Jingbo , journal=. The Price of Format: Diversity Collapse in

  49. [49]

    arXiv preprint arXiv:2410.15096 , year=

    GDPO: Learning to Directly Align Language Models with Diversity Using GFlowNets , author=. arXiv preprint arXiv:2410.15096 , year=

    work page arXiv

  50. [50]

    Echo chamber:

    Zhao, Rosie and Meterez, Alexandru and Kakade, Sham and Pehlevan, Cengiz and Jelassi, Samy and Malach, Eran , booktitle=. Echo chamber:

  51. [51]

    Evaluating the diversity and quality of

    Shypula, Alexander and Li, Shuo and Zhang, Botong and Padmakumar, Vishakh and Yin, Kayo and Bastani, Osbert , booktitle=. Evaluating the diversity and quality of

  52. [52]

    Proceedings of the Conference on Language Modeling , year=

    Base models beat aligned models at randomness and creativity , author=. Proceedings of the Conference on Language Modeling , year=

  53. [53]

    arXiv preprint arXiv:2509.06941 , year=

    Outcome-based exploration for LLM reasoning , author=. arXiv preprint arXiv:2509.06941 , year=

    work page arXiv

  54. [54]

    Scaling Laws for Reward Model Overoptimization

    Scaling laws for reward model overoptimization , author=. arXiv preprint arXiv:2210.10760 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  55. [55]

    Confronting reward model overoptimization with constrained

    Moskovitz, Ted and Singh, Aaditya K and Strouse, DJ and Sandholm, Tuomas and Salakhutdinov, Ruslan and Dragan, Anca D and McAleer, Stephen , booktitle=. Confronting reward model overoptimization with constrained

  56. [56]

    Advances in Neural Information Processing Systems , year=

    Scaling laws for reward model overoptimization in direct alignment algorithms , author=. Advances in Neural Information Processing Systems , year=

  57. [57]

    Reinforcement Learning from Human Feedback , year =

    Lambert, Nathan , title =. Reinforcement Learning from Human Feedback , year =

  58. [58]

    Advances in Neural Information Processing Systems , year=

    Convergence Theorems for Entropy-Regularized and Distributional Reinforcement Learning , author=. Advances in Neural Information Processing Systems , year=

  59. [59]

    Connection Science , volume=

    Function optimization using connectionist reinforcement learning algorithms , author=. Connection Science , volume=. 1991 , publisher=

    1991

  60. [60]

    Balashankar, Ananth and Sun, Ziteng and Berant, Jonathan and Eisenstein, Jacob and Collins, Michael and Hutter, Adrian and Lee, Jong and Nagpal, Chirag and Prost, Flavien and Sinha, Aradhana and Ananda Theertha Suresh and Ahmad Beirami , booktitle=

  61. [61]

    Proceedings of the International Conference on Learning Representations , year=

    Post-training Large Language Models for Diverse High-Quality Responses , author=. Proceedings of the International Conference on Learning Representations , year=

  62. [62]

    Proceedings of the International Conference on Machine Learning , year=

    A distributional view on multi-objective policy optimization , author=. Proceedings of the International Conference on Machine Learning , year=

  63. [63]

    Proceedings of the International Conference on Autonomous Agents and Multiagent Systems , year=

    Multi-objective reinforcement learning with non-linear scalarization , author=. Proceedings of the International Conference on Autonomous Agents and Multiagent Systems , year=

  64. [64]

    Econometrica: Journal of the Econometric Society , pages=

    The dual theory of choice under risk , author=. Econometrica: Journal of the Econometric Society , pages=. 1987 , publisher=

    1987

  65. [65]

    ASTIN Bulletin: The Journal of the IAA , volume=

    Premium calculation by transforming the layer premium density , author=. ASTIN Bulletin: The Journal of the IAA , volume=. 1996 , publisher=

    1996

  66. [66]

    Proceedings of the International Conference on Machine Learning , year=

    Implicit quantile networks for distributional reinforcement learning , author=. Proceedings of the International Conference on Machine Learning , year=

  67. [67]

    Journal of Risk and Uncertainty , volume=

    Advances in prospect theory: Cumulative representation of uncertainty , author=. Journal of Risk and Uncertainty , volume=. 1992 , publisher=

    1992

  68. [68]

    Algorithms for

    Chow, Yinlam and Ghavamzadeh, Mohammad , booktitle=. Algorithms for

  69. [69]

    2023 , publisher=

    Distributional reinforcement learning , author=. 2023 , publisher=

    2023

  70. [70]

    Vector Policy Optimization: Training for Diversity Improves Test-Time Search

    Vector Policy Optimization: Training for Diversity Improves Test-Time Search , author=. arXiv preprint arXiv:2605.22817 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  71. [71]

    Understanding Performance Gap Between Parallel and Sequential Sampling in Large Reasoning Models

    Understanding Performance Gap Between Parallel and Sequential Sampling in Large Reasoning Models , author=. arXiv preprint arXiv:2604.05868 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  72. [72]

    1999 , publisher=

    Nonlinear multiobjective optimization , author=. 1999 , publisher=

    1999

  73. [73]

    2005 , publisher=

    Multicriteria optimization , author=. 2005 , publisher=

    2005

  74. [74]

    On the relationship of the

    Bowman Jr, V Joseph , booktitle=. On the relationship of the. 1976 , organization=

    1976

  75. [75]

    An interactive weighted

    Steuer, Ralph E and Choo, Eng-Ung , journal=. An interactive weighted. 1983 , publisher=

    1983

  76. [76]

    Lin, Xi and Zhang, Xiaoyuan and Yang, Zhiyuan and Liu, Fei and Wang, Zhenkun and Zhang, Qingfu , booktitle=. Smooth

  77. [77]

    Operations Research , volume=

    Solving bicriterion mathematical programs , author=. Operations Research , volume=. 1967 , publisher=

    1967

  78. [78]

    A closer look at drawbacks of minimizing weighted sums of objectives for

    Das, Indraneel and Dennis, John E , journal=. A closer look at drawbacks of minimizing weighted sums of objectives for. 1997 , publisher=

    1997

  79. [79]

    Zhang, Yiming and Diddee, Harshita and Holm, Susan and Liu, Hanchen and Liu, Xinyue and Samuel, Vinay and Wang, Barry and Ippolito, Daphne , booktitle=

  80. [80]

    Friedman, Dan and Dieng, Adji Bousso , journal=. The

Showing first 80 references.