Title: Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models

URL Source: https://arxiv.org/html/2605.25189

Published Time: Tue, 26 May 2026 01:11:28 GMT

Markdown Content:
Jiaji Huang Kaan Ozkara Yushu Li Christos Thrampoulidis Xiaoxiao Li Youngsuk Park

###### Abstract

Reward hacking arises when a model improves a proxy reward by exploiting shortcuts rather than solving the intended task. We study this failure mode through the geometry of reinforcement learning updates in language models and argue that hacking emerges when optimization drifts away from a stable low-dimensional learning trajectory. We analyze this drift through dominant singular directions of parameter updates and show that reward-hacking runs exhibit substantially larger directional change than clean runs. Motivated by this observation, we introduce trusted-direction projection, which constrains gradients to remain within a clean reference subspace. Across reward-hacking experiments on mathematical reasoning, the proposed approach delays shortcut exploitation and better preserves task performance.

reward hacking, reinforcement learning, language models, optimization dynamics

## 1 Introduction

Reinforcement learning (RL) has become a widely used approach for improving the reasoning capabilities of large language models (LLMs)(Guo et al., [2025](https://arxiv.org/html/2605.25189#bib.bib12 "Deepseek-r1: incentivizing reasoning capability in llms via reinforcement learning"); Deng et al., [2025b](https://arxiv.org/html/2605.25189#bib.bib13 "On the effect of negative gradient in group relative deep reinforcement optimization")). However, RL training can suffer from reward hacking(Skalse et al., [2022](https://arxiv.org/html/2605.25189#bib.bib9 "Defining and characterizing reward gaming")), where a model achieves high reward by exploiting unintended shortcuts in the training environment rather than genuinely solving the target task(Wang et al., [2026](https://arxiv.org/html/2605.25189#bib.bib11 "Is it thinking or cheating? detecting implicit reward hacking by measuring reasoning effort"); Li et al., [2025](https://arxiv.org/html/2605.25189#bib.bib7 "Generalist reward models: found inside large language models")). This failure mode is particularly concerning for LLM reasoning, as the proxy reward may indicate improvement even while true task performance degrades. For instance, when a dataset or evaluation pipeline contains exploitable artifacts(Wang et al., [2026](https://arxiv.org/html/2605.25189#bib.bib11 "Is it thinking or cheating? detecting implicit reward hacking by measuring reasoning effort")), the model may learn to depend on these artifacts instead of developing the intended reasoning ability. 

Prior work has largely framed reward hacking as a problem of reward misspecification(Turpin et al., [2025](https://arxiv.org/html/2605.25189#bib.bib10 "Teaching models to verbalize reward hacking in chain-of-thought reasoning")). From this perspective, failures arise because reward functions or learned reward models do not fully capture the true objective, allowing models to optimize proxy signals in unintended ways. Existing approaches therefore focus on improving reward modeling(Turpin et al., [2025](https://arxiv.org/html/2605.25189#bib.bib10 "Teaching models to verbalize reward hacking in chain-of-thought reasoning"); Li et al., [2025](https://arxiv.org/html/2605.25189#bib.bib7 "Generalist reward models: found inside large language models"); He et al., [2025](https://arxiv.org/html/2605.25189#bib.bib4 "GARDO: reinforcing diffusion models without reward hacking")) or introducing additional regularization toward a reference model(Laidlaw et al., [2024](https://arxiv.org/html/2605.25189#bib.bib6 "Correlated proxies: a new definition and improved mitigation for reward hacking")). While these methods are valuable, they face a fundamental limitation: constructing a perfect reward model is inherently difficult, particularly for complex reasoning tasks where the true objective is only partially specified. Strong regularization can also constrain the model’s ability to learn beyond the reference policy. 

Recent work(Ackermann et al., [2026](https://arxiv.org/html/2605.25189#bib.bib1 "Gradient regularization prevents reward hacking in reinforcement learning from human feedback and verifiable rewards")) has begun to investigate the learning mechanisms underlying reward hacking, showing that it is closely associated with sharp local minima and can therefore be mitigated by smoothing the optimization landscape(Kwon et al., [2021](https://arxiv.org/html/2605.25189#bib.bib5 "Asam: adaptive sharpness-aware minimization for scale-invariant learning of deep neural networks")). However, these approaches primarily regulate the magnitude of parameter updates, without explicitly enforcing their directional alignment with the true objective. Meanwhile, RL updates in language models appear to exhibit a striking linear structure: a large fraction of the performance gain is captured by the leading singular direction of the parameter update matrix, and this dominant direction evolves along an approximately linear trajectory throughout training(Cai et al., [2026](https://arxiv.org/html/2605.25189#bib.bib2 "On predictability of reinforcement learning dynamics for large language models")). Building on this observation, we study reward hacking through the lens of optimization dynamics. We argue that reward hacking is not merely a consequence of imperfect reward design, but more fundamentally arises when gradient updates drift away from the model’s intrinsic learning trajectory and enter directions that improve proxy reward while remaining misaligned with true task performance. 

To mitigate reward hacking, we propose trusted-direction gradient alignment (TDGA), which constructs a reliable optimization subspace by applying SVD to the parameter changes induced by a small number of clean supervised training steps. During RL training, we project gradients onto this trusted learning subspace, constraining updates to remain within a safer region of the parameter space. Our contributions are threefold: 

\bullet We characterize reward hacking as directional drift in the dominant singular subspace of RL updates. 

\bullet We empirically show that clean training preserves directional consistency, whereas reward-hacking runs exhibit sharp rotations away from trusted directions. 

\bullet We introduce a trusted-direction gradient alignment that anchors RL updates to a clean reference subspace and substantially delays reward hacking.

## 2 Related Work

Reward hacking in language-model RL. Reward hacking occurs when an agent achieves high proxy reward by exploiting a mismatch between the reward signal and the intended objective(Skalse et al., [2022](https://arxiv.org/html/2605.25189#bib.bib9 "Defining and characterizing reward gaming")). Existing mitigations mainly improve the reward signal itself, through stronger reward models(Li et al., [2025](https://arxiv.org/html/2605.25189#bib.bib7 "Generalist reward models: found inside large language models"); Liu et al., [2025](https://arxiv.org/html/2605.25189#bib.bib8 "Inference-time scaling for generalist reward modeling")), task-specific anti-hacking schemes(He et al., [2025](https://arxiv.org/html/2605.25189#bib.bib4 "GARDO: reinforcing diffusion models without reward hacking")), or formal treatments of correlated proxies(Laidlaw et al., [2024](https://arxiv.org/html/2605.25189#bib.bib6 "Correlated proxies: a new definition and improved mitigation for reward hacking")). However, perfect reward specification is often difficult. A complementary line of work studies hacking through optimization geometry: gradient regularization smooths unstable updates(Ackermann et al., [2026](https://arxiv.org/html/2605.25189#bib.bib1 "Gradient regularization prevents reward hacking in reinforcement learning from human feedback and verifiable rewards")), while sharpness-aware methods favor flatter and more robust minima(Foret et al., [2020](https://arxiv.org/html/2605.25189#bib.bib3 "Sharpness-aware minimization for efficiently improving generalization"); Kwon et al., [2021](https://arxiv.org/html/2605.25189#bib.bib5 "Asam: adaptive sharpness-aware minimization for scale-invariant learning of deep neural networks")). These methods control local smoothness or update magnitude, but do not explicitly preserve alignment with task-relevant learning directions. 

Optimization geometry and learning dynamics. The learning dynamics of LLMs have recently received increasing attention. For example, (Deng et al., [2025c](https://arxiv.org/html/2605.25189#bib.bib15 "Efficient forward-only data valuation for pretrained llms and vlms")) studies learning dynamics for identifying valuable fine-tuning data, while (Deng et al., [2025b](https://arxiv.org/html/2605.25189#bib.bib13 "On the effect of negative gradient in group relative deep reinforcement optimization"), [a](https://arxiv.org/html/2605.25189#bib.bib14 "On grpo collapse in search-r1: the lazy likelihood-displacement death spiral")) analyze likelihood dynamics to diagnose RL training collapse. Recent work on language-model RL further shows that parameter updates often exhibit a low-rank and approximately linear structure, where the leading singular directions explain much of the performance change induced by training(Cai et al., [2026](https://arxiv.org/html/2605.25189#bib.bib2 "On predictability of reinforcement learning dynamics for large language models")). Our work connects these lines of research by interpreting reward hacking as directional drift away from a trusted low-dimensional learning trajectory, and by projecting RL gradients back onto that trajectory.

## 3 Dominant Update Directions

![Image 1: Refer to caption](https://arxiv.org/html/2605.25189v1/images/rank1_cca_similarity_same_row_lightcolor.png)

Figure 1: Rank-1 CCA similarity of dominant update directions in clean and reward-hacking training. We compare \mathrm{CCA}_{1}(U_{20},U_{80}), the cosine similarity between the dominant update directions at checkpoints 20 and 80, across different weight matrices. Higher values indicate stronger directional consistency over training. Left: mean rank-1 CCA across layers for each module. Right: worst-layer rank-1 CCA, corresponding to the least aligned layer within each module. Across both views, the reward-hacking model exhibits consistently lower similarity than the clean model, indicating substantially stronger directional drift away from the stable learning trajectory.

Let W_{t} denote the model parameters at training step t, and define the parameter update as

\displaystyle\Delta W_{t}=W_{t}-W_{0}.(1)

We analyze the structure of \Delta W_{t} via singular value decomposition (SVD):

\displaystyle\Delta W_{t}=\sum_{i=1}^{r}\sigma_{i}^{(t)}u_{i}^{(t)}{v_{i}^{(t)}}^{\top},(2)

where \sigma_{i}^{(t)} are the singular values in descending order, and u_{i}^{(t)} and v_{i}^{(t)} are the corresponding left and right singular vectors.

###### Definition 3.1(Rank-K Dominant Direction).

Let \Delta W_{t}=\sum_{i=1}^{r}\sigma_{i}^{(t)}u_{i}^{(t)}{v_{i}^{(t)}}^{\top} be the singular value decomposition of the parameter update at training step t. We define the _rank-K dominant update_ as the truncated SVD,

\Delta W_{t}^{(K)}=\sum_{i=1}^{K}\sigma_{i}^{(t)}u_{i}^{(t)}{v_{i}^{(t)}}^{\top}.

We further define the corresponding output-direction subspace as

U_{t}^{(K)}:=\bigl[u_{1}^{(t)},\ldots,u_{K}^{(t)}\bigr],

which represents the principal output-space directions along which the update acts.

Interpretation. The subspace U_{t}^{(K)} captures the dominant modes of change induced by RL training, corresponding to the directions that explain the largest variation in the parameter update. Empirically, these dominant directions account for a substantial portion of the performance gain and evolve smoothly throughout training(Cai et al., [2026](https://arxiv.org/html/2605.25189#bib.bib2 "On predictability of reinforcement learning dynamics for large language models")). From a functional perspective, the rank-K update induces the following transformation on a hidden representation h:

\Delta y^{(K)}=\Delta W_{t}^{(K)}h=\sum_{i=1}^{K}\sigma_{i}^{(t)}u_{i}^{(t)}\langle v_{i}^{(t)},h\rangle.(3)

This can be interpreted as a superposition of K key-value operations, where each v_{i}^{(t)} selects a relevant input feature and each u_{i}^{(t)} determines the corresponding direction of the output update. We focus on controlling the output-direction subspace U_{t}^{(K)}, while leaving the input-side directions \{v_{i}^{(t)}\}_{i=1}^{K} unconstrained. 

Measuring Directional Change via CCA. Given the dominant directions defined above, we quantify how they evolve throughout training. Because dominant update directions may form a low-dimensional subspace when aggregated across layers or checkpoints, we use Canonical Correlation Analysis (CCA) to measure subspace similarity in a geometry-aware manner. For two checkpoints t and s, let

U_{t}=\bigl[u_{t}^{(1)},\dots,u_{t}^{(k)}\bigr],\qquad U_{s}=\bigl[u_{s}^{(1)},\dots,u_{s}^{(k)}\bigr],

denote the subspaces spanned by their top-k singular directions, where k is the number of retained dominant components. We define their CCA similarity as \mathrm{CCA}_{k}(U_{t},U_{s})=\frac{1}{k}\sum_{i=1}^{k}\sigma_{i}, where \sigma_{i} are the canonical correlations between the two subspaces. Values close to 1 indicate that the two subspaces are highly aligned, whereas smaller values reflect increasingly strong directional drift.

## 4 Directional Shift in Reward Hacking

We analyze how the dominant update direction evolves during training through the rank-1 CCA similarity, \mathrm{CCA}_{1}(U_{20},U_{80}), between checkpoints 20 and 80 (details in [Section 6](https://arxiv.org/html/2605.25189#S6 "6 Experimental Setting ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models")) (Analysis of Rank 5 see[Section A.2](https://arxiv.org/html/2605.25189#A1.SS2 "A.2 More Directional Shift ‣ Appendix A Appendix ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models")). Larger values indicate stronger directional consistency. 

Small Direction Shift in Non-Hacking Models. We first examine checkpoints that do not exhibit reward hacking. Empirically, their dominant update directions evolve smoothly and consistently over training. As shown in [Figure 1](https://arxiv.org/html/2605.25189#S3.F1 "In 3 Dominant Update Directions ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"), the clean run maintains high mean CCA values around 0.8 across nearly all modules, indicating that the dominant update direction is largely preserved throughout training. Even in the worst-layer view, the similarities remain substantially higher than those of the hacking model, suggesting limited layer-wise drift. Thus, non-hacking training exhibits only a small directional shift. 

Large Direction Shift in Hacking Models. We next examine models that exhibit reward hacking. In contrast to the clean run, the hacking model shows a pronounced loss of directional consistency over training. As shown in [Figure 1](https://arxiv.org/html/2605.25189#S3.F1 "In 3 Dominant Update Directions ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"), its mean CCA decreases across nearly all modules by roughly 0.2, indicating much stronger drift in the dominant update direction. The effect is more severe in the worst-layer view, where several modules reach very low similarity values, sometimes below 0.1. Overall, the results demonstrate that reward hacking is associated with a substantial departure from the model’s intrinsic learning direction.

## 5 Method

Motivated by this observation, we constrain RL updates to the rank-K dominant subspace, keeping optimization aligned with intrinsic dynamics and preventing drift into hacking directions. 

Trusted Direction Gradient Alignment.

![Image 2: Refer to caption](https://arxiv.org/html/2605.25189v1/x1.png)

(a)Proxy reward during training.

![Image 3: Refer to caption](https://arxiv.org/html/2605.25189v1/images/true_reward_improved.png)

(b)True reward under loophole-free evaluation.

Figure 2: Delayed reward hacking and improved preservation of true performance. The left panel shows the evolution of the proxy reward, which reflects the model’s ability to exploit the loophole, while the right panel reports the true reward measured under loophole-free evaluation. Faint curves denote raw trajectories and bold curves denote smoothed trends. Compared with vanilla RL, SAM, and gradient regularization, trusted-direction projection slows the rise of proxy reward and maintains substantially more stable true reward throughout training.

At training step t, let G_{t}\in\mathbb{R}^{d_{\mathrm{out}}\times d_{\mathrm{in}}} denote the gradient of the objective with respect to a model weight matrix. Following [Section 3](https://arxiv.org/html/2605.25189#S3 "3 Dominant Update Directions ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"), we first estimate a trusted rank-K output subspace from a short clean-data warmup phase:

U_{\mathrm{clean}}^{(K)}=\bigl[u_{1}^{\mathrm{clean}},\dots,u_{K}^{\mathrm{clean}}\bigr],(4)

together with the associated singular values \{\sigma_{i}^{\mathrm{clean}}\}_{i=1}^{K}. To preserve the relative importance of the dominant clean directions, we define the diagonal weight matrix

\Lambda_{\mathrm{clean}}^{(K)}=\operatorname{diag}(\alpha_{1},\dots,\alpha_{K}),\qquad\alpha_{i}=\frac{\sigma_{i}^{\mathrm{clean}}}{\sum_{j=1}^{K}\sigma_{j}^{\mathrm{clean}}}.(5)

We then project the gradient onto the trusted output-direction subspace using singular-value weighting:

G_{t}^{\parallel}=U_{\mathrm{clean}}^{(K)}\Lambda_{\mathrm{clean}}^{(K)}{U_{\mathrm{clean}}^{(K)}}^{\top}G_{t}.(6)

Our update uses only the trusted component, \widetilde{G}_{t}=G_{t}^{\parallel}. This retains the top-K clean directions and emphasizes each one according to its singular value. As a result, the update remains aligned with the intrinsic clean learning trajectory while suppressing off-subspace components that may introduce instability or encourage reward-hacking behavior.

## 6 Experimental Setting

Training Settings. We follow Wang et al.(Wang et al., [2026](https://arxiv.org/html/2605.25189#bib.bib11 "Is it thinking or cheating? detecting implicit reward hacking by measuring reasoning effort")) and evaluate on Big-Math-RL-Verified under the in-context loophole setting. We use Qwen2.5-3B-Instruct and train on 24,379 examples, with 1,498 examples held out for validation and evaluation. All methods are trained on 8 GPUs with per-device batch size 4, 64 gradient accumulation steps, learning rate 10^{-5}, constant scheduling, and KL coefficient 10^{-3}. We sample 8 rollouts per prompt during training and 1 during evaluation, with a maximum completion length of 512 tokens. Unless otherwise stated, all methods use the same configuration. 

Baselines. We compare against representative RL-stabilization baselines under reward hacking. Gradient Regularization(Ackermann et al., [2026](https://arxiv.org/html/2605.25189#bib.bib1 "Gradient regularization prevents reward hacking in reinforcement learning from human feedback and verifiable rewards")) smooths optimization by penalizing large or unstable gradients, but mainly controls update magnitude. SAM(Foret et al., [2020](https://arxiv.org/html/2605.25189#bib.bib3 "Sharpness-aware minimization for efficiently improving generalization"); Kwon et al., [2021](https://arxiv.org/html/2605.25189#bib.bib5 "Asam: adaptive sharpness-aware minimization for scale-invariant learning of deep neural networks")) improves robustness by favoring flatter minima, but targets local loss smoothness.

## 7 Results

Delayed Reward Hacking. As shown in [Figure 2(a)](https://arxiv.org/html/2605.25189#S5.F2.sf1 "In Figure 2 ‣ 5 Method ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"), vanilla RL rapidly enters the hacking regime, with proxy reward saturating near 0.9 within about 50 steps; gradient regularization and SAM provide only limited delay. In contrast, trusted-direction projection substantially slows saturation: rank-1 does not reach this regime within 400 steps, while rank-5 and rank-10 remain unhacked till 200 steps. Consistently, [Figure 2(b)](https://arxiv.org/html/2605.25189#S5.F2.sf2 "In Figure 2 ‣ 5 Method ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models") shows that our methods preserve a higher true reward. [Table 1](https://arxiv.org/html/2605.25189#S7.T1 "In 7 Results ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models") further summarizes this effect by comparing peak, epoch-level true reward across methods.

Table 1: Peak and epoch-level true reward. Values are mean true rewards from the curves in [Figure 2(b)](https://arxiv.org/html/2605.25189#S5.F2.sf2 "In Figure 2 ‣ 5 Method ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"). The peak is computed over the displayed training horizon.

Epoch-Level Performance.[Table 1](https://arxiv.org/html/2605.25189#S7.T1 "In 7 Results ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models") quantifies the stability gains from trusted-direction projection. While non-TDGA methods improve early true reward, they collapse by the second epoch, with vanilla RL, gradient regularization, and SAM all falling to 0.000. In contrast, TDGA improves both peak and long-horizon performance: rank-10 achieves the highest peak at 0.541 and one-epoch rewards, while rank-5 obtains the best two-epoch value of 0.529. These results show that trusted-direction projection not only delays proxy-reward saturation but also improves and preserves genuine task performance over longer training. 

Trade-off with Projection Rank. The trusted-subspace rank K controls the trade-off between robustness and flexibility. Smaller ranks enforce stronger alignment with the clean trajectory, suppressing reward hacking more aggressively but limiting adaptation. Larger ranks provide more optimization freedom and better task performance, but weaken the constraint against shortcut directions. As shown in [Figure 2](https://arxiv.org/html/2605.25189#S5.F2 "In 5 Method ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"), rank-1 is the most conservative, while rank-5 and rank-10 better preserve true reward while still delaying hacking relative to baselines.

## 8 Conclusion

We studied reward hacking in language-model reinforcement learning through the geometry of parameter updates. Our analysis shows that clean training preserves a stable dominant update direction, whereas reward-hacking runs undergo a pronounced directional shift away from this trajectory. Motivated by this finding, we introduced TDGA, which projects RL gradients onto a trusted subspace estimated from clean supervised updates. Experiments show that TDGA delays reward hacking and preserves true reward. 

Future Work: We will explore more precise constraints to better unlock model performance while preventing reward hacking. One promising direction, for which we have already observed positive results, is iteratively updating the trusted learning directions. Additional directions are discussed in [Section A.1](https://arxiv.org/html/2605.25189#A1.SS1 "A.1 Future Work ‣ Appendix A Appendix ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models").

## Acknowledgments

The authors sincerely thank Yida Wang and Xuanqi Zhang for their support. This work was partially funded by the NSERC Discovery Grant RGPIN-2021-03677, Alliance Grant ALLRP 581098-22, the Natural Science and Engineering Research Council of Canada (NSERC), the Canada CIFAR AI Chairs program, the Canada Research Chair program, an IITP grant funded by MSIT, and the Digital Research Alliance of Canada.

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*   X. Wang, N. Joshi, B. Plank, R. Angell, and H. He (2026)Is it thinking or cheating? detecting implicit reward hacking by measuring reasoning effort. ICLR. Cited by: [§1](https://arxiv.org/html/2605.25189#S1.p1.3 "1 Introduction ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"), [§6](https://arxiv.org/html/2605.25189#S6.p1.2 "6 Experimental Setting ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"). 

## Appendix A Appendix

### A.1 Future Work

A natural next step is to investigate reward hacking in multi-turn reinforcement learning(Deng et al., [2025a](https://arxiv.org/html/2605.25189#bib.bib14 "On grpo collapse in search-r1: the lazy likelihood-displacement death spiral")). Recent work on inference-time scaling for reward modeling(Liu et al., [2025](https://arxiv.org/html/2605.25189#bib.bib8 "Inference-time scaling for generalist reward modeling")) suggests that reward exploitation may become more pronounced in longer-horizon and agentic settings, where a model can exploit the reward through a sequence of intermediate actions rather than a single shortcut. Extending our framework to analyze trajectory-level directional drift may therefore provide a clearer understanding of how reward hacking emerges and accumulates over long-horizon interactions.

Another important direction is to choose the projection rank and training schedule more systematically. Our results suggest a clear trade-off: small ranks suppress hacking more strongly but may over-constrain learning, while larger ranks improve flexibility but weaken robustness. Future work could adapt the rank, RL steps, and clean fine-tuning schedule online using signals such as singular-value decay, directional drift, or validation performance.

### A.2 More Directional Shift

![Image 4: Refer to caption](https://arxiv.org/html/2605.25189v1/images/rank5_cca_similarity_same_row_lightcolor.png)

Figure 3: Rank-5 CCA similarity of dominant update directions in clean and reward-hacking training. We compare \mathrm{CCA}_{5}(U_{20},U_{80}), the similarity between the top-5 dominant update subspaces at checkpoints 20 and 80, across different weight matrices. Higher values indicate stronger directional consistency over training. Left: mean rank-5 CCA across layers for each module. Right: worst-layer rank-5 CCA, corresponding to the least aligned layer within each module. Across both views, the reward-hacking model exhibits consistently lower similarity than the clean model, indicating stronger directional drift away from the stable learning trajectory.

Figure[Figure 3](https://arxiv.org/html/2605.25189#A1.F3 "In A.2 More Directional Shift ‣ Appendix A Appendix ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models") shows that the rank-5 analysis leads to the same qualitative conclusion as the rank-1 result in[Figure 1](https://arxiv.org/html/2605.25189#S3.F1 "In 3 Dominant Update Directions ‣ Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models"): reward-hacking training deviates more strongly from the clean learning trajectory. Across both the mean-layer and worst-layer views, the hacking run maintains lower CCA similarity than the clean run, indicating that the larger trusted subspace still captures a clear difference in directional stability.

At the same time, moving from rank-1 to rank-5 slightly reduces the absolute CCA values for both clean and hacking runs. This suggests that the approximate linearity of the update trajectory weakens somewhat as additional singular directions are included, since those weaker components are less stable than the leading one. Nevertheless, the clean–hacking gap remains pronounced, showing that the directional-drift phenomenon is robust beyond the single dominant direction.

### Impact Statement

This paper studies reward hacking in reinforcement learning for language models and proposes a mitigation strategy aimed at improving training reliability. Better understanding and controlling reward hacking may reduce unsafe shortcut-seeking behavior in downstream systems, but the same insights could also be used to design stronger proxy objectives or more effective attacks if misapplied.
