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A robust max-min optimization over all r-correlated proxy rewards produces RL policies that resist reward hacking.

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T0 review · grok-4.3

2026-05-10 15:13 UTC

load-bearing objection This paper adds a max-min robust layer on top of the r-correlation model to guard policies against worst-case proxies, but the tractability step likely restricts the guarantee. the 2 major comments →

arxiv 2604.12086 v1 submitted 2026-04-13 cs.LG

Robust Optimization for Mitigating Reward Hacking with Correlated Proxies

Zixuan Liu , Xiaolin Sun , Zizhan Zheng This is my paper
classification cs.LG
keywords reward hackingrobust reinforcement learningproxy rewardsr-correlationmax-min optimizationpolicy optimizationreward misspecification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows how to train reinforcement learning agents that are robust to imperfect proxy rewards by optimizing against the worst possible proxy that still correlates with the true reward at a given level r. This addresses reward hacking, where agents exploit flaws in the proxy to achieve high scores without fulfilling the intended goal. By deriving a tractable formulation that maximizes the minimum performance across all such proxies, the method provides guarantees against a broader set of misspecifications than optimizing a single proxy. Experiments demonstrate improved worst-case performance and stability compared to prior approaches like ORPO.

Core claim

We formulate reward hacking as a robust policy optimization problem over the space of all r-correlated proxy rewards. We derive a tractable max-min formulation, where the agent maximizes performance under the worst-case proxy consistent with the correlation constraint. When the reward is a linear function of known features, our approach can be adapted to incorporate this prior knowledge, yielding both improved policies and interpretable worst-case rewards.

What carries the argument

The max-min robust policy optimization formulation over the class of r-correlated proxy rewards, which finds the policy that performs best in the worst case within that class.

Load-bearing premise

The correlation level r between proxy and true reward is known and fixed, and the worst-case proxy in this class meaningfully models real reward misspecifications.

What would settle it

Observe whether the learned policy's return under the computed worst-case proxy equals or exceeds the value predicted by the max-min optimization; failure to do so in controlled environments with known true rewards would falsify the robustness claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Agents achieve higher worst-case returns than those optimized against a fixed proxy.
  • Performance remains robust across varying levels of proxy-true reward correlation.
  • When rewards are linear in known features, the worst-case proxies become interpretable.
  • Training yields more stable policies under uncertain reward designs.

Where Pith is reading between the lines

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

  • This method could be extended by learning the correlation level r from data rather than assuming it fixed.
  • It may apply to other forms of reward uncertainty beyond linear correlations.
  • Integration with online reward learning could further mitigate misspecification in deployed systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper claims to address reward hacking in RL by formulating it as a robust policy optimization problem over the space of all r-correlated proxy rewards. It derives a tractable max-min formulation in which the agent optimizes against the worst-case proxy consistent with the correlation constraint, adapts the approach for linear rewards using known features to yield interpretable worst-case proxies, and reports that the resulting algorithms outperform ORPO in worst-case returns with improved robustness and stability across environments and correlation levels.

Significance. If the max-min formulation provides a valid robustness guarantee over the full class of r-correlated proxies and the experimental claims hold with proper statistical support, the work would offer a principled method for handling reward misspecification with both theoretical grounding and practical interpretability in the linear case. The open-sourced code is a positive factor for reproducibility.

major comments (2)
  1. [§3 (Derivation of the max-min formulation)] §3 (Derivation of the max-min formulation): The claim that the formulation optimizes against the worst-case proxy over the full (infinite-dimensional) class of r-correlated reward functions is not supported if the practical algorithm restricts to a parameterized family (e.g., linear or neural); nothing in the derivation shows that the restricted optimum coincides with the global worst-case, so the robustness guarantee may fail even while satisfying the correlation constraint.
  2. [Experiments section] Experiments section and abstract claims: The manuscript reports consistent outperformance in worst-case returns across environments and correlation levels, but provides no details on the number of independent runs, error bars, statistical significance tests, or the precise protocol for evaluating worst-case proxies, undermining verification of the empirical support for the central robustness claims.
minor comments (2)
  1. [§2] The definition of r-correlation and the precise form of the constraint should be stated explicitly in the introduction or early in §2 to aid readability before the max-min derivation.
  2. [Figures] Figure captions and axis labels in the experimental results could be expanded to include the exact correlation values and environment names for quicker interpretation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive comments on our manuscript. We provide point-by-point responses to the major comments below and indicate the revisions we will make to address them.

read point-by-point responses

  1. Referee: §3 (Derivation of the max-min formulation): The claim that the formulation optimizes against the worst-case proxy over the full (infinite-dimensional) class of r-correlated reward functions is not supported if the practical algorithm restricts to a parameterized family (e.g., linear or neural); nothing in the derivation shows that the restricted optimum coincides with the global worst-case, so the robustness guarantee may fail even while satisfying the correlation constraint.

    Authors: We appreciate this important clarification. The derivation in Section 3 formulates the robust optimization problem over the entire space of r-correlated proxy rewards, resulting in a tractable max-min objective that does not inherently restrict the class. However, as the referee notes, our practical algorithms parameterize the proxy rewards (using linear functions for the interpretable case or neural networks otherwise) to compute the worst-case proxy. This parameterization means the implemented method optimizes against an approximate worst-case within the parameterized family rather than the global worst-case over all possible functions. We agree that this limits the strict robustness guarantee to the parameterized class. In the revised manuscript, we will explicitly state this distinction, explain that the theoretical guarantee applies to the exact formulation, and discuss the implications and potential gaps introduced by parameterization. We will also explore adding analysis on the expressiveness of the chosen parameterization. revision: yes

  2. Referee: Experiments section and abstract claims: The manuscript reports consistent outperformance in worst-case returns across environments and correlation levels, but provides no details on the number of independent runs, error bars, statistical significance tests, or the precise protocol for evaluating worst-case proxies, undermining verification of the empirical support for the central robustness claims.

    Authors: We acknowledge that the current manuscript lacks sufficient details on the experimental setup to allow full verification of the results. In the revised version, we will add comprehensive information including: the number of independent runs performed for each experiment, error bars (e.g., standard deviation across runs), statistical significance tests comparing our methods to ORPO, and a precise description of the protocol used to evaluate worst-case proxies, such as how the worst-case is selected or approximated within the correlation constraint. These additions will strengthen the empirical support for our claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct formulation from the correlation constraint

full rationale

The paper defines reward hacking as robust optimization over the class of all r-correlated proxy rewards and derives a max-min objective directly from that definition. No equations or steps reduce a claimed prediction or worst-case result to a fitted parameter, self-citation, or ansatz that was smuggled in. The tractability claim and linear adaptation are presented as consequences of the formulation rather than tautological renamings. The parameterization concern raised in the skeptic note pertains to whether the guarantee is complete (a correctness issue), not to any self-referential reduction in the derivation itself. The approach is self-contained against the stated r-correlation inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of r-correlation as a constraint on proxy rewards and the assumption that a tractable worst-case proxy exists within that set; no new entities are postulated.

free parameters (1)
  • r (correlation level)
    User-specified bound on proxy-true reward correlation; treated as an input hyperparameter rather than fitted inside the derivation.
axioms (2)
  • domain assumption Proxy rewards are drawn from the set of all functions r-correlated with the true reward.
    Invoked in the problem formulation to define the uncertainty set for the robust optimization.
  • domain assumption When reward is linear in known features, the worst-case proxy can be computed by optimizing over feature weights.
    Used to obtain the interpretable linear-reward variant.

pith-pipeline@v0.9.0 · 5535 in / 1456 out tokens · 21051 ms · 2026-05-10T15:13:45.894085+00:00 · methodology

0 comments
read the original abstract

Designing robust reinforcement learning (RL) agents in the presence of imperfect reward signals remains a core challenge. In practice, agents are often trained with proxy rewards that only approximate the true objective, leaving them vulnerable to reward hacking, where high proxy returns arise from unintended or exploitative behaviors. Recent work formalizes this issue using r-correlation between proxy and true rewards, but existing methods like occupancy-regularized policy optimization (ORPO) optimize against a fixed proxy and do not provide strong guarantees against broader classes of correlated proxies. In this work, we formulate reward hacking as a robust policy optimization problem over the space of all r-correlated proxy rewards. We derive a tractable max-min formulation, where the agent maximizes performance under the worst-case proxy consistent with the correlation constraint. We further show that when the reward is a linear function of known features, our approach can be adapted to incorporate this prior knowledge, yielding both improved policies and interpretable worst-case rewards. Experiments across several environments show that our algorithms consistently outperform ORPO in worst-case returns, and offer improved robustness and stability across different levels of proxy-true reward correlation. These results show that our approach provides both robustness and transparency in settings where reward design is inherently uncertain. The code is available at https://github.com/ZixuanLiu4869/reward_hacking.

Figures

Figures reproduced from arXiv: 2604.12086 by Xiaolin Sun, Zixuan Liu, Zizhan Zheng.

Figure 1
Figure 1. Figure 1: Mean reward and standard deviation under sampled θ and true reward features at different proxy–true reward correlation levels r for the Traffic and Pandemic environments. Our methods (Max-Min and Linear Max-Min) yield more stable and higher average performance across all choices of r. setup mimics a more realistic real-world scenario in which the true reward function may depend on features not explicitly m… view at source ↗
Figure 2
Figure 2. Figure 2: Discriminator loss across RL iterations in the [PITH_FULL_IMAGE:figures/full_fig_p036_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows that in our implementation, the discriminator loss consistently decreases within each iteration, e.g., from an initial value around 1.4 to below 0.2 in the Traffic environment, indicating effective optimization and more accurate occupancy-ratio estimation. In the Glucose and Pandemic environments, however, we observe that training the discriminator for too long leads to slower con￾vergence and little… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of adversarial reward weights θ over training epochs for different environments using the Linear Maxmin method. H ADDITIONAL EXPERIMENT RESULTS H.1 FEATURE WEIGHTS IN LINEAR MAX-MIN OPTIMIZATION DURING TRAINING [PITH_FULL_IMAGE:figures/full_fig_p053_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean reward and standard deviation under sampled reward functions at different proxy [PITH_FULL_IMAGE:figures/full_fig_p057_5.png] view at source ↗

discussion (0)

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Forward citations

Cited by 1 Pith paper

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

  1. Multimodal Reward Hacking in Reinforcement Learning

    cs.AI 2026-07 conditional novelty 6.5

    Imperfect multimodal RL rewards systematically create new failures (NRFR > RHR); scaling and answer-aware rewards help but do not eliminate hacking, and unreliable visual verifiers actively increase it.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · cited by 1 Pith paper

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    (L(s, a)−λ 1 Rproxy(s,a) V −λ 2)2 4λ3 # +λ1r+λ 2M+λ 3(M2+V 2) (20) We now compute the gradients of the dual objective with respect to the dual variables: ∂l0 ∂λ1 =−E µπref

    to handle high-dimensional, continious state spaces and to use the method for transfer learn- ing. Specifically, (Barreto et al., 2017) formalize this idea into the successor features (SF) framework for transfer learning, assuming that tasks share dynamics but differ only in their reward functions parameterized as linear combinations of features. This yie...

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    For fixed (λ 1, λ2, λ3) withλ 3 <0, the Lagrangian is a strictly convex quadratic inR, so its stationary pointR ⋆(λ)is the unique global minimizer of the inner problem with those multipliers

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    Solving the dual and enforcing feasibility recovers the specific choice of multipliersλ ⋆ for whichR ⋆(λ⋆)lies on the sphere defined by the norm constraint

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    sprinkler state

    Because the reduced problem is linear over a sphere, this feasible stationary pointR ⋆(λ⋆) must be the global minimizer of the original inner problem. E.3 PROOF THATχ 2(µπ ∥µ πREF )≥E 2 µπ[RPROXY ] To ensure that the inner term of the square root in Equation 25 remains non-negative, we need to show that χ2(µπ ∥µ πref)≥E 2 µπ[Rproxy] Proof.Recall that Eµπ[...

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    could be adapted to solve the problem by sampling multiplerand solving Equation 27 in our paper. We should note that the reason these frameworks are potentially applicable is that our formulation admits a closed-form solution for the inner minimization. However, the main challenge lies in estimiating the occupancy measure. An interesting direction for fut...

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