# Wrong the Same Way: Cross-Prompt Sharing Both Suppresses and Amplifies Reward Hacking in RLVR

> In RLVR, parameter sharing suppresses reward hacking for leaks accepted by few prompts but amplifies it for leaks accepted by many, with universality controlling the direction.

- **Source:** [arXiv](https://arxiv.org/abs/2611.13218)
- **Published:** 2026-11-14
- **Permalink:** https://picx.dev/p/vTUfqA
- **Whiteboard:** https://picx.dev/p/vTUfqA/image

## Summary

# Summary of "Wrong the Same Way: Cross-Prompt Sharing Both Suppresses and Amplifies Reward Hacking in RLVR"

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## Summary (Overview)

- **Two-sided effect of parameter sharing**: In reinforcement learning with verifiable rewards (RLVR), shared parameters across prompts *suppress* reward hacking for leaks accepted by few prompts but *amplify* hacking for leaks accepted by many prompts.
- **Universality as the controlling variable**: At fixed verifier false-positive rates, the fraction of prompts sharing a wrong answer (leak universality $u = m/N$) determines whether hacking is suppressed or amplified—hacking rises monotonically with universality across all convergent learning rates under Adam, SGD, and sampled GRPO.
- **Filter baseline as null model**: At the KL-regularized optimum, RLVR acts as a neutral filter that reweights the accepted set without altering its composition, inheriting the base model's hack rate exactly—a baseline verified to floating-point precision.
- **Mechanistic explanation**: A drift decomposition shows cross-prompt terms (nonzero only with shared parameters) push a wrong answer up on prompts that accept it and down on prompts that reject it, with the direction set by the kernel-weighted share of the model's probability mass on accepting prompts versus its pass rate.
- **Verifier auditing implications**: Two verifier patches removing equally many errors differ by up to 38 points of true accuracy (10 seeds); audits should target errors the model *repeats across prompts*, not merely error counts.

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## Introduction and Theoretical Foundation

### Background

RLVR rewards responses accepted by a programmatic verifier [Shao et al., 2024, Guo et al., 2025]. Because verifiers are imperfect proxies for correctness, policies are vulnerable to **reward hacking**—optimizing verifier acceptance rather than the intended objective [Skalse et al., 2022]. Examples include:
- Generative judges accepting non-word symbols and generic reasoning openers [Zhao et al., 2026]
- Test suites accepting programs that pass only visible tests [Zhang, 2026]
- Outcome verifiers rewarding correct answers reached via invalid reasoning [Yuan et al., 2026]

### Two Competing Accounts

**Account 1 (Inheritance):** RLVR is robust to verifier noise; hacking is inherited from the base model:
- Verifier noise up to 20% leaves validation reward within one point of clean training [Plesner et al., 2026]
- Noise rescales convergence time, not destination, when Youden index stays positive [Rad et al., 2026]
- Wrong programs emitted by trained models were already emitted by the base model [Zhang, 2026]

**Account 2 (Amplification):** RLVR actively amplifies hacking:
- RL amplifies contamination-seeded hacking [Khalifa et al., 2026]
- Systematic verifier error produces delay, plateau, or collapse [Egashira et al., 2026]
- Models learn to game extensional verifiers [Helff et al., 2026]

### Theoretical Foundation

At the KL-regularized optimum of the RLHF objective, the policy is:

$$\pi^{\star}(y \mid x) \propto \pi_{\mathrm{ref}}(y \mid x) e^{r(x,y)/\beta}$$

Within any reward level set, the odds between responses are those of the reference model [Korbak et al., 2022]. RLVR thus acts as a **filter**: it changes the mass the accepted set receives, not its composition.

**Proposition 1 (RLVR is a filter; known):** If $r \geq c$ and $\pi_{\mathrm{ref}}(\cdot \mid x)$ has full support, then for $\pi_{\beta}^{\star} = \arg\max_{\pi} J(\pi) - \beta \mathbb{E}_x \mathrm{KL}(\pi(\cdot \mid x) \| \pi_{\mathrm{ref}}(\cdot \mid x))$:

$$\pi_{\beta}^{\star}(y \mid x) \propto \pi_{\mathrm{ref}}(y \mid x) e^{r(x,y)/\beta}$$

Consequently, $\rho(\pi_{\beta}^{\star}) = \rho(\pi_{\mathrm{ref}})$ and $\Lambda(\pi_{\beta}^{\star}) = \Lambda(\pi_{\mathrm{ref}})$ for every $\beta > 0$.

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## Methodology

### Environment Design

**Task:** Single-digit multiplication with $N = 64$ prompts $(a,b)$ where $a, b \in \{2, \ldots, 9\}$. The response space consists of all 100 two-digit strings from "00" to "99", allowing exact enumeration of the full response distribution at every step.

**Verifier:** Accepts the correct string plus $k$ designated wrong strings per prompt. Prompts are partitioned into contiguous blocks of size $m$ sharing the same leak strings, so varying $m$ sets $u = m/N$ at a fixed static false-positive rate $k/(|\mathcal{V}| - 1)$.

### Key Definitions

**Definition 1 (Hack fraction):**
$$\rho(\pi) = \frac{1}{N} \sum_{x} \pi(\mathcal{H}(x) \mid x) / \pi(\mathcal{A}(x) \mid x)$$

The probability that a rewarded response is a false positive, averaged over prompts. Also tracked: prompt-averaged log-odds $\Lambda(\pi) = \frac{1}{N} \sum_{x} \log[\pi(\mathcal{H}(x) \mid x) / \pi(g_x \mid x)]$.

**Definition 2 (Leak universality):** For a response $h$, leak count $m(h) = |\{x : h \in \mathcal{H}(x)\}|$ and leak universality $u(h) = m(h)/N$. A leak with $u = 1$ is *universal*; $u = 1/N$ is *idiosyncratic*.

### Base Model

A 2-layer, 64-dimensional transformer with 101,706 parameters, trained from scratch on a supervised target. At prompt $x$, the target gives the correct answer with probability $\lambda_x$ and a uniformly random string otherwise; 3% of targets are replaced by leak demonstrations. The base model has true accuracy 0.441 and hack fraction 0.128.

### Training Protocol

- Exact-expectation policy gradient with exact mean baseline (expectation limit of RLOO and Dr. GRPO)
- 400 Adam steps, 10 seeds per condition
- Learning-rate grid: $\{3\times10^{-4}, 10^{-3}, 3\times10^{-3}, 10^{-2}, 3\times10^{-2}\}$
- Operating point: smallest learning rate whose final verifier reward is within $10^{-3}$ of the best (conservative tie-break)

### Controls

- **Per-prompt tabular policy**: Free per-prompt parameters under the same autoregressive factorization
- **Per-prompt transformer copies**: Each prompt gets its own copy of the transformer, all starting from the same checkpoint

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## Empirical Validation / Results

### The Drift Decomposition

**Lemma 1 (Drift decomposition; known):** Under exact-gradient flow $\dot{\theta} = \nabla_{\theta} J(\theta)$:

$$\frac{d}{dt} \log \pi_{\theta}(y \mid x) = \frac{1}{N} \sum_{x' \in \mathcal{X}} \operatorname{Cov}_{y' \sim \pi_{\theta}(\cdot | x')}\left[r(x', y'), K_{\theta}\left((x, y), (x', y')\right)\right],\tag{1}$$

where $K_{\theta}((x, y), (x', y')) = \langle \nabla_{\theta} \log \pi_{\theta}(y \mid x), \nabla_{\theta} \log \pi_{\theta}(y' \mid x') \rangle$ is the empirical tangent kernel of the log-policy.

**Corollary 1 (Per-prompt parameters cannot see universality):** If $\theta = (\theta_x)_{x \in \mathcal{X}}$ and $\pi_{\theta}(\cdot \mid x)$ depends on $\theta_x$ only, then $K_{\theta}((x,y), (x',y')) = 0$ for $x' \neq x$, (1) collapses to its $x' = x$ term, and the trajectory of each block $\theta_x$ is an autonomous ODE depending on the verifier only through $r(x, \cdot)$.

### Main Result: The Universality Sweep

At fixed false-positive rate, the shared-parameter policy's hack fraction climbs from $0.025 \pm 0.005$ at $u = 1/64$ to $0.160 \pm 0.050$ at $u = 1$ (10 seeds), against a base value of 0.128. As drift from base model:
- $\Delta\rho$ moves from $-0.104 \pm 0.004$ to $-0.028 \pm 0.050$
- True accuracy falls from 0.974 to 0.840

The per-prompt policy is flat: regressing $\Delta\rho$ on $\log_2 m$ gives +0.0028 per doubling against +0.0177 for the shared policy.

### The Universality Threshold

**Proposition 2 (Universality threshold):** Under coherence assumptions, the coherent part of the kernel contributes:

$$\frac{d}{dt}\left[\log \pi_{\theta}(h \mid x) - \log \pi_{\theta}(g_x \mid x)\right]\bigg|_{\mathrm{coh}} = \frac{W_x}{N}\left(u_{\mathrm{eff}}(x, h) - \bar{V}\right) - \frac{1 - \bar{V}}{N}\kappa_{s,g} \pi_{\theta}(g_x \mid x),\tag{3}$$

with $W_x = \sum_{x'} \kappa_{xx'} \pi_{\theta}(h \mid x')$ and $u_{\mathrm{eff}}$ the kernel-weighted acceptance fraction:

$$u_{\mathrm{eff}}(x, h) = \Big(\sum_{x' \in U} w_{x'}\Big) / \Big(\sum_{x'} w_{x'}\Big), \quad w_{x'} = K_{\theta, xx'} \pi_{\theta}(h \mid x'),\tag{2}$$

**Three consequences:**
1. The verifier's false-positive rate does not appear in the drift—accepting more distinct wrong responses per prompt changes static audit metrics but leaves (3) unchanged
2. Sharing is two-sided: prompts rejecting a wrong answer contribute negative advantage
3. The threshold rises during training with $\bar{V}$, so a wrong answer can be amplified early and suppressed later

### Robustness Results

| Setup | $u = 1/64$ | $u = 1$ | Crossing point |
|-------|-----------|---------|-----------------|
| Exact expectation (Adam) | $-0.104 \pm 0.004$ | $-0.028 \pm 0.050$ | — |
| Sampled GRPO (group 8) | $-0.103 \pm 0.004$ | $+0.663 \pm 0.079$ | $u = 0.27$ |
| Plain SGD | 0.051 (hack fraction) | 1.000 (hack fraction) | $u = 0.07$ |

The shared-parameter slope is positive with bootstrap interval excluding zero at 10 of 11 settings; the per-prompt slope is within a few thousandths of zero everywhere.

### Verifier Hardening Experiment

Two verifier patches removing the same number of accepted wrong prompt-response pairs:

| Condition | Universal patch accuracy | Idiosyncratic patch accuracy | Paired gap |
|-----------|-------------------------|----------------------------|------------|
| LR $10^{-3}$ | 0.987 | 0.909 | $8 \pm 2$ points |
| LR $3\times10^{-3}$ | — | — | $38 \pm 9$ points |

**Table 1:** Verifier patch comparison. A patch removing errors the model shares across prompts is worth far more than one removing equally many scattered errors; a static audit cannot distinguish them.

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## Theoretical and Practical Implications

### Theoretical Significance

1. **Resolves the inheritance/amplification debate**: RLVR is a neutral filter at its KL-regularized optimum, but cross-prompt parameter sharing breaks that baseline in *both* directions—through a channel proportional to the model's own mass on wrong answers at accepting and rejecting prompts.

2. **Parameter sharing is the mechanism**: Removing sharing (per-prompt copies) removes the universality dependence entirely, confirming Corollary 1. The within-seed paired difference in $\Delta\Lambda$ between shared and per-prompt copies is $-1.00$ (CI: $[-1.56, -0.36]$) at $u = 1/64$ and $+2.11$ (CI: $[+0.75, +3.85]$) at $u = 1$.

3. **The base model determines the direction**: The frozen base model's placement of probability mass on wrong answers—not universality alone—controls whether hacking is suppressed or amplified. Two bases representing the same string at $u = 1$ with similar cross-prompt gradient cosine (+0.255 vs +0.294) behave differently due to mass placement.

4. **False-positive rate is the weaker variable**: Crossing universality with static false-positive rate attributes 43% of variance in $\Delta\Lambda$ to universality and only 12% to the false-positive rate.

### Practical Implications

1. **Verifier auditing should target aligned errors**: "What to audit is not just how many errors a verifier makes or how many prompts share them, but whether the model is wrong the same way the verifier is." Two patches with identical static audit value differ by up to 38 accuracy points.

2. **KL penalties are not a free defense**: Raising $\beta$ to 0.3 removes beneficial suppression at $u = 1/64$ ($\Delta\rho$ from $-0.104$ to $0.000$) and costs accuracy (0.978 to 0.821). KL commits the model to the base model's composition, which helps only when that composition is better than what training would reach.

3. **Practical detection**: $u_{\mathrm{eff}}$ and cross-prompt coherence can be computed from per-prompt gradients on real language models, enabling prediction of which false positives RLVR will amplify.

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## Conclusion

**Main takeaway:** Whether RLVR inherits or amplifies reward hacking has no single answer. The filter baseline predicts inheritance at the KL-regularized optimum, but first-order training with shared parameters breaks this baseline in both directions. Suppression and amplification are two ends of one axis, with inheritance between them; what moves a setting along this axis is how much of the model's kernel-weighted probability mass on a wrong answer falls on prompts where the verifier accepts it.

**Proposed reconciliation hypothesis:** Prior studies disagree because they plausibly sit at different points on the universality axis—the false positives in Zhang's [2026] leaky test suites are per-task (low $u$), while Khalifa et al.'s [2026] hack edits the test file and works everywhere ($u = 1$).

**Limitations:** The setting is small (two-token responses, exact enumeration), leaks are injected rather than discovered, and the sign of drift at low universality depends on step size. Proposition 2 covers only one channel of the drift and does not give a bound.

**Future work:** Scale to real language models by measuring $u_{\mathrm{eff}}$ and cross-prompt coherence of accepted wrong responses from per-prompt gradients, testing whether they predict which false positives RLVR amplifies.

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