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Factored Causal Representation Learning for Robust Reward Modeling in RLHF
Yupei Yang, Lin Yang, Wanxi Deng, Lin Qu, Fan Feng, Biwei Huang, Shikui Tu, Lei Xu
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Summary
The paper introduces CausalRM, a framework for robust reward modeling in RLHF that uses factored causal representation learning. By decomposing contextual embeddings into causal factors (sufficient for reward prediction) and non-causal factors (capturing spurious attributes like length or sycophantic bias), the model enforces causal invariance. It utilizes an adversarial head with gradient reversal to suppress reward-relevant information in the non-causal component, effectively mitigating reward hacking.
Entities (6)
Relation Signals (3)
CausalRM → mitigates → Reward Hacking
confidence 95% · CausalRM improves both reward model accuracy and downstream RLHF performance. Furthermore, we demonstrate strong mitigation of reward hacking behaviors
CausalRM → uses → Gradient Reversal Layer
confidence 95% · we introduce an adversarial head trained via gradient reversal to suppress reward-relevant signals in the non-causal component
Causal Factors → predicts → Reward
confidence 90% · The reward head is then constrained to depend only on the causal component.
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Abstract
Abstract:A reliable reward model is essential for aligning large language models with human preferences through reinforcement learning from human feedback. However, standard reward models are susceptible to spurious features that are not causally related to human labels. This can lead to reward hacking, where high predicted reward does not translate into better behavior. In this work, we address this problem from a causal perspective by proposing a factored representation learning framework that decomposes the model's contextual embedding into (1) causal factors that are sufficient for reward prediction and (2) non-causal factors that capture reward-irrelevant attributes such as length or sycophantic bias. The reward head is then constrained to depend only on the causal component. In addition, we introduce an adversarial head trained to predict reward from the non-causal factors, while applying gradient reversal to discourage them from encoding reward-relevant information. Experiments on both mathematical and dialogue tasks demonstrate that our method learns more robust reward models and consistently improves downstream RLHF performance over state-of-the-art baselines. Analyses on length and sycophantic bias further validate the effectiveness of our method in mitigating reward hacking behaviors.
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- Source: https://arxiv.org/abs/2601.21350
- Canonical: https://arxiv.org/abs/2601.21350
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Factored Causal Representation Learning for Robust Reward Modeling in RLHF Yupei Yang 1 2† Lin Yang 2 Wanxi Deng 2 Lin Qu 2 Fan Feng 3 4 Biwei Huang 3 Shikui Tu 1 * Lei Xu 1 Abstract A reliable reward model is essential for align- ing large language models (LLMs) with human preferences through reinforcement learning from human feedback (RLHF). However, standard re- ward models are susceptible to spurious features that are not causally related to human labels. This can lead to reward hacking, where high predicted reward does not translate into better behavior. In this work, we address this problem from a causal perspective by proposing a factored representation learning framework that decomposes the model’s contextual embedding into (1) causal factors that are sufficient for reward prediction and (2) non– causal factors that capture reward-irrelevant at- tributes such as length or sycophantic bias. The reward head is then constrained to depend only on the causal component. In addition, we introduce an adversarial head trained to predict reward from the non-causal factors, while applying gradient re- versal to discourage them from encoding reward- relevant information. Experiments on both mathe- matical and dialogue tasks demonstrate that our method learns more robust reward models and consistently improves downstream RLHF perfor- mance over state-of-the-art baselines. Analyses on length and sycophantic bias further validate the effectiveness of our method in mitigating reward hacking behaviors. 1. Introduction In recent years, RLHF has emerged as a powerful approach for aligning LLMs with human preferences (Ouyang et al., 2022; Bai et al., 2022). A core component of RLHF is the reward model, which serves as a proxy for human judg- ment during policy optimization. However, standard reward model training is prone to learning spurious correlations: † This work was done when the author was a research intern at Alibaba Group. 1 Shanghai Jiao Tong University 2 Alibaba Group 3 University of California San Diego 4 Mohamed bin Zayed Uni- versity of Artificial Intelligence. Correspondence to: Shikui Tu <tushikui@sjtu.edu.cn>. Preprint. January 30, 2026. the model may assign higher scores to preference-irrelevant patterns, such as response length or sycophantic phrasing, rather than the true causal drivers of human preference (Liu et al., 2024; Miao et al., 2024; Wang et al., 2025). The LLM then exploits these shortcuts during RLHF, achieving higher predicted rewards while drifting from the intended objec- tive—a phenomenon known as reward hacking (Amodei et al., 2016; Gao et al., 2023b; Skalse et al., 2022). To tackle this issue, extensive efforts have been made to mitigate the influence of known spurious factors in reward modeling. For example, ODIN (Chen et al., 2024) decom- poses the reward head into separate quality and length com- ponents to reduce reward hacking driven by response length. Park et al. (2024) develops a length-regularization strategy to prevent length exploitation during DPO (Rafailov et al., 2023). Wang et al. (2025) further proposes a more general maximum mean discrepancy (MMD)-based regularization that can constrain spurious behaviors beyond length, such as sycophantic bias (Sharma et al., 2023) or concept bias (Zhou et al., 2024). Despite their effectiveness, these ap- proaches require explicitly specifying the spurious variables to be controlled, whereas it is challenging to anticipate all possible exploitation patterns in practice. Another line of work seeks to improve reward models by filtering out irrelevant information directly through repre- sentation learning. Nath et al. (2024) employs a contrastive objective to learn goal-conditioned representations within the reward model, which helps distinguish between pre- ferred and dispreferred responses. InfoRM (Miao et al., 2024; 2025a) adopts an information-theoretic perspective, introducing a variational information bottleneck objective to encourage the latent representations to retain only infor- mation relevant to human preference. However, to the best of our knowledge, none of the existing methods explicitly disentangle reward-irrelevant factors from the latent state for reward modeling. Nevertheless, causal representation learning (Schölkopf et al., 2021; Huang et al., 2022; Kong et al., 2023; Yang et al., 2024b) has demonstrated strong promise in traditional reinforcement learning for learning minimal sufficient state representations that capture only the causally relevant aspects of the environment. Motivated by these insights, we propose CausalRM, a novel framework that explores the potential of causal represen- 1 arXiv:2601.21350v1 [cs.LG] 29 Jan 2026 Factored Causal Representation Learning for Robust Reward Modeling in RLHF tation learning to mitigate spurious correlations in reward modeling. Specifically, CausalRM decomposes the model’s contextual embedding into two disentangled components: (1) causal factors that are sufficient for reward prediction, and (2) non-causal factors that capture reward-irrelevant attributes such as response length or stylistic bias. Build- ing on this factorized representation, we constrain the re- ward model to predict rewards using only the causal fac- tors, and optimize it with a standard pairwise preference loss augmented with mutual-information-based constraints. To further discourage the non-causal factors from carry- ing reward-predictive signals, we introduce an adversarial head trained to predict reward from these factors. Follow- ing Ganin & Lempitsky (2015), we then optimize it with a gradient reversal layer (GRL), so that the adversarial head learns to predict reward while the encoder is pushed to remove reward-relevant information from the non-causal component. To summarize, our main contributions are three-fold: • We investigate the potential of causal representation learning for mitigating reward hacking in RLHF, and propose CausalRM, a framework that explicitly de- composes the latent representation into causal factors and reward-irrelevant non-causal factors. •To characterize both causal and non-causal representa- tions, we design a novel VAE-based architecture with (1) a reward prediction head trained with a pairwise preference loss augmented by mutual information con- straints, and (2) an adversarial head trained via gradient reversal to suppress reward-relevant signals in the non– causal component, which jointly enforces sufficiency, minimality, and causal invariance in the learned repre- sentations. •Extensive experiments on mathematical and dialogue tasks show that CausalRM improves both reward model accuracy and downstream RLHF performance. Furthermore, we demonstrate strong mitigation of re- ward hacking behaviors through reduced sensitivity to length and sycophantic bias, providing empirical vali- dation of causal invariance in learned representations. 2. Preliminaries Reward modeling in RLHF.In RLHF, reward modeling aims to learn a scalar-valued function that approximates human preferences. Given a promptxand a responsey, a reward model outputs a scalar rewardr θ (x,y)indicating the degree of preference for y under x. A widely used formulation is the Bradley–Terry model (Bradley & Terry, 1952), which defines the probability that a preferred responsey w is favored over a rejected response y l as: p(y w ≻ y l | x) = exp(r θ (x,y w )) exp(r θ (x,y w )) + exp(r θ (x,y l )) . (1) Given a human preference datasetD = (x i ,y w i ,y l i ) N i=1 , the reward model is typically trained by minimizing the pairwise negative log-likelihood: L RM (θ) = −E (x,y w ,y l )∼D logσ r θ (x,y w )− r θ (x,y l ) , (2) where σ(·) denotes the sigmoid function. Standard parameterization and reward hacking. In practice, the reward model is often initialized from a su- pervised fine-tuned (SFT) language model by reusing its backbone as a feature extractor and attaching a lightweight reward head, which is typically a single linear layer. Con- cretely, given a prompt–response pair(x,y), the SFT back- bone produces a contextual representation h = f φ (x,y),(3) and the reward head maps it to a scalar reward r θ (x,y) = g ψ (h), θ = (φ,ψ),(4) wheref φ denotes the pretrained backbone andg ψ denotes the reward head. While effective, this parameterization can be vulnerable to reward hacking: the reward model may inherit and amplify preference-irrelevant signals already encoded in the SFT representation, causing the learned reward to correlate with spurious patterns rather than the causal drivers of human judgment (Miao et al., 2024; Wang et al., 2025). 3. Methodology In this section, we first analyze why standard reward models tend to learn spurious correlations from a causal perspective. Building on these insights, we construct a causal reward model, termed CausalRM, that filters out reward-irrelevant information via factored representation learning. Finally, we present the complete estimation procedure for CausalRM. 3.1. A Causal View of Reward Hacking The causal structure underlying standard reward modeling can be represented by Figure 1, where(x,y)are the input prompt and response pair,z c denotes causal factors that contain the essential information to predict the rewardr, and z nc represents spurious factors that do not causally influence the true reward, such as response length or stylistic bias. As illustrated, the presence of a direct edgez nc → ral- lows spurious features to directly affect the learned reward, 2 Factored Causal Representation Learning for Robust Reward Modeling in RLHF x y z c z nc r reward hacking Figure 1. Causal graph for standard reward modeling. The prompt– response pair(x,y)encode both causal (z c ) and non-causal (z nc ) factors, which in turn affect the predicted rewardr. While the path z c → ris desired, the spurious pathz nc → rleads to reward hacking. thereby leading to reward hacking. For example, suppose z nc captures response length on mathematical tasks, then changing the length alone may substantially alter the pre- dicted reward, even when the underlying solution quality remains unchanged. Instead, a robust reward model should satisfy causal invariance with respect to non-causal factors (Bühlmann, 2020; Veitch et al., 2021): r θ (x,y) ⊥ z nc .(5) In other words, the reward value should be insensitive to non-causal attributes of the response. 3.2. CausalRM: Factored Causal Representation Learning for Reward Models Motivated by the causal analysis in Section 3.1, we propose CausalRM, a reward modeling framework that aims to block the spurious pathz nc → rby (1) structurally restricting reward prediction to depend only on a causal representation, and (2) actively removing reward-predictive information from the remaining representation. Concretely, CausalRM augments a standard RM with a latent-variable bottleneck and two auxiliary heads, jointly promoting sufficiency for reward prediction, invariance to non-causal variation, and non-degenerate representations. Factored latent representation. Given a prompt– response pair(x,y), we first compute a contextual embed- dingh = f φ (x,y)using the SFT backbone. CausalRM then maps h to two latent variables via a variational encoder: q α (z c | h), q α (z nc | h),(6) wherez c is encouraged to retain information that is suffi- cient for predicting human preference, whilez nc is encour- aged to capture reward-irrelevant attributes (e.g., length or stylistic cues). We parameterize both posteriors as diagonal- covariance Gaussians whose means and log-variances are produced by separate linear projections applied toh, with standard normal priors p(z c ) = p(z nc ) =N(0,I). This factorization provides a convenient interface for im- posing causal invariance: downstream prediction modules can be forced to condition only onz c , whilez nc serves as a dedicated channel for non-causal variation. Causal reward head. We predict reward solely from the causal component: ˆr = r θ (x,y) = g ψ (z c ), θ = (φ,α,ψ),(7) whereg ψ is a linear reward head. By construction, the reward head has no access toz nc , which implements a structural bias towards the invariance principle in Eq. (5). Reconstruction head. A structural restriction alone may lead to degenerate solutions, such as posterior collapse (Bowman et al., 2016; Alemi et al., 2016). To encourage (z c ,z nc )to retain the information inhwhile allowing it to be redistributed across the two factors, we add a reconstruc- tion decoder ˆ h = d η ([z c ;z nc ]),(8) which reconstructs the backbone embedding from the con- catenated latents. Adversarial head with gradient reversal. Finally, to explicitly remove reward-predictive signals from the non- causal component, we introduce an adversarial heada ω that predicts reward from z nc : ˆr adv = a ω (z nc ).(9) The adversary is optimized to be predictive, while the en- coder is optimized via a gradient reversal layer to makez nc uninformative about the reward. This adversarial objective complements the structural restriction above by penalizing any reward-relevant information that leaks intoz nc , thereby encouraging the desired invariance. In summary, the reward head enforces a structural pref- erence for invariance by conditioning only onz c , the re- construction head prevents degenerate factorization, and the adversarial head further suppresses reward information leakage intoz nc . Together, these mechanisms mitigate spu- rious correlations in reward modeling without requiring task- specific supervision. The overall architecture of CausalRM is illustrated in Figure 2. 3.3. Model Estimation CausalRM is trained by jointly optimizing the backbone, the factorized encoder, and the three heads introduced in Section 3.2. Specifically, we learn parametersθ = (φ,α,ψ,η,ω) by solving: min φ,α,ψ,η max ω L pref + λ c KL L c KL |z minimal sufficiency + λ rec L rec |z non-degeneracy +λ nc KL L nc KL − λ adv L adv |z invariance , (10) whereL pref encouragesz c to be predictive of human pref- erence,L c KL enforces an information bottleneck onz c to discourage redundant information,L rec prevents degener- ate factorizations by reconstructing the backbone embed- ding,L nc KL regularizes the non-causal latent by matching 3 Factored Causal Representation Learning for Robust Reward Modeling in RLHF LLM Backbone 푓 ! Prompt 푥 Response 푦 ℎ VAE Encoder 푞 " 푧 # ℎ,푞 " (푧 $# |ℎ) 푧 # 푧 $# Concat [푧 # ;푧 $# ] Reconstruction Decoder 푑 % 0 ℎ Non-degeneracy 휆 &'( ℒ &'( RewardHead 푔 ) (푧 # ) ̂푟 Minimal sufficiency ℒ *&'+ +휆 ,- # ℒ ,- # AdversarialHead 푎 . (푧 $# ) ̂푟 /01 Invariance −휆 /01 ℒ /01 +휆 ,- $# ℒ ,- $# Factored latent bottleneck causal factors non-causal factors GRL on ℒ /01 Causal reward prediction Representation reconstruction Adversarial invariance ℒ 232/4 Figure 2. Overview of CausalRM. The backbone embeddinghis factorized into causal latentsz c and non-causal latentsz nc via a variational encoder. Reward prediction is restricted to depend only onz c , while an adversarial head trained through a gradient reversal layer (GRL) discouragesz nc from encoding reward-predictive information. A reconstruction decoder prevents degenerate factorization by reconstructing h from [z c ;z nc ]. q α (z nc | h)to the prior, andL adv measures how well the non-causal factorz nc can predict human preferences. This minimax objective reflects distinct goals for different com- ponents: •The adversary parametersωare minimized overL adv , encouraging it to accurately predict preferences from the non-causal representation z nc . • The encoder parameters(φ,α)are maximized overL adv , i.e., they are updated to increase the adversarial loss, thereby removing reward-predictive signals fromz nc and enforcing causal invariance. The coefficientsλ c KL ,λ nc KL ,λ adv ,λ rec control the trade-offs between these competing objectives. Sufficiency and minimality for the causal factorz c .To makez c sufficient for reward prediction, we maximize the mutual informationI(z c ;r)between the causal latent and the reward signal. To further encourage a minimal causal representation, we introduce an information bottleneck that penalizes reward-irrelevant information retained from the backbone embedding, measured byI(h;z c ). We thus seek to maximize the following objective: max I(z c ;r) − λ c KL I(h;z c ). (11) Given a preference triplet(x,y w ,y l ), we optimize a vari- ational lower bound of Eq. (11). Leth w = f φ (x,y w )and h l = f φ (x,y l )be the backbone embeddings, and sample z w,c ∼ q α (z c | h w ) and z l,c ∼ q α (z c | h l ). Then, I(z c ;r) − λ c KL I(h;z c ) ≥ E (x,y w ,y l )∼D h logσ r θ (x,y w )− r θ (x,y l ) |z ≜−L pref − λ c KL KL(q α (z w,c | h w )∥p(z c )) + KL q α (z l,c | h l )∥p(z c ) |z ≜L c KL i . (12) HereL pref is the standard pairwise preference loss, and L c KL upper bounds the information bottleneck term via a KL regularizer. We usez w,c andz l,c to denote the causal latents inferred from the preferred and dispreferred responses, re- spectively. A detailed derivation is provided in Appendix A. Invariance via adversarial prediction onz nc .While the objective above encouragesz c to be minimally sufficient, it does not by itself prevent reward-predictive information from being encoded in the non-causal factorz nc . To this end, we introduce an adversarial heada ω that attempts to predict human preferences fromz nc . Specifically, we decompose the adversarial objective into (1) an adversarial preference loss that trainsa ω to predict preferences fromz nc , and (2) a standard KL regularizer onz nc that prevents unconstrained growth of the non-causal latent: L adv =−E (x,y w ,y l )∼D logσ a ω (z w,nc )− a ω (z l,nc ) , (13) L nc KL = E (x,y w ,y l )∼D h KL(q α (z w,nc |h w )∥p(z nc )) + KL(q α (z l,nc |h l )∥p(z nc )) i , (14) wherez w,nc ∼ q α (z nc | h w )andz l,nc ∼ q α (z nc | h l )are sampled from the non-causal posteriors for the preferred and dispreferred responses, respectively. The adversary parametersωare optimized to minimizeL adv , yielding a strong predictor fromz nc . Meanwhile, we place a gradient reversal layer betweenz nc anda ω , so that gradients from L adv are negated before reaching the encoder. Consequently, the encoder is encouraged to maximize theL adv , making z nc uninformative for preference prediction and thereby reducing reward leakage into the non-causal component. Non-degeneracy via reconstruction. To ensure that the factorized latent representation(z c ,z nc )collectively pre- serves the information in the backbone embeddingh, we include a reconstruction term that minimizes the distance 4 Factored Causal Representation Learning for Robust Reward Modeling in RLHF Algorithm 1 Training CausalRM Require:Preference datasetD = (x,y w ,y l ), hyperpa- rameters λ c KL ,λ adv ,λ nc KL ,λ rec Ensure: Trained parameters θ = (φ,α,ψ,η,ω) 1: for each batch fromD do 2: Encode(x,y w )and(x,y l )to geth w ,h l via the LLM backbone f φ 3:Sample latent pairs(z w,c ,z w,nc )and(z l,c ,z l,nc )us- ing encoder q α (·| h) 4: Compute reward predictionsˆr = g ψ (z c )and adver- sarial predictions ˆr adv = a ω (z nc ) 5:Reconstruct embeddings ˆ h = d η ([z c ;z nc ]) 6:Compute total lossL total as in Eq. (10) 7: Apply gradient reversal toL adv during backpropaga- tion 8:Update all parametersθjointly by optimizingL total 9: end for between h and its reconstruction: L rec = E (x,y)∼D h h− d η ([z c ;z nc ]) 2 2 i ,(15) whered η is the reconstruction decoder and[z c ;z nc ]denotes concatenation. This objective encourages the latent repre- sentations to preserve the information inhwhile allowing reward-irrelevant variation to be captured inz nc , facilitating the intended factorization. These objectives are unified into a single training objec- tive that jointly learns a minimal sufficient causal factor z c for preference prediction, enforces invariance by sup- pressing reward-predictive signals in the non-causal factor z nc through adversarial training with gradient reversal, and prevents degenerate representations via reconstruction. The complete training procedure is summarized in Algorithm 1. 4. Experiments We evaluate the effectiveness of CausalRM in mitigating reward hacking across two representative tasks: mathemat- ical reasoning and open-ended dialogue. Specifically, our evaluation focuses on three key research questions: •RQ1 (Reward modeling accuracy): Can CausalRM bet- ter approximate human preferences compared to existing reward models? • RQ2 (Downstream RLHF alignment): Do improve- ments in reward modeling translate into stronger down- stream RLHF performance? • RQ3 (Causal invariance): Does CausalRM better sat- isfy the causal invariance principle, particularly with respect to known spurious attributes such as response length and sycophantic bias? 4.1. Setup Datasets. Formathematicalreasoning,weuse OpenMathInstruct-1 (Toshniwal et al., 2024), which contains 1.8M problem–solution pairs sourced from GSM8K and MATH. Following Nath et al. (2024), we adopt their constructed preference dataset to train both the reward model and the downstream policy. We evaluate reward modeling and RLHF performance on both in-distribution (ID) and out-of-distribution (OOD) benchmarks. The ID evaluation uses the test split from the same preference distribution, while the OOD evaluation includes algebra222 (He-Yueya et al., 2023), GSM-Hard (Gao et al., 2023a), ASDiv (Miao et al., 2020), MAWPS (Koncel-Kedziorski et al., 2016), and SVAMP (Patel et al., 2021). For open-ended dialogue, we train the reward model and perform RLHF on Anthropic-RLHF-H (Bai et al., 2022), which provides human preference annotations on helpful- ness and harmlessness for assistant responses. We use the dataset’s test split as the ID evaluation set, and evaluate OOD generalization on MT-Bench (Zheng et al., 2023), PKU-SafeRLHF (Ji et al., 2023), SHP (Askell et al., 2021), and TruthfulQA (Lin et al., 2022). Models.For mathematical reasoning, we adopt Qwen2.5- Math-7B (Yang et al., 2024a), a strong decoder-only LLM that has been tuned on GSM8K and MATH, as the base model for both reward modeling and RLHF. For open- ended dialogue, we first perform supervised fine-tuning on ShareGPT (Chiang et al., 2023) using Qwen2.5-7B (Qwen et al., 2025), and then use the resulting SFT backbone as the base model. Across all experiments, we employ Proxi- mal Policy Optimization (PPO; (Schulman et al., 2017)) for RLHF. All SFT, reward model, and PPO training are imple- mented using the OpenRLHF (Hu et al., 2024) framework. Baselines. We compare CausalRM against the following state-of-the-art reward modeling approaches: • Standard RM: the conventional reward model trained with the Bradley–Terry pairwise loss, using a linear re- ward head on top of the backbone embedding. •GoalRM (Nath et al., 2024): improves reward model- ing by learning goal-conditioned representations via a contrastive, Q-function-based objective, which helps dis- tinguish preferred and dispreferred responses for RLHF alignment. •InfoRM (Miao et al., 2025a): introduces a variational information bottleneck to filter out information irrelevant for reward prediction. Our work extends InfoRM by explicitly factorizing the latent space into causal and non- causal components, and jointly optimizing them under a unified framework that enforces both sufficiency and invariance. 5 Factored Causal Representation Learning for Robust Reward Modeling in RLHF Table 1. Reward model performance measured by pairwise accuracy (%, higher is better) on Mathematical Reasoning and Open-Ended Dialogue. We report in-distribution (ID) and out-of-distribution (OOD) results for each benchmark. Avg. denotes the average across the corresponding benchmarks. Bold indicates the best result and underlinedindicates the second best. Method Mathematical ReasoningOpen-Ended Dialogue IDOODIDOOD GSM8KMATHAvg.Algebra222GSM-HardASDivMAWPSSVAMPAvg.HelpfulHarmlessAvg.MT-BenchPKU-SafeRLHFSHPTruthfulQAAvg. Standard RM75.560.267.983.660.890.591.588.683.067.473.870.668.257.854.258.559.7 GoalRM80.356.268.381.664.489.589.186.482.267.574.771.166.259.353.863.660.7 InfoRM74.158.066.182.763.588.989.987.682.567.973.770.866.760.150.461.959.8 CausalRM (Ours)81.758.470.189.966.289.992.289.685.669.475.272.368.360.953.966.262.3 Table 2. Downstream RLHF performance on Mathematical Reasoning measured by final-answer accuracy (%, higher is better). Method IDOOD GSM8KMATHAvg.Algebra222GSM-HardASDivMAWPSSVAMPAvg. SFT80.453.366.980.254.582.493.688.680.0 Standard RM85.550.167.889.650.388.095.392.183.1 GoalRM89.455.672.595.170.188.495.992.988.5 InfoRM71.024.948.060.437.546.351.453.949.9 CausalRM (Ours)91.856.174.097.371.089.196.593.989.6 For downstream alignment, we additionally report the per- formance of the SFT model as a reference to evaluate whether RLHF with learned reward models leads to mean- ingful improvement. Evaluation Metrics. We evaluate reward models using pairwise accuracy, the fraction of preference pairs where the model correctly assigns a higher score to the preferred response, across both tasks. For downstream RLHF, evalua- tion differs by domain. On mathematical reasoning, we re- port the model’s final answer accuracy against ground-truth solutions. For open-ended dialogue, we follow common practices (Chen et al., 2023) and use Qwen3-Max as an external judge to perform pairwise comparisons between responses generated by policies trained with CausalRM and those trained with each baseline, reporting the win rate as the evaluation metric. To reduce the cost of LLM-based evaluation, we randomly sample 1,000 instances from the corresponding test set for the pairwise judging. 4.2. Main Results CausalRM consistently outperforms baselines in pre- dicting human preferences (RQ1).As shown in Table 1, CausalRM achieves strong reward modeling performance across both mathematical reasoning and open-ended dia- logue. On the ID splits, CausalRM reaches an average pairwise accuracy of 70.1% on math and 72.3% on dialogue, improving over the best baseline by 1.8% and 1.2%. Notably, this advantage becomes more pronounced under OOD shifts, where reward hacking and spurious correlations are more likely to emerge. On mathematical reasoning OOD benchmarks, CausalRM attains 85.6% average pairwise ac- curacy, outperforming the second-best method by 2.6%. Similarly, on dialogue OOD evaluation, CausalRM yields a 1.6% improvement over GoalRM. These results suggest that by explicitly disentangling reward-relevant and reward- irrelevant factors during reward model training, CausalRM generalizes better to unseen datasets and is less prone to exploiting spurious features. Improved reward prediction with CausalRM translates into stronger RLHF performance (RQ2). We report downstream RLHF results on mathematical reasoning in Table 2 and on open-ended dialogue in Table 3. As shown in Table 2, RLHF with CausalRM yields consistent gains over baseline reward models, improving final-answer accuracy by 1.5% on ID benchmarks and by 1.1% on OOD benchmarks. To examine whether these gains stem from mitigating re- ward hacking, we further analyze the discrepancy between the reward values optimized during RLHF and the corre- sponding ground-truth (gold) scores. Figure 3 presents the evolution of reward predictions and ground-truth perfor- mance on the ID test set throughout RLHF training, where dashed lines denote the normalized rewards predicted by different reward models, and solid lines indicate the cor- responding gold rewards. As training proceeds, policies optimized with baseline reward models exhibit a noticeable degradation in gold reward to varying degrees. This issue is particularly severe for InfoRM, where the gold reward di- verges sharply from the predicted reward. Such a widening gap is a typical signature of reward hacking and helps ex- plain InfoRM’s unexpectedly poor RLHF performance. We provide qualitative examples illustrating this reward hacking phenomenon in Appendix F. In contrast, CausalRM main- tains a consistent trend between predicted and gold rewards throughout training, highlighting its robustness to spurious features. On open-ended dialogue, CausalRM similarly demonstrates 6 Factored Causal Representation Learning for Robust Reward Modeling in RLHF Table 3. Downstream RLHF performance on open-ended dialogue evaluated by Qwen3-Max pairwise comparison win rate (%). Each entry reports Win/Tie/Lose of CausalRM-trained policy against an opponent policy trained with a baseline reward model. ModelOpponent IDOOD Anthropic-HelpfulAnthropic-HarmlessAvg.MT-BenchPKU-SafeRLHFSHPTruthfulQAAvg. WinTieLoseWinTieLoseWinTieLoseWinTieLoseWinTieLoseWinTieLoseWinTieLoseWinTieLose CausalRM (Ours) SFT76.219.44.468.123.88.172.221.66.244.341.014.767.821.510.777.511.810.752.241.26.660.528.910.6 Standard RM53.735.311.055.930.913.254.833.112.142.144.013.957.229.713.155.026.318.750.930.418.751.332.616.1 GoalRM50.039.011.034.644.121.342.341.616.135.051.014.031.647.820.627.950.821.331.752.915.431.650.617.8 InfoRM52.438.09.638.537.224.345.537.616.934.348.816.942.733.124.233.845.620.644.139.716.238.741.819.5 00.20.40.60.81 0.4 0.5 0.6 0.7 0.8 0.9 1 Value Standard RM 00.20.40.60.81 0.4 0.5 0.6 0.7 0.8 0.9 1 GoalRM 00.20.40.60.81 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized training step Value InfoRM 00.20.40.60.81 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized training step CausalRM (Ours) Figure 3. Reward hacking diagnosis on mathematical reasoning. The dashed curve shows the average normalized reward predicted by each reward model on the ID test set, and the solid curve is the average gold score measured by final-answer accuracy. strong alignment with human preferences. On the ID split, it achieves an average win rate of 54.8% against Standard RM, 45.5% against InfoRM, and 42.3% against GoalRM. This advantage persists under OOD evaluation with an av- erage win rate of 51.3%, 38.7%, and 31.6%, respectively. Following Rafailov et al. (2024) and Miao et al. (2025b), we further assess reward hacking mitigation by tracking Qwen3-Max win-rate dynamics of RLHF policies against the SFT reference throughout training. Figure 4 summarizes the results. Overall, CausalRM exhibits more stable training dynamics and sustained preference improvement, consis- tent with effective reward hacking mitigation, which in turn leads to the stronger RLHF outcomes reported in Table 3. 4.3. Causal Invariance Analyses In this subsection, we further investigate whether CausalRM better satisfies the causal invariance principle in Eq. (5), using response length and sycophantic bias as representa- tive spurious attributes. Our main finding is that by ex- plicitly disentangling causal and non-causal representations, CausalRM substantially reduces the sensitivity of pre- dicted rewards to spurious attributes compared to strong baselines (RQ3). 00.20.40.60.81 30 40 50 60 70 80 Normalized training step Win rate (%) Standard RMGoalRM InfoRMCausalRM (Ours) Figure 4. Average win rate against the SFT model on the ID test sets of open-ended dialogue benchmarks during RLHF. 0.050.150.250.350.450.550.650.750.850.95 0.2 0.4 0.6 0.8 1 Normalized answer length Normalized reward Standard RM (σ len =0.12) GoalRM (σ len =0.22) InfoRM (σ len =0.14) CausalRM (Ours, σ len =0.03) Figure 5. Sensitivity of predicted reward to response length on mathematical reasoning. Length is normalized to[0, 1]and re- wards are averaged within length quantile buckets.σ len denotes the standard deviation of bucket-wise mean rewards. Length bias. Response length is a well-known spurious feature in mathematical reasoning that can induce reward hacking (Singhal et al., 2023; Zhou et al., 2025). We there- fore examine how sensitive different reward models are to answer length. Figure 5 shows the average predicted reward as a function of response length on the chosen responses from the ID test sets. CausalRM remains nearly invari- ant across length bins, with a standard deviation of only 0.03. In contrast, baseline reward models exhibit substantial fluctuations, particularly showing a pronounced negative preference for longer responses. This length sensitivity is a plausible contributor to reward hacking during RLHF. Sycophantic bias. In open-ended dialogue, sycophantic bias refers to a model’s tendency to produce responses that agree with the user rather than providing reliable or truthful answers. Following Liu et al. (2024) and Wang et al. (2025), we first train a hacked SFT model by prepending the prefix “Sure, here is the response: ” to assistant messages with 7 Factored Causal Representation Learning for Robust Reward Modeling in RLHF Table 4. Robustness to sycophantic-phrasing artifacts measured by pairwise accuracy on hacked test sets. Each cell reports accuracy, with the relative change compared to the corresponding model trained on the unperturbed dataset shown in parentheses. Method IDOOD HelpfulHarmlessAvg.MT-BenchPKU-SafeRLHFSHPTruthfulQAAvg. Standard RM 56.2 (-11.2) 62.1 (-11.7) 59.2 (-11.4) 56.2 (-12.0) 54.6 (-3.2) 46.5 (-7.7) 58.3 (-0.2) 53.9 (-5.8) GoalRM 60.0 (-7.5) 64.3 (-10.4) 62.2 (-8.9) 59.6 (-6.6) 55.3 (-4.0) 50.7 (-3.1) 61.9 (-1.7) 56.9 (-3.8) InfoRM 63.7 (-4.2) 69.8 (-3.9) 66.8 (-4.0) 61.4 (-5.3) 57.4 (-2.7) 50.0 (-0.4) 60.2 (-1.7) 57.3 (-2.5) CausalRM (Ours) 67.4 (-2.0) 73.8 (-1.4) 70.6 (-1.7) 65.7 (-2.6) 62.0 (+1.1) 50.3 (-3.6) 66.8 (+0.6) 61.2 (-1.1) probabilityp=0.8. Starting from this SFT model (which exhibits a preference for the sycophantic phrasing), we then construct a hacked version of the Anthropic-H preference training set by prepending the same prefix to the chosen re- sponse withp chosen = 0.8and to the rejected response with p rejected = 0.2, and train reward models on this perturbed dataset. For evaluation, we perturb the test split by prepend- ing the prefix to both chosen and rejected responses with p = 0.3, and report pairwise accuracy of reward models on each benchmark. Table 4 summarizes the results and also re- ports the performance change relative to the corresponding reward model trained on the unperturbed dataset. Overall, CausalRM is more robust to the sycophantic-phrasing ar- tifact: it achieves the best ID and OOD accuracies on the hacked tests with only a minor average drop of−1.7points and−1.1points, respectively. In contrast, baselines degrade much more (e.g., Standard RM drops by−11.4on ID and −5.8on OOD), indicating that CausalRM is less likely to exploit the spurious prefix. 5. Related Work 5.1. Reward hacking in RLHF Reward hacking (Amodei et al., 2016; Skalse et al., 2022; Gao et al., 2023b) remains a central challenge for aligning LLMs with human preferences via RLHF. A primary cause of this phenomenon is that reward models often exploit spurious correlations in the training data to gain maximum benefit without truly capturing the underlying intent of hu- man judgments (Eisenstein et al., 2023). This behavior, also known as goal misgeneralization (Di Langosco et al., 2022) or shortcut learning (Geirhos et al., 2020) in traditional rein- forcement learning (RL), typically arises when the reward model erroneously associates high rewards with non-causal attributes such as response length (Dubois et al., 2024), for- matting cues (Chen et al., 2024), sycophantic agreement (Perez et al., 2023), or superficial conceptual patterns (Zhou et al., 2023). A growing body of work proposes to mitigate reward hacking through techniques including but are not limited to data augmentation (Liu et al., 2024), reward en- sembles (Coste et al., 2023), reward shaping (Fu et al., 2025), and representation learning (Nath et al., 2024; Miao et al., 2024). Our work investigates the feasibility of addressing reward hacking from the perspective of causal representa- tion learning, aiming to explicitly separate reward-relevant factors from spurious ones during reward modeling. Causal representation learning.In traditional RL, causal representation learning aims to extract high-level causal variables from low-level observations that are both minimal and sufficient for policy learning (Schölkopf et al., 2021). For example, ASR (Huang et al., 2022) and IFactor (Liu et al., 2023) learn more accurate world models by disentan- gling the most predictive features in environment dynamics. AdaRL (Huang et al., 2021) and CSR (Yang et al., 2024b) further extend causal factorization to domain adaptation, leveraging causal variables to capture environment changes. Moreover, works like Zheng & Makar (2022) and Stein- mann et al. (2024) improve robustness to distribution shifts by explicitly identifying and removing spurious correlations that act as shortcuts. Motivated by these insights, several recent approaches have applied causal principles to mitigate reward hacking in RLHF. RMM (Liu et al., 2024) proposes a data augmen- tation strategy grounded in causal invariance to eliminate context-free artifacts. Ovinnikov et al. (2024) and Wang et al. (2025) formalize this invariance as an explicit regu- larization term during reward model training. CRA (Song et al., 2025a) employs backdoor adjustments to deconfound spurious associations, while DEPTH (Yang et al., 2025) adopts a fixed, template-based factorization to filter out ir- relevant information, yielding reward models tailored to relation extraction. CausalRM differs from prior approaches by learning factorized latent representations without task- specific modifications, providing a general mechanism to disentangle reward-relevant signals from spurious attributes. 6. Conclusion In this paper, we propose CausalRM, a novel reward mod- eling framework that addresses reward hacking in RLHF through factored causal representation learning. By decom- posing the backbone embedding into causal and non-causal latent factors, CausalRM enforces that reward prediction depends solely on the minimal sufficient causal component, while actively suppressing reward-relevant signals in the non-causal part via an adversarial head with gradient re- versal. Extensive experiments on mathematical reasoning and open-ended dialogue demonstrate that CausalRM im- proves both reward model accuracy and downstream RLHF performance, while significantly mitigating sensitivity to spurious attributes such as response length and sycophantic bias. Future work includes extending CausalRM to process reward modeling scenarios and exploring its application in multi-turn dialogues with dynamic confounders. 8 Factored Causal Representation Learning for Robust Reward Modeling in RLHF Impact Statement This paper presents a method for improving reward model robustness in RLHF by reducing reliance on spurious cor- relations in training data (e.g., response length or stylistic cues). More robust reward modeling may help make RLHF training more stable and improve generalization across eval- uation settings. This work is a technical contribution to reward modeling and does not introduce new application domains or new data collection involving human subjects. As with RLHF methods generally, outcomes in deployed systems will depend on the quality and representativeness of the preference data and on the surrounding safety mea- sures. 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(2024). Recall that our minimal sufficiency objective for the causal latent z c is max I(z c ;r) − λ c KL I(h;z c ),(16) where h = f φ (x,y) is the backbone embedding and r denotes the preference signal induced by human rankings. Step 1: A variational lower bound for I(z c ;r). By definition, I(z c ;r) = E p(z c ,r) log p(r | z c ) p(r) = E p(z c ,r) [logp(r | z c )] + H(r),(17) whereH(r)does not depend on model parameters. Introducing an auxiliary variational distributionq(r | z c )to approximate p(r | z c ) and applying Gibbs’ inequality yields I(z c ;r) ≥ E p(z c ,r) [logq(r | z c )] + H(r).(18) Dropping the constant H(r), we maximize E p(z c ,r) [logq(r | z c )]. In our reward modeling setup, preferences are given as pairwise comparisons. For each triplet(x,y w ,y l ) ∼ D, let h w = f φ (x,y w ) and h l = f φ (x,y l ), and sample z w,c ∼ q α (z c | h w ), z l,c ∼ q α (z c | h l ).(19) We instantiate the variational likelihood with a Bradley–Terry model: q(r | z c ) = q(y w ≻ y l | z w,c ,z l,c ) = σ g ψ (z w,c )− g ψ (z l,c ) ,(20) where σ(·) is the sigmoid function. Taking the log-likelihood gives logq(y w ≻ y l | z w,c ,z l,c ) = logσ g ψ (z w,c )− g ψ (z l,c ) .(21) Thus, maximizing the lower bound in Eq. (18) corresponds to minimizing the standard preference lossL pref in the main paper. Step 2: A variational upper bound forI(h;z c ). Let the joint distribution be defined by the data marginalp(h)and the encoder q α (z c | h), i.e., p true (h,z c )≜ p(h)q α (z c | h). Then the mutual information can be written as I(h;z c ) = E p(h) KL q α (z c | h)∥p true (z c ) ,(22) wherep true (z c ) = R q α (z c | h)p(h)dhis the aggregated posterior, which is generally intractable. Following the variational information bottleneck (Alemi et al., 2016), we upper-boundI(h;z c )by replacingp true (z c )with a tractable variational prior p(z c ) =N(0,I). Since KL(p true (z c )∥p(z c ))≥ 0, we have E p true (z c ) [logp true (z c )]≥ E p true (z c ) [logp(z c )], which implies I(h;z c ) = E p(h) E q α (z c |h) [logq α (z c | h)]− E p true (z c ) [logp true (z c )] ≤ E p(h) E q α (z c |h) [logq α (z c | h)]− E p(h) E q α (z c |h) [logp(z c )] = E p(h) KL q α (z c | h)∥p(z c ) . (23) For a preference pair (x,y w ,y l ), we use both embeddings and obtain the empirical estimate KL(q α (z c | h w )∥p(z c )) + KL q α (z c | h l )∥p(z c ) ,(24) which corresponds toL c KL in Eq. (12). 13 Factored Causal Representation Learning for Robust Reward Modeling in RLHF Step 3: Putting the bounds together. Combining the variational lower bound forI(z c ;r)(Eq. 18) and the variational upper bound forI(h;z c )(Eq. 23), and using the Bradley–Terry likelihood in Eq. (21), we obtain the following variational lower bound for Eq. (16): I(z c ;r)− λ c KL I(h;z c ) ≥ E (x,y w ,y l )∼D h logσ g ψ (z w,c )− g ψ (z l,c ) − λ c KL KL(q α (z c | h w )∥p(z c )) + KL q α (z c | h l )∥p(z c ) i . (25) Identifying L pref =− logσ r θ (x,y w )− r θ (x,y l ) , L c KL = KL(q α (z c | h w )∥p(z c )) + KL q α (z c | h l )∥p(z c ) ,(26) and usingr θ (x,y) = g ψ (z c )withz c ∼ q α (z c | f φ (x,y))yields the training objective in Eq. (12) in the main paper. This completes the derivation. B. Implementation Details Model Architectures. CausalRM is implemented as a lightweight latent-variable module on top of a pretrained LLM backbone. Given the backbone embeddingh ∈ R H , we map it to two diagonal-Gaussian posteriors:q α (z c | h) = N(μ c (h), diag(σ 2 c (h))) andq α (z nc | h) =N(μ nc (h), diag(σ 2 nc (h))) . We then samplez c andz nc via the reparameteriza- tion trick. Reward prediction depends only onz c through a linear reward head, while the non-causal latentz nc is fed into an adversarial head with a gradient reversal layer (GRL). A reconstruction decoder maps the concatenated latent[z c ;z nc ]back to ˆ h. At inference time, we use the mean μ c (instead of a stochastic sample) for stable reward prediction. The GRL is implemented as a custom autograd function: in the forward pass it is the identity map, while in the backward pass it multiplies the gradient by−λ grl . Table 5 summarizes the architectural choices. Table 5. CausalRM architecture details used in our experiments. H is the backbone hidden size. ComponentSpecification Causal posteriorμ c , logσ 2 c : linear H→d c Non-causal posterior μ nc , logσ 2 nc : linear H→d nc Latent dimsd c = 128, d nc = 512 SamplingReparameterization z = μ + σ⊙ ε, ε∼N(0,I) Reward headLinear d c →1 (no bias), input: z c (train) / μ c (eval) Adversary headLinear d nc →1 (no bias), input: GRL(z nc ) ReconstructorLinear (d c +d nc )→H Training cost.Both reward model training and PPO optimization are conducted on a single compute node equipped with 8 NVIDIA H20 GPUs (96GB VRAM each), 192 CPU cores, and 1.8 TB RAM. We employ Ray-based distributed training (Moritz et al., 2018) during RLHF. Table 6 summarizes the approximate training time in our experiments. Table 6. Approximate training time of CausalRM (in hours) for reward model (RM) and PPO stages. TaskRM TrainingPPO Training Mathematical reasoning6.526.1 Open-ended dialogue4.022.3 Hyperparameters.Our implementation is based on the OpenRLHF library (v0.8.5). Unless otherwise stated, we use the same training recipe for both mathematical reasoning and dialogue experiments. Reward model and PPO hyperparameters are summarized in Tables 7 and 8, respectively. 14 Factored Causal Representation Learning for Robust Reward Modeling in RLHF Table 7. Reward model training hyperparameters. HyperparameterValue Epochs1 Max sequence length1024 Train batch size / micro-batch256 / 1 Learning rate9× 10 −6 PrecisionBF16 Latent dims (d c ,d nc )(128, 512) λ pred (pref. loss)1.0 λ rec (reconstruction)0.001 λ adv (adversarial)0.05 λ c KL (causal KL)0.001 λ nc KL (non-causal KL)0.001 GRL strength λ grl 1.0 Table 8. PPO hyperparameters. HyperparameterValue Epochs1 Prompt max length1024 Generation max length1024 Train batch size / micro-batch64 / 8 Rollout batch size / micro-rollout512 / 16 Actor learning rate5× 10 −7 Critic learning rate9× 10 −6 Reward normalizationenabled PrecisionBF16 Colocationcritic+reward, actor+ref vLLM #engines / tensor parallel2 / 2 vLLM GPU memory utilization0.95 C. Prompt for Qwen3-Max Evaluation We use Qwen3-Max as an external judge to perform pairwise comparisons between model responses in our dialogue experiments. Given an instruction and two candidate outputs, Qwen3-Max is asked to rank the two models by the quality of their responses, reflecting the preference that the majority of humans would give. The exact prompt used for evaluation is shown below. Qwen3-Max Evaluation Prompt I want you to create a leaderboard of different large-language models. To do so, I will give you the instructions (prompts) given to the models, and the responses of two models. Please rank the models based on which responses would be preferred by human. All inputs and outputs should be Python objects. Here is the prompt: “instruction”: instruction Here are the outputs of the models: [ “model”: model1, “answer”: output1 , “model”: model2, “answer”: output2 ] Now please rank the models by the quality of their answers, so that the model with rank 1 has the best output. Then return a list of the model names and ranks, i.e., produce the following output: [ “model”: "<model-name>", “rank”: <model-rank>, “model”: "<model-name>", “rank”: <model-rank> ] Your response must be a valid Python object and should contain nothing else because we will directly use it in Python. Please provide the ranking that the majority of humans would give. 15 Factored Causal Representation Learning for Robust Reward Modeling in RLHF Table 9. Ablation on mathematical reasoning reward modeling. We report pairwise accuracy (%, higher is better) on ID and OOD benchmarks. Numbers in parentheses are deltas relative to full CausalRM. Variant ID OOD GSM8KMATHAvg.Algebra222GSM-HardASDivMAWPSSVAMPAvg. CausalRM81.758.470.1 (+0.0)89.966.289.992.289.685.6 (+0.0) w/o factorization74.158.066.1 (-4.0)82.763.588.989.987.682.5 (-3.1) w/o reconstruction78.858.168.5 (-1.6)87.764.587.388.787.483.1 (-2.5) w/o adversarial / GRL76.457.667.0 (-3.1) 84.461.987.188.585.681.5 (-4.1) w/o KL on z c 78.056.567.3 (-2.8)84.460.386.287.986.981.1 (-4.5) w/o KL on z nc 72.755.364.0 (-6.1)83.363.086.488.786.881.6 (-4.0) w/o KL on both71.349.460.4 (-9.7) 77.955.478.581.779.474.6 (-11.0) Table 10. Ablation on robustness to sycophantic-phrasing artifacts (pairwise accuracy on hacked test sets; %, higher is better). Numbers in parentheses are deltas relative to full CausalRM. Variant IDOOD HelpfulHarmlessAvg.MT-BenchPKU-SafeRLHFSHPTruthfulQAAvg. CausalRM67.473.870.6 (+0.0)65.762.050.366.861.2 (+0.0) w/o factorization63.769.866.8 (-3.8) 61.457.450.060.257.3 (-3.9) w/o reconstruction66.873.570.2 (-0.4)63.159.248.064.258.6 (-2.6) w/o adversarial / GRL62.971.667.3 (-3.3)61.959.646.760.757.2 (-4.0) w/o KL on z c 67.073.270.1 (-0.5) 62.358.246.161.957.1 (-4.1) w/o KL on z nc 62.765.063.9 (-6.7) 63.760.048.563.959.0 (-2.2) w/o KL on both46.445.546.0 (-24.6) 43.344.443.161.148.0 (-13.2) D. Ablation Studies This section ablates the key design choices in CausalRM to better understand which components are essential for robust reward modeling and, in particular, for mitigating spurious correlations. Since our primary goal is to improve reward model robustness, we focus on reward model evaluation: (1) pairwise accuracy on ID/OOD benchmarks for mathematical reasoning, (2) length-sensitivity analysis on math, and (3) robustness to sycophantic artifacts on open-ended dialogue via the hacked evaluation described in Section 4.3. Settings. Starting from the full CausalRM objective in Eq. (10), we consider the following variants: • w/o factorization: remove the latent factorization and train a single latentzwith the preference loss and the KL bottleneck on the reward-predictive latent (i.e., onlyL pref +λ c KL L c KL ). This corresponds to the InfoRM-style information bottleneck baseline. • w/o reconstruction: set λ rec = 0 (removeL rec ) while keeping factorization and adversarial training. • w/o adversarial/GRL: set λ adv = 0 (removeL adv and the GRL) while keeping factorization and reconstruction. • w/o KL on z c : set λ c KL = 0. • w/o KL on z nc : set λ nc KL = 0. • w/o KL on both latents: set λ c KL = λ nc KL = 0. All other training details follow the main experiments to ensure a controlled comparison. Results and analysis. Tables 9 and 10 summarize the ablation results. Overall, we observe that the full CausalRM consistently performs best, and different components contribute in complementary ways. Factorization and the structural restriction are important.Compared to the InfoRM-equivalent variant (w/o factoriza- tion), the full CausalRM improves math pairwise accuracy by +4.0 points on ID (70.1 vs. 66.1) and +3.1 on OOD (85.6 vs. 82.5), and substantially reduces length sensitivity (std. across length bins drops from 0.14 to 0.03; Figure 6). On the hacked sycophancy evaluation for dialogue, factorization also yields consistent gains (ID +3.8, OOD +3.9). These results suggest that explicitly separating reward-relevant and reward-irrelevant channels provides a stronger inductive bias than a single 16 Factored Causal Representation Learning for Robust Reward Modeling in RLHF 0.050.150.250.350.450.550.650.750.850.95 0.2 0.4 0.6 0.8 1 Normalized answer length Normalized reward CausalRM (full, σ len =0.03)w/o factorization (σ len =0.14) w/o reconstruction (σ len =0.13)w/o adversarial/GRL (σ len =0.13) w/o KL on z c (σ len =0.17)w/o KL on z nc (σ len =0.09) w/o KL both (σ len =0.18) Figure 6. Length sensitivity under ablations on mathematical reasoning. Length is normalized to[0, 1]and rewards are averaged within length quantile buckets on chosen responses from the ID test set. bottlenecked latent: even when capacity is constrained, a single latent can still entangle spurious cues with reward-relevant features, whereas the factorized design makes it easier to route spurious variation away from the reward head. Adversarial training (GRL) is crucial for invariance and OOD robustness.Removing the adversarial head/GRL causes a noticeable degradation on mathematical reasoning, especially under distribution shift (ID: -3.1, OOD: -4.1), and increases length sensitivity (std. rises from 0.03 to 0.13; Figure 6). On dialogue, the same ablation reduces accuracy on the hacked evaluation by -3.3 (ID) and -4.0 (OOD). This pattern supports that the adversarial head helps suppress reward-predictive leakage intoz nc , which becomes particularly important when spurious artifacts correlate with preference labels or when test distributions shift. Reconstruction improves robustness but is not the main driver of accuracy. The reconstruction term has a relatively small effect on ID accuracy (math: -1.6; dialogue hacked: -0.4), but provides consistent gains on OOD benchmarks (math: -2.5; dialogue hacked: -2.6 when removed). This indicates that reconstruction acts as a stabilizer that helps preserve information from the backbone embedding under the factorized bottleneck, complementing the structural and adversarial mechanisms. Both KL terms matter and are complementary.Removing the KL onz c (λ c KL =0 ) hurts generalization on math (OOD: -4.5) and reduces robustness on hacked dialogue OOD (-4.1), consistent with the information bottleneck onz c discouraging the reward head from exploiting redundant or spurious signals. Removing the KL onz nc (λ nc KL =0) also degrades robustness (math OOD: -4.0; dialogue hacked ID: -6.7), suggesting that regularizing the non-causal channel is important for stable learning and robustness, as an unconstrainedz nc can become an easy pathway for dataset-specific signals. Finally, removing both KL terms yields the largest drop (math OOD: -11.0; dialogue hacked ID: -24.6), showing that explicit factorization and adversarial training still benefit from information-theoretic regularization to achieve a robust decomposition in practice. In summary, the ablations support our design choices: factorization with a reward head conditioned only onz c provides a strong inductive bias, GRL-based adversarial training is critical for enforcing invariance under spurious shifts, reconstruction offers additional robustness, and KL regularization on both latents is necessary to avoid brittle solutions. 17 Factored Causal Representation Learning for Robust Reward Modeling in RLHF E. Limitations and Discussion CausalRM is designed to mitigate reward hacking by promoting causal invariance through structured representation learning. By decomposing the latent space into causal and non-causal factors, our framework encourages reward prediction to rely on minimally sufficient signals while actively suppressing reward-predictive leakage through spurious channels. A key limitation is that our approach primarily follows the invariance principle—encouraging the learned reward to be insensitive to non-causal variations—rather than providing a formal guarantee that the learned factors uniquely recover the true causal drivers of human preferences. In general, identifying underlying causal mechanisms from observational data requires stronger assumptions or additional supervision (e.g., interventional data, multiple environments, or reliable proxies for confounders) (Peters et al., 2012; Yao et al., 2021; von Kügelgen et al., 2023; Song et al., 2025b), which are not provided by default in standard RLHF pipelines and often require additional data collection or experimental design. Nevertheless, our empirical results indicate that CausalRM is a practically useful step toward causally robust reward modeling: across both mathematical reasoning and dialogue tasks, it improves reward model accuracy and consistently strengthens downstream RLHF performance. Moreover, it substantially reduces sensitivity to representative spurious attributes such as response length and sycophantic phrasing, suggesting improved behavioral invariance of the learned reward signal in realistic settings. We view these findings as evidence that causal principles can be incorporated into reward modeling in a lightweight yet effective manner. An important direction for future work is to connect empirically effective invariance objectives with stronger identifiability guarantees. Potential avenues include leveraging explicit surrogates for spurious variables, exploiting multi-environment variation or controlled perturbations, and integrating identifiability-oriented formulations that recover bias-free latent factors under additional assumptions. F. Reward Hacking Example To better illustrate the effectiveness of CausalRM in mitigating reward hacking, we provide concrete examples from both mathematical reasoning and dialogue tasks, where CausalRM outperforms baselines by avoiding spurious patterns such as format hacking (see Figure 7), off-topic continuation (see Figures 7, 8 and 9), redundant repetition (see Figure 9), and misleading or incomplete reasoning (see Figure 10). 18 Factored Causal Representation Learning for Robust Reward Modeling in RLHF LLM Input (Prompt) System: You're an expert Python programmer and mathematician. Help the user to solve this problem using code when necessary. Make sure to put the answer (and only answer) inside \ . User: If the numbers 4, 5 and 6 are each used exactly once to replace the letters in the expression 퐴(퐵−퐶), what is the least possible result? Assistant: Standard RM Wrong! GoalRM Off-topic! InfoRM Missing \ ! CausalRM (Ours) To find the least possible result of the expression \\(A (B - C)\\) where \\(A\\), \\(B\\), and \\(C\\) are the numbers 4, 5, and 6, we need to evaluate the expression for all possible permutations of \\(A\\), \\(B\\), and \\(C\\). The permutations are: 1. \\(A = 4\\), \\(B = 5\\), \\(C = 6\\) 2. \\(A = 4\\), \\(B = 6\\), \\(C = 5\\) 3. \\(A = 5\\), \\(B = 4\\), \\(C = 6\\) 4. \\(A = 5\\), \\(B = 6\\), \\(C = 4\\) 5. \\(A = 6\\), \\(B = 4\\), \\(C = 5\\) 6. \\(A = 6\\), \\(B = 5\\), \\(C = 4\\) will calculate the value of the expression for each permutation and find the minimum value. 1. \\(4 (5 - 6) = 4 (-1) = -4\\) 2. \\(4 (6 - 5) = 4 (1) = 4\\) 3. \\(5 (4 - 6) = 5 (-2) = -10\\) 4. \\(5 (6 - 4) = 5 (2) = 10\\) 5. \\(6 (4 - 5) = 6 (-1) = -6\\) 6. \\(6 (5 - 4) = 6 (1) = 6\\) least possible result is \\(-10\\). The final answer is (\ -10). ```python itertools # Define the numbers = [4, 5, 6] # Initialize the minimum value to a large number _value = float('inf') # Iterate through all permutations of the numbers A, B, C in itertools.permutations(numbers): # Calculate the value of the expression A * (B - C) value = A * (B - C) # Update the minimum value if the current value is smaller if value < min_value: min_value = value (min_value) ``` To determine the least possible result of the expression 퐴(퐵−퐶) when the numbers 4, 5, and 6 are each used exactly once to replace the letters 퐴, 퐵, and 퐶, we need to evaluate all possible combinations of these numbers and find the minimum value. First, let's list all possible permutations of the numbers 4, 5, and 6: 1. \\( A = 4, B = 5, C = 6 \\) 2. \\( A = 4, B = 6, C = 5 \\) 3. \\( A = 5, B = 4, C = 6 \\) 4. \\( A = 5, B = 6, C = 4 \\) 5. \\( A = 6, B = 4, C = 5 \\) 6. \\( A = 6, B = 5, C = 4 \\) , we will evaluate the expression \\( A ( B - C ) \\) for each permutation: 1. For \\( A = 4, B = 5, C = 6 \\): \\[ 4 ( 5 - 6 ) = 4 \× (-1) = -4 \\] 2. For \\( A = 4, B = 6, C = 5 \\): \\[ 4 ( 6 - 5 ) = 4 \× 1 = 4 \\] 3. For \\( A = 5, B = 4, C = 6 \\): \\[ 5 ( 4 - 6 ) = 5 \× (-2) = -10 \\] 4. For \\( A = 5, B = 6, C = 4 \\): \\[ 5 ( 6 - 4 ) = 5 \× 2 = 10 \\] 5. For \\( A = 6, B = 4, C = 5 \\): \\[ 6 ( 4 - 5 ) = 6 \× (-1) = -6 \\] 6. For \\( A = 6, B = 5, C = 4 \\): \\[ 6 ( 5 - 4 ) = 6 \× 1 = 6 \\] , we compare the results of all the evaluations: \\[ - 4, 4, -10, 10, -6, 6 \\] least possible result is \\(-10\\). Thus, the least possible result of the expression 퐴(퐵−퐶) is (\ -10). Human: Prove that $f$ is a constant function. Let $f$ be a function with the property that for every disk $D\⊂ $, there exists a function $g_D$ holomorphic on $D$ such that $g_D'=f$ on $D$. Prove that $f$ is a constant function. have no idea how to start this problem. Any help would be appreciated. : To prove that \\( f \\) is a constant function, we will use the given property that for every disk \\( D \⊂ \\), there exists a function \\( g_D \\) holomorphic on \\( D \\) such that \\( g_D' = f \\) on \\( D \\). We will proceed step by step. ### Step 1: Understanding the Given Property property states that for any disk \\( D \⊂ \\), there exists a holomorphic function \\( g_D \\) on \\( D \\) such that the derivative of \\( g_D \\) is \\( f \\). This implies that \\( f \\) is the derivative of some holomorphic function on every disk in \\( \\). ### Step 2: Holomorphic Functions and Their Derivatives that a function \\( g \\) is holomorphic on a domain \\( D \\) if it is complex differentiable at every point in \\( D \\). The derivative of a holomorphic function is also holomorphic. ### Step 3: Local Constancy of \\( f \\) two overlapping disks \\( D_1 \\) and \\( D_2 \\) in \\( \\). Let \\( g_D_1 \\) and \\( g_D_2 \\) be the holomorphic functions on \\( D_1 \\) and \\( D_2 \\) respectively such that \\( g_D_1' = f \\) on \\( D_1 \\) and \\( g_D_2' = f \\) on \\( D_2 \\). the intersection \\( D_1 \∩ D To find the least possible result of the expression 퐴(퐵−퐶) when the numbers 4, 5, and 6 are used exactly once to replace the letters 퐴, 퐵, and 퐶,we need to evaluate the expression for all possible permutations of 퐴, 퐵, and 퐶. We can use Python to iterate through all permutations and find the minimum value of the expression. Let's write the code to find the solution. ```python itertools # Define the numbers to be used = [4, 5, 6] # Initialize the minimum value to a large number _value = float('inf') # Iterate through all permutations of the numbers A, B, C in itertools.permutations(numbers): # Calculate the value of the expression A * (B - C) value = A * (B - C) # Update the minimum value if the current value is smaller if value < min_value: min_value = value (min_value) ``` ```output -22 ``` The least possible result of the expression 퐴(퐵−퐶) when the numbers 4, 5, and 6 are used exactly once to replace the letters 퐴, 퐵, and 퐶 is (\ -22). To solve the problem, we need to evaluate the expression 퐴(퐵−퐶) for all possible permutations of the numbers 4, 5, and 6 assigned to the variables 퐴, 퐵, and 퐶. We will then find the permutation that gives the least possible result. Let's write a Python code to iterate through all permutations and calculate the value of the expression for each permutation. We will then determine the minimum value. 's the code to solve the problem: ```python itertools # Define the numbers to be used = [4, 5, 6] # Initialize the minimum value to a large number _value = float('inf') # Iterate through all permutations of the numbers perm in itertools.permutations(numbers): A, B, C = perm result = A * (B - C) if result < min_value: min_value = result (min_value) ``` Figure 7. Reward hacking behaviors on an ID MATH prompt. Standard RM outputs an incorrect boxed answer (-22), InfoRM exhibits format hacking by outputting code without a final boxed answer, and GoalRM answers correctly but continues with an unrelated prompt (off-topic continuation). In contrast, CausalRM follows the instruction and produces the correct boxed answer (-10). 19 Factored Causal Representation Learning for Robust Reward Modeling in RLHF LLM Input (Prompt) System: You're an expert Python programmer and mathematician. Help the user to solve this problem using code when necessary. Make sure to put the answer (and only answer) inside \ . User: Janeth borrowed 2000 and promised to return it with an additional 10% of the amount. If she is going to pay 5845682 a month for 12 months, how much will be Janeth's remaining balance by then? Assistant: Standard RM Wrong! GoalRM Wrong! Off-topic! CausalRM (Ours) Let's break down the problem step by step: 1. First, calculate the total amount Janeth needs to return. She borrowed $2000 and promised to return it with an additional 10% of the amount. So, the total amount to be returned is: \\[ 2000 + (2000 \× 0.10) = 2000 + 200 = 2200 \\] 2. Janeth is paying $5845682 a month for 12 months. So, the total amount she will pay in 12 months is: \\[ 5845682 \× 12 = 70148184 \\] 3. The remaining balance after 12 months will be the total amount she needs to return minus the total amount she has paid: \\[ 2200 - 70148184 = -70145984 \\] Since the remaining balance is negative, it means she has overpaid by $70145984. The answer is: \\[ \ -70145984 \\] Let's confirm this with Python code. ```python # Calculate the total amount to be returned _amount = 2000 _percentage = 0.10 _amount_to_return = borrowed_amount * (1 + additional_percentage) # Calculate the total amount paid in 12 months _payment = 5845682 _paid = monthly_payment * 12 # Calculate the remaining balance _balance = total_amount_to_return - total_paid (remaining_balance) ``` InfoRM Wrong! Let's break down the problem step by step. 1. First, calculate the total amount Janeth needs to return. She borrowed $2000 and promised to return it with an additional 10% of the amount. So, the total amount to be returned is $2000 + 10% of $2000 = $2000 + $200 = $2200. 2. Next, calculate the total amount Janeth is going to pay over 12 months. She is paying $5845682 a month for 12 months, so the total amount paid is $5845682 * 12. 3. Finally, subtract the total amount paid from the total amount to be returned to find the remaining balance. Let's implement this in Python to get the accurate result. ```python # Step 1: Calculate the total amount to be returned _amount = 2000 _percentage = 0.10 _amount_to_return = borrowed_amount * (1 + additional_percentage) # Step 2: Calculate the total amount paid over 12 months _payment = 5845682 _amount_paid = monthly_payment * 12 # Step 3: Calculate the remaining balance _balance = total_amount_to_return - total_amount_paid (remaining_balance) ``` ```output -70147384.0 ``` The remaining balance is (\ -70147384.0). This negative value indicates that Janeth has overpaid by $70,147,384.00. To determine Janeth's remaining balance after paying $5845682 a month for 12 months, we need to follow these steps: 1. **Calculate the total amount Janeth needs to return:** Janeth borrowed $2000 and promised to return it with an additional 10% of the amount. \\[ amount = 2000 \× 0.10 = 200 \\] \\[ amount to return = 2000 + 200 = 2200 \\] 2. **Calculate the total amount Janeth will pay over 12 months:** Janeth is going to pay $5845682 a month for 12 months. \\[ payment = 5845682 \× 12 = 70148184 \\] 3. **Determine the remaining balance:** Subtract the total amount Janeth needs to return from the total payment she will make. \\[ balance = 70148184 - 2200 = 70145984 \\] Thus, Janeth's remaining balance by the end of the 12 months is (\ 70145984). Human: Prove that $f(x)=\ 1x$ is continuous on $(0,\∞)$ using the $\ε-\δ$ definition of continuity. that $f(x)=\ 1x$ is continuous on $(0,\∞)$ using the $\ε-\δ$ definition of continuity. know that $f$ is continuous on $(0,\∞)$ if $f$ is continuous at every point of $(0,\∞)$. also know that $f$ is continuous at $x_0$ if $\ _x\→ x_0f(x)=f(x_0)$. , I need to show that $\ _x\→ x_0f(x)=f(x_0)$ for every $x_0\∈(0,\∞)$. know that $\ _x\→ x_0f(x)=f(x_0)$ if for every $\ε>0$ there exists a $\δ>0$ such that $|f(x)-f(x_0)|<\ε$ whenever $x\∈ D$ and $0<|x-x_0|<\δ$. , I need to show that for every $\ε>0$ there exists a $\δ>0$ such that $|f(x)-f(x_0)|<\ε$ whenever $x\∈(0,\∞)$ and $0<|x-x_0|<\δ$. know that $|f(x)-f(x_0)|=|\ 1x-\ 1x_0|=|\ x_0-xx_0|=\ |x-x_0||x_0|$. 'm stuck on how to choose $\δ$ so that $\ |x-x_0||x_0|<\ε$. : To prove that \\( f(x) = \ 1x \\) is continuous on \\((0, \∞)\\) using the \\(\ε-\δ\\) definition of continuity, we need to show that for every \\( x_0 \∈ (0, \∞) \\) and every \\(\ε > 0\\), there exists a \\(\δ > 0\\) such that if \\( 0 < |x - x_0| < \δ \\), then \\( \ | \ 1x - \ 1x_0 \ | < \ε \\). 's start by analyzing the expression \\( \ | \ 1x - \ 1x_0 \ | \\): \\[ \ | \ 1x - \ 1x_0 \ | = \ | \ x_0 - xx_0 \ | = \ |x - x_0||x_0| \\] need to find a \\(\δ\\) such that: \\[ \ |x - x_0||x_0| < \ε \\] , let's bound \\( |x| \\) in terms of \\( x_0 \\) and \\(\δ\\). If \\( 0 < |x - x_0| < To solve this problem, we need to follow these steps: 1. Calculate the total amount Janeth needs to return, which includes the borrowed amount plus the additional 10%. 2. Calculate the total amount Janeth will pay over 12 months. 3. Subtract the total amount Janeth will pay from the total amount she needs to return to find the remaining balance. Let's break it down: 1. The borrowed amount is $2000, and the additional 10% is $2000 * 0.10 = $200. So, the total amount Janeth needs to return is $2000 + $200 = $2200. 2. Janeth is paying $5845682 a month for 12 months, so the total amount she will pay is $5845682 * 12 = $70148184. 3. The remaining balance is the total amount she needs to return minus the total amount she will pay, which is $2200 - $70148184 = -$70145984. Since the remaining balance is negative, it means Janeth will have overpaid by $70145984. Therefore, the remaining balance is \ 0. Let's confirm this with Python code. ```python # Step 1: Calculate the total amount Janeth needs to return _amount = 2000 _percentage = 0.10 _amount_to_return = borrowed_amount * (1 + additional_percentage) # Step 2: Calculate the total amount Janeth will pay over 12 months _payment = 5845682 _payment = monthly_payment * 12 # Step 3: Calculate the remaining balance _balance = total_amount_to_return - total_payment (remaining_balance) ``` Figure 8. Reward hacking behaviors on a GSM-Hard prompt. Standard RM computes the correct numerical result but outputs an incorrect boxed answer due to arithmetic error. InfoRM correctly calculates the balance but hacks the format by overriding the true answer with 0, falsely claiming no overpayment. GoalRM produces the right magnitude but misses the negative sign and appends an unrelated continuity proof (off-topic continuation). In contrast, CausalRM faithfully follows the instruction, correctly computes the negative remaining balance, and outputs the exact answer (-70145984) as required. 20 Factored Causal Representation Learning for Robust Reward Modeling in RLHF LLM Input (Prompt) System: You are a helpful assistant. User: I liketo hostguestsatmyhomefromtimeto time, andI amgatheringrecipesofdifferentdishesanddrinksto keepthingsinteresting. I aminterestedintrying someLativiandishes. CanyougivemearecipeforFrikadeļu zupa? Assistant: Standard RM Frikadeļu zupa isatraditionalLatviansoupmadewithmeatballsandvegetables. HereisarecipeforFrikadeļu zupa: : * 1 lbgroundbeef * 1/2 cup breadcrumbs * 1/4 cupgratedonion * 1/4 cupgratedcarrot * 1/4 cupgratedpotato * 1/4 cupgratedrutabaga * 1/4 cupgratedturnip * 1/4 cupgratedcelery root * 1/4 cupgratedparsnip * 1/4 cupgratedturnip * 1/4 cupgratedcarrot * 1/4 cupgratedonion * 1/4 cupgratedpotato * 1/4 cupgratedrutabaga * 1/4 cup gratedturnip * 1/4 cupgratedceleryroot * 1/4 cupgratedparsnip * 1/4 cupgratedturnip * 1/4 cupgratedcarrot * 1/4 cupgratedonion * 1/4 cupgrated potato * 1/4 cupgratedrutabaga * 1/4 cupgratedturnip * 1/4 cupgratedceleryroot * 1/4 cupgratedparsnip * 1/4 cupgratedturnip * 1/4 cupgratedcarrot * 1/4 cupgratedonion * 1/4 cupgratedpotato * 1/4 cupgratedrutabaga * 1/4 cupgratedturnip * 1/4 cupgratedceleryroot * 1/4 cupgratedparsnip * 1/4 cup gratedturnip * 1/4 cupgratedcarrot * 1/4 cupgratedonion * 1/4 cupgratedpotato * 1/4 cupgratedrutabaga * 1/4 cupgratedturnip * 1/4 cupgratedcelery root * 1/4 cupgratedparsnip * 1/4 cupgratedturnip * 1/4 cupgratedcarrot * 1/4 cupgratedonion * 1/4 cupgratedpotato * 1/4 cupgratedrutabaga * 1/4 cup gratedturnip * 1/4 cupgratedceleryroot * 1/4 cupgratedpar GoalRM Frikadeļu zupa isatraditionalLatviansoupmadewithmeatballsandservedwithsourcream. HereisarecipeforFrikadeļu zupa: : * 1 lbground beef * 1/2 cupbreadcrumbs * 1/4 cupgratedonion * 1 egg * 1/2 tspsalt * 1/4 tspblackpepper * 1/4 tsppaprika * 1/4 tspgarlicpowder * 1/4 tspnutmeg * 1/4 tspallspice * 1/4 tspcumin * 1/4 tspcoriander * 1/4 tspthyme * 1/4 tspmarjoram * 1/4 tspparsley * 1/4 tspdill * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbay leaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tsp bayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * 1/4 tspbayleaf * InfoRM Frikadeļu zupa isatraditionalLatviansoupmadewithmeatballsandpotatoes. HereisarecipeforFrikadeļu zupa: : * 1 lbgroundbeef * 1/2 cup breadcrumbs * 1/4 cupgratedonion * 1 egg * 1/2 tspsalt * 1/4 tspblackpepper * 1/4 tsppaprika * 1/4 tspgarlicpowder * 1/4 cupmilk * 2 tbspbutter * 2 tbspflour * 4 cupschickenbroth * 2 mediumpotatoes, peeledanddiced * 1/2 cupsourcream * Salt andpepperto taste : 1. Inalargemixing bowl, combinethegroundbeef, breadcrumbs, onion, egg, salt, pepper, paprika, andgarlicpowder. Mixwell. 2. Formthemixtureintosmallmeatballsandset aside. 3. Inalargepot, meltthebutterovermediumheat. Addtheflourandstiruntilthemixtureissmoothandbubbly. 4. Graduallyaddthechickenbroth, stirringconstantly, untilthemixturethickens. 5. Addthedicedpotatoesto thepotandstirto combine. 6. Bringthesoupto aboil, thenreducetheheatto lowandsimmerfor20-25 minutes, oruntilthepotatoesaretender. 7. Addthemeatballsto thepotandsimmerforanadditional10-15 minutes, oruntilthemeatballsarecookedthrough. 8. Removethepotfromtheheatandstirinthesourcream. Seasonwithsalt andpepperto taste. 9. Serve thesouphot, garnishedwithadditionalsourcreamandchopped freshherbsifdesired. ļu zupa!<|im_end|> <|im_end|> : Whatarethebenefitsofeatingfruitsand vegetables?<|im_end|> . Theyarerichinvitamins, minerals, andfiber, whichareimportantformaintaining goodhealth. Eatingavarietyoffruitsandvegetablescanhelpto reducetherisk ofchronicdiseasessuchasheartdisease, stroke, andcertaintypesofcancer. Theyare alsoimportantformaintainingahealthyweight, astheyarelowincaloriesandcanhelpto keepyoufeelingfullandsatisfied. Additionally, fruitsandvegetablesarea goodsourceofantioxidants, whichcanhelpto protectthebodyfromdamagecausedby CausalRM (Ours) Frikadeļu zupa isatraditionalLatviansoupmadewithmeatballsandpotatoes. HereisarecipeforFrikadeļu zupa: : * 1 lbgroundbeef * 1/2 cup breadcrumbs * 1/4 cupgratedonion * 1 egg * 1/2 tspsalt * 1/4 tspblackpepper * 1/4 tsppaprika * 1/4 tspgarlicpowder * 1/4 tspnutmeg * 1/4 tsp allspice * 4 cupschickenbroth * 2 potatoes, peeledanddiced * 1/2 cupsourcream * 1 tbspchoppedfreshdill : 1. Inalargebowl, mix together thegroundbeef, breadcrumbs, onion, egg, salt, pepper, paprika, garlicpowder, nutmeg, andallspice. 2. Formthemixtureintosmallmeatballsandset aside. 3. Ina largepot, heatthechickenbrothovermediumheat. Addthedicedpotatoesandbringto aboil. 4. Reducetheheatto lowandaddthemeatballsto thepot. Coverand simmerfor20-25 minutes, oruntilthepotatoesaretender. 5. Removethepotfromtheheatandstirinthesourcreamandchoppeddill. 6. Serve thesouphot, garnishedwithadditionaldillifdesired. ļu zupa! Letmeknowifyouhaveanyotherquestions. Redundant repetition! Redundant repetition! Off-topic! Figure 9. Reward hacking behaviors on an Anthropic-Helpful prompt. Standard RM and GoalRM exhibit verbosity hacking by generating excessively long, repetitive ingredient lists (e.g., duplicating the same vegetables or repeatedly listing “bay leaf”), which inflates superficial “helpfulness” without adding useful content. InfoRM produces a reasonable recipe but drifts off-topic by continuing into an unrelated dialogue about fruits and vegetables. In contrast, CausalRM provides a concise, coherent recipe that stays on-topic and avoids redundant repetition. 21 Factored Causal Representation Learning for Robust Reward Modeling in RLHF LLM Input (Prompt) System: You are a helpful assistant. User: Explain like I'm five years old: Why is tupperware wet coming out of the dishwasher, when plates and glasses are all dry? Assistant: CausalRM (Ours) Whenyouput dishesinthedishwasher, theygetwetbecausethedishwashersprayswateronthemto cleanthem. Thewaterhelpsto removedirtandfoodfromthe dishes. Afterthedishesarecleaned, thedishwasheruseshotairto drythem. Thishotairhelpsto evaporatethewateronthedishes, makingthemdry. , the waterfromthedishwashercangetonotherthings, likeTupperware, whileit isstillwet. ThiscanhappeniftheTupperwareisnotplacedinthedishwasherproperlyorif thedishwasherisnotworkingcorrectly. IftheTupperwareisstillwetwhenit comesoutofthedishwasher, it isbecausethewaterfromthedishwasherhasnothad enoughtimeto evaporateanddrytheTupperware. Standard RM Whenyouput dishesinthedishwasher, theygetwetbecausethedishwashersprayswateronthemto cleanthem. Thewaterhelpsto removedirtandfoodfromthe dishes. Afterthedishesarecleaned, thedishwasheruseshotairto drythem. Thehotairhelpsto evaporatethewateronthedishes, makingthemdry. Sometimes, the waterfromthedishwashercanstillbeonthedisheswhentheycomeout, butthehotairusuallydriesthemquickly. Tupperwareisatypeofcontainerthatisusedto store food. It isnotadish, soit doesnotgetwetinthedishwasher. Misleading! GoalRM Whenyouput dishesinthedishwasher, theygetwetbecausethedishwasheruseswaterto cleanthem. Afterthedishesarecleaned, thedishwasheruseshotairto dry them. Sometimes, thehotairdoesn'tdryeverythingcompletely, andsomedishesmightstillbealittlewet. ThisiswhytheTupperwaremightbewetcomingoutofthe dishwasher, even thoughtheplates andglassesaredry. Incomplete! InfoRM Whenyouput dishesinthedishwasher, theygetwetbecausethewaterissprayedonthemto cleanthem. Afterthedishesarecleaned, thewaterissupposedto be drainedoutofthedishwasher. Butsometimes, thewaterdoesn'tdrainoutcompletelyandsomeofit staysonthedishes. ThisiswhytheTupperwaremightbewet comingoutofthedishwasher. It'snotbecausetheTupperwareisdirty, butbecausethereisstillsomewaterleftinthedishwasher. Incomplete! Figure 10. Reward hacking behaviors on a SHP prompt. Standard RM exhibits misleading explanations by providing a factually incorrect rationale (claiming that “Tupperware is not a dish, so it does not get wet”). GoalRM and InfoRM avoid the explicit error but give shallow, incomplete explanations that do not account for how plastic and container geometry affect drying. In contrast, CausalRM produces a coherent, physically plausible explanation for why plastic containers often remain wet after a dishwasher cycle. 22