Robust Preference Optimization via Dynamic Target Margins
Authors: Jie Sun, Junkang Wu, Jiancan Wu, Lintao Ma, Zhibo Zhu, Xingyu Lu, JUN ZHOU, Xiang Wang
Accepted at: acl25

Robust Preference Optimization via Dynamic Target Margins

Jie Sun1, Junkang Wu1, Jiancan Wu2, Zhibo Zhu1, Xingyu Lu1, Jun Zhou1, Lintao Ma1*, Xiang Wang3*
1 Ant Group   2 Shanghai Key Laboratory of Data Science   3 National University of Singapore
{kangji.sj, gavin.zzb, lintao.mlt, sing.lxy}@antgroup.com
{jkwu0909, wujcan}@gmail.com
jun.zhoujun@antfin.com   xiangwang@u.nus.edu

Abstract

The alignment of Large Language Models (LLMs) is crucial for ensuring their safety and reliability in practical applications. Direct Preference Optimization (DPO) has emerged as an efficient method that directly optimizes models using preference pairs, significantly reducing resource demands. However, the effectiveness of DPO heavily depends on the data quality, which is frequently compromised by noise. In this work, we propose \( \gamma \)-PO, a dynamic target margin preference optimization algorithm that adjust reward margins at the pairwise level. By introducing instance-specific margin calibration, \( \gamma \)-PO strategically prioritizes high-confidence pairs (those demonstrating higher reward margins) while suppressing potential noise from ambiguous pairs. Moreover, \( \gamma \)-PO is a plug-and-play method, compatible with variants of DPO that rely on reward margin between preference pairs. Across benchmarks such as AlpacaEval2 and ArenaHard, \( \gamma \)-PO achieves an average 4.4% improvement over other baselines, setting new benchmarks for state-of-the-art performance. Additionally, \( \gamma \)-PO requires minimal code changes and has a negligible impact on training efficiency, making it a robust solution for enhancing LLMs alignment. Our codes are available at https://github.com/sunjie279/gammaPO.

1 Introduction

The alignment of Large Language Models (LLMs) with human values remains critical for their safe deployment (Dong et al., 2024). While Reinforcement Learning from Human Feedback (RLHF) (Ouyang et al., 2022a) pioneered model alignment through reward modeling and Proximal Policy Optimization (PPO) (Schulman et al., 2017), its computational intensity and training instability (Zheng et al., 2023b) motivate simpler alternatives. Direct Preference Optimization (DPO) (Rafailov et al., 2023) offers an efficient alternative by directly optimizing preference probabilities with human preference pairs, bypassing the need for explicit reward modeling.

However, DPO exhibits significant performance degradation when dealing with data that presents uncertain or ambiguous reward signals (see Figure 1, left panel). In such cases, narrow reward margins reflect inherent ambiguity in response-reward comparisons, which can lead models to overfit to weak or conflicting preference signals rather than capturing consistent human preferences. This challenge is particularly critical as our empirical analysis in Figure 2 reveals that a substantial majority of training pairs are concentrated around zero reward margin, necessitating more attention and methodological improvements to effectively handle such ambiguous instances.

Existing robust alignment approaches typically adopt two distinct paradigms: (1) applying soft labels through label smoothing techniques to reduce overconfidence in potentially noisy annotations (Chowdhury et al., 2024), and (2) implementing rule-based data filtering mechanisms to eliminate ambiguous preference pairs (Wu et al., 2024). Both categories face critical limitations:

Prompt: How to relieve work stress?

<table>
  <tr>
    <th></th>
    <th>ambiguous pair</th>
    <th>unambiguous pair</th>
  </tr>
  <tr>
    <td>\( y_w \): Use your weekends to go outdoors for hiking, cycling and other sports to get closer to nature. (reward=0.9)</td>
    <td>ambiguous pair</td>
    <td>unambiguous pair</td>
  </tr>
  <tr>
    <td>\( y_i \): Do some simple stretching exercises during work breaks to relax your body. (reward=0.8)</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>\( y_u \): You can talk to your friends or family and share your worries and stress. (reward=0.8)</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>\( y_t \): Drink more alcohol. When you are drunk, you don’t have to think about anything and the stress will naturally disappear. (reward=0.1)</td>
    <td></td>
    <td></td>
  </tr>
</table>

Figure 1: Comparison of ambiguous and unambiguous sample pairs. Ambiguous pairs exhibit narrow reward margins, indicating low confidence in model predictions, whereas unambiguous pairs demonstrate wide reward margins, reflecting high prediction confidence.
Figure 2: Distribution of the reward margin for the Mistral model using the SimPO objective on the Ultrafeedback Binarized dataset. The blue violin plot represents the full range of the original distribution, while the brown plot provides a zoomed-in view of the distribution, focusing on the central values close to zero.

• One-Size-Fits-All Objective: Current methods (Chowdhury et al., 2024; Meng et al., 2024; Amini et al., 2024; Rafailov et al., 2023) employ uniform loss functions that treat all preference pairs equally, disregarding varying confidence scores across samples.

• Strong Theoretical Assumptions: Prior work requires either known label flip rates (Chowdhury et al., 2024) or specific reward margin distributions (Wu et al., 2024), coupled with complex auxiliary mechanisms such as \( \beta \)-guided filtering that significantly limit practical adoption.

For modeling preference pair ambiguity, the Bradley-Terry model (Bradley and Terry, 1952) provides a probabilistic foundation through \( P(y_w > y_l) = \sigma(r_w - r_l) \), where smaller reward margin indicates higher uncertainty. Building on this insight, recent work SimPO (Meng et al., 2024) introduces a fixed target margin \( \gamma_0 \) to reshape the reward landscape via \( \sigma(r_w - r_l - \gamma_0) \). While effective in clean data scenarios, this rigid margin fails to adapt to the spectrum of uncertainty inherent in real-world preference pairs. This limitation motivates our key research question:

Can adaptive target margin adjustment based on reward margin enhance alignment robustness?

We propose \( \gamma \)-PO, a plug-and-play method that dynamically adjusts target margins \( \gamma_i \) for each preference pair. Our approach employs confidence-aware margin scaling: for unambiguous pairs with a significant difference in rewards, we increase the target margins to strengthen gradient signals during training. Conversely, for ambiguous pairs with small reward margins, we reduce the target margins to mitigate the influence of uncertain label. This dual mechanism provides two complementary benefits: (1) Algorithm-agnostic design compatible with any reward margin-based method (\emph{e.g.}, DPO, SimPO), and (2) Theoretical connections to dynamic label smoothing (Su et al., 2022). We derive that \( \gamma \)-PO inherently implements \emph{dynamic label smoothing} through its dynamic target margins. Specifically, smaller reward margins induce larger \( \varepsilon \) (\emph{i.e.}, smoother labels), effectively smoothing labels for ambiguous pairs. Conversely, larger margins reduce \( \varepsilon \) (\emph{i.e.}, sharper labels), sharpening supervision for unambiguous samples.

Experiments across four base models (LLaMA-3-8B-Instruct, Mistral-7B-Instruct, Gemma-2-9B-Instruct, and Qwen-2.5-7B-Instruct) show \( \gamma \)-PO achieves 4.4% average gains on AlpacaEval2 (Li and Zhang, 2023), Arena-Hard (Li et al., 2024), and MT-Bench (Zheng et al., 2023a), with minimal computational overhead. Additionally, our plug-and-play approach requires only a few lines of code and has a negligible impact on training efficiency, yet significantly enhances performance.

2 Preliminaries

2.1 Reinforcement Learning from Human Feedback

While LLMs are pre-trained on extensive datasets, those undergoing only supervised fine-tuning may output harmful or helpless content. To better align LLMs with human preferences, a widely adopted method is Reinforcement Learning from Human Feedback (Ouyang et al., 2022a), which involves three main steps: supervised fine-tuning (Zhou et al., 2023; Ding et al., 2023), reward model training (Chen et al., 2024; Gao et al., 2024), and policy model optimization. Specifically, the policy model is trained using Proximal Policy Optimization (PPO) (Anthony et al., 2017) that maximizes response rewards with Kullback-Leibler (KL) regularization (Kullback and Leibler, 1951) between the policy model \( \pi_\theta \) and a reference model \( \pi_{\text{ref}} \):

\[
\max_{\pi_\theta} \mathbb{E}_{x \sim \mathcal{D}, y \sim \pi_\theta(y|x)} \\
\left[ r_\phi(x, y) - \beta \mathbb{D}_{\text{KL}}[\pi_\theta(y|x)||\pi_{\text{ref}}(y|x)] \right],
\]

where response \( y \) is generated from policy \( \pi_\theta \) based on the prompt \( x \) in dataset \( \mathcal{D} \), and \( r_\phi \) is the reward
model. To estimate the labeling uncertainty of tuple \((x, y_w, y_l)\), Bradley-Terry (BT) model (Bradley and Terry, 1952) defines the success probability:

\[
\mathbb{P}(y_w > y_l \mid x) = \sigma(r_\phi(x, y_w) - r_\phi(x, y_l)),
\]

where \( \sigma \) is the sigmoid function.

2.2 Direct Preference Optimization

DPO (Rafailov et al., 2023) is a prominent alignment method that directly optimizes the policy model, eliminating the need for training an explicit reward model and significantly reducing the resources required for alignment:

\[
\mathcal{L}_{\text{DPO}}(\pi_\theta; \pi_{\text{ref}}) = -\mathbb{E}_{(x, y_w, y_l) \sim \mathcal{D}}
\left[ \log \sigma \left( \beta \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} - \beta \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)} \right) \right],
\]

where \( y_w \) is the preferred response over \( y_l \), and \( \beta \) is the hyperparameter.

2.2.1 Preference Optimization with Margin

Recent studies such as ODPO (Amini et al., 2024) and SimPO (Meng et al., 2024) propose to modulate the reward difference between preferred and less preferred samples by introducing a target margin \( \gamma_0 \). We express the generalized preference optimization with a target margin through a unified loss function:

\[
\mathcal{L}_{\text{margin PO}} = -\mathbb{E}_{(x, y_w, y_l) \sim \mathcal{D}} [\log \sigma \left( r_w - r_l - \gamma_0 \right)].
\]

The definition of \( r_w \) and \( r_l \) in each method are summarized in Table 1.

2.2.2 Robust Preference Optimization

To address the challenge of noisy preference data, Robust DPO (rDPO) (Chowdhury et al., 2024) has been proposed. rDPO enhances robustness by explicitly incorporating the label flip rate into its formulation, thereby reducing sensitivity to erroneous preference flips. The loss function for rDPO is defined as follows:

\[
\mathcal{L}_{\text{rDPO}} = (1 - \varepsilon) \mathcal{L}_{\text{DPO}}(y_w > y_l) + \varepsilon \mathcal{L}_{\text{DPO}}(y_l > y_w),
\]

where \( \varepsilon \) represents the flip rate of labels. This formulation ensures that when there is no label flip (\( \varepsilon = 0 \)), the rDPO loss reduces to the standard DPO loss. rDPO shares similarities with label smoothing techniques, which introduce regularization to soften predictions’ confidence. In rDPO, the parameter \( \varepsilon \) can be seen as a smoothing factor that controls the degree of confidence for the predictions. This provides a mechanism analogous to label smoothing, specifically tailored for handling noisy preferences optimization.

<table>
  <tr>
    <th>Methods</th>
    <th>\( r_w \)</th>
    <th>\( r_l \)</th>
  </tr>
  <tr>
    <td>ODPO</td>
    <td>\( \beta \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} \)</td>
    <td>\( \beta \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)} \)</td>
  </tr>
  <tr>
    <td>SimPO</td>
    <td>\( \frac{\beta}{|y_w|} \log \pi_\theta(y_w|x) \)</td>
    <td>\( \frac{\beta}{|y_l|} \log \pi_\theta(y_l|x) \)</td>
  </tr>
</table>

Table 1: Definitions of \( r_w \) and \( r_l \) for alignment method ODPO and SimPO.

![Diagram showing the Dynamic Target Margin Module adjusting adaptive target margin γi through reward-driven optimization, guided by dual mechanisms described in Section 3.1.](page_1016_1012_388_246.png)

Figure 3: The Dynamic Target Margin Module dynamically adjusts the adaptive target margin (\( \gamma_i \)) through reward-driven optimization, guided by the dual mechanisms described in Section 3.1. The optimized \( \gamma_i \) subsequently replaces the static margin (\( \gamma_0 \)) in the policy optimization loss function, enabling adaptive margin adjustment throughout the training process.

3 Method

In this section, we first present the methodology for deriving the adaptive target margin. Next, we present our proposed method, \( \gamma \)-PO, along with its two variants, \( \gamma \)-DPO and \( \gamma \)-SimPO. Finally, we conduct a comprehensive comparative analysis with rDPO (Chowdhury et al., 2024), highlighting its superior robustness and alignment capabilities.

3.1 Derivation of Adaptive Target Margin

The core objective of preference optimization with margin is to enforce \( r_w > r_l + \gamma_0 \) where \( \gamma_0 \) is a fixed margin. Our framework extends this by introducing instance-specific margins to address ambiguous reward signal in preference optimization, ensuring that:

\[
r_w - r_l > \gamma_i \iff r_l - r_w + \gamma_i < 0.
\]

To optimize \( \gamma_i \), we focus on instances that violate this condition (where \( r_l - r_w + \gamma_i \geq 0 \)). This leads
to the following optimization objective:

\[
\min_{\gamma_i} \max \{0, r_l - r_w + \gamma_i\}.
\] (7)

For computational efficiency, we reformulate this objective using the logarithmic approximation (\emph{i.e.}, \( \max\{0, y - x\} \approx -\log \frac{\exp(x)}{\exp(x) + \exp(y)} \)), yielding:

\[
\begin{aligned}
&\min_{\gamma_i} -\log \frac{\exp(r_w)}{\exp(r_w) + \exp(r_l) \exp(\gamma_i)} \\
&= \min_{\gamma_i} -\log \sigma(r_w - r_l - \gamma_i).
\end{aligned}
\] (8)

This logarithmic formulation enhances numerical stability during gradient computation while maintaining the theoretical guarantees.

Our dynamic target margin \( \gamma_i \) are guided by two principles: (1) \textbf{Adaptive Margin}: The preference pairs with larger reward margins represent higher-confidence annotations and should receive stronger learning signals, while ambiguous pairs require more conservative treatment, and (2) \textbf{Regularization}: To mitigate the risk of margin collapse (\emph{i.e.}, \( \gamma_i \to -\infty \)), the adaptive target margin distribution must preserve a reasonable scale relative to a baseline target margin \( \gamma_0 \). To achieve this, we regularize the target margin distribution by minimizing the KL divergence with respect to a uniform prior. This dual-objective formulation is expressed as:

\[
\gamma_i = \arg\min_{\gamma_i} -\mathbb{E}_D[\log \sigma(r_w - r_l - \gamma_i) \\
+ \tau \mathbb{D}_{KL}(p_i \| p_0)],
\] (9)

where \( p_i = \gamma_i / \sum_j \gamma_j \) and \( p_0 \) represent the normalized \( \gamma_i \) distribution and the uniform distribution, respectively. Regularization coefficient \( \tau \) controls the trade-off between adaptation flexibility and stability. The complete derivation and computational details of \( \gamma_i \) are provided in Appendix C. For clarity, we summarize the key notations used in this paper in Table 9.

3.2 Generalizability of \( \gamma \)-PO

Building upon the dynamic target margin \( \gamma_i \) derived in Section 2.2.1, we extend the conventional fixed margin loss in Sec 2.2.1 to a dynamic, instance-specific formulation. Consequently, we propose \( \gamma \)-PO, a method that adapts the margin for each instance. The final policy optimization objective is consequently defined as:

\[
\mathcal{L}_{\gamma\text{-PO}} = -\mathbb{E}_D[\log \sigma(r_w - r_l - \gamma_i)].
\] (10)

Algorithm 1 \( \gamma \)-PO Algorithm

1: \textbf{Input}: SFT model \( \pi_{SFT} \), dataset \( \mathcal{D} \), total number of iterations \( T \), learning rate \( \alpha \),
2: Initial policy model \( \pi_\theta = \pi_{SFT} \),
3: \textbf{for} \( t = 0 \) to \( T \) \textbf{do}
4: \hspace{0.5em} Sample a batch of tuples \( (x, y_w, y_l) \) from \( \mathcal{D} \),
5: \hspace{0.5em} Calculate the rewards \( (r_w, r_l) \) via Table 1,
6: \hspace{0.5em} Calculate the adaptive margin \( \gamma_i \) using Equation (9),
7: \hspace{0.5em} Update the policy parameters \( \theta \) for \( \pi_\theta \) by applying gradient descent to Equation (10):
\( \theta_{t+1} = \theta_t - \alpha \nabla_{\theta_t} \mathcal{L}_{\gamma\text{-PO}} \).
8: \textbf{end for}

The proposed \( \gamma \)-PO framework demonstrates remarkable generalizability as an algorithm-agnostic method. It can be seamlessly integrated with existing approaches, yielding enhanced variants such as \( \gamma \)-DPO:

\[
\mathcal{L}_{\gamma\text{-DPO}} = -\mathbb{E}_D[\log \sigma(r_w - r_l - \gamma_i)], \quad \text{where}
\]
\[
r_w = \beta \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)}, \quad r_l = \beta \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)},
\] (11)

and \( \gamma \)-SimPO:

\[
\mathcal{L}_{\gamma\text{-SimPO}} = -\mathbb{E}_D[\log \sigma(r_w - r_l - \gamma_i)], \quad \text{where}
\]
\[
r_w = \frac{\beta}{|y_w|} \log \pi_\theta(y_w|x), \quad r_l = \frac{\beta}{|y_w|} \log \pi_\theta(y_w|x).
\] (12)

3.3 Relation of \( \gamma \)-PO and rDPO

To bridge \( \gamma \)-PO with existing robust methods, we analyze its connection to rDPO through their loss formulations. The rDPO objective applies label smoothing to the standard DPO loss:

\[
\mathcal{L}_{\text{rDPO}} = -\varepsilon \mathbb{E}_D[\log \sigma(r_l - r_w - \gamma_0)] \\
-(1-\varepsilon) \mathbb{E}_D[\log \sigma(r_w - r_l - \gamma_0)],
\] (13)

where \( \varepsilon \) controls label smoothing intensity. Our key insight is that \( \gamma \)-PO implicitly implements \emph{adaptive label smoothing} through dynamic margins. We formalize this connection via:

**Theorem 3.1.** *Let \( \delta = \gamma_i - \gamma_0 \) and \( m = r_w - r_l \). When \( |\delta| \ll |m| \), equating \( \mathcal{L}_{\text{rDPO}} \) and \( \mathcal{L}_{\gamma\text{-PO}} \) yields the approximation:*

\[
\varepsilon \approx \frac{\delta [1 - \sigma(m - \gamma_0)]}{\log(1 + e^{m+\gamma_0}) - \log(1 + e^{\gamma_0-m})}.
\] (14)
This analysis demonstrates that \( \gamma \)-PO inherently implements *adaptive label smoothing* through its dynamic margins. The derived relationship between \( \delta \) (margin adjustment) and \( \varepsilon \) (smoothing parameter) reveals that smaller reward margin \( m = r_w - r_l \) automatically induce larger \( \varepsilon \) values, effectively smoothing labels for ambiguous pairs. Conversely, a larger margin reduces \( \varepsilon \), sharpening supervision for high-confidence samples. Crucially, the negative gradient \( \partial \varepsilon / \partial d < 0 \) formalizes this inverse relationship, enabling \( \gamma \)-PO to self-regulate label certainty based on reward margin magnitudes. By contrast, rDPO’s static \( \varepsilon \) requires pre-defined label flip rates and lacks this instance-level adaptation. This perspective provides new insights into the interpretation of \( \gamma \)-PO. The detailed proof of this theorem is provided in Appendix A.

4 Experiments

This section starts with an overview of the experimental setup, including various base models, training datasets, evaluation methods, and baselines. We then present the experimental results and visualize the dynamic target reward margin along with the hyperparameter analysis.

4.1 Experiment Setup

**Base Model.** Our approach is tested on four different base models: LLaMA-3-8B-Instruct (Dubey et al., 2024), Mistral-7B-Instruct (Jiang et al., 2023), Gemma-2-9B-Instruct (Rivière et al., 2024), and Qwen-2.5-7B-Instruct (Yang et al., 2024). The diversity of these base models provides a solid foundation for our experiments.

**Dataset.** Following the approach in SimPO (Meng et al., 2024), a strong reward model is used to re-annotate the preference datasets. In this work, five different responses are generated for each prompt from the Ultrafeedback Binarized dataset* using the base models. These responses are then re-annotated using a more powerful reward model, RLHFFlow/ArmoRM-Llama3-8B-v0.1*. Then, the responses with the highest and lowest scores are selected to combine the preference data.

**Evaluation.** We use three well-established metrics, AlpacaEval2, Arena-Hard, and MT-Bench to assess LLMs alignment with human preferences. AlpacaEval2 (Dubois et al., 2024) is an enhanced automated metric that evaluates LLMs performance using Win Rate (WR) and Length-Controlled Win Rate (LC) with a dataset of 805 instructions. Both the reference model and judge model are GPT-4-1106-preview (OpenAI, 2023). The length control improves robustness to length manipulation and better aligns with human evaluations. Arena-Hard (Li et al., 2024) assesses LLMs by applying 500 prompts derived from the Chatbot Arena dataset. It benchmarks the LLMs against GPT-4-0314 (OpenAI, 2023) as the baseline, and GPT-4-1106-preview evaluates the responses. Each response is rated using a five-point Likert scale (Liang et al., 2020), which facilitates a fair comparison and emphasizes performance disparities between models. MT-bench (Zheng et al., 2023a) evaluates chatbots by using 80 multi-turn dialogue prompts that are specifically designed to assess their abilities in multi-round interactions and following instructions. It benchmarks the chatbots against GPT-4, which serves as both the baseline and the evaluator for the responses. Each response is scored based on specific criteria, allowing for objective comparisons and highlighting the differences in how well various models handle complex conversational tasks.

**Baselines.** We compare \( \gamma \)-PO with several state-of-the-art preference optimization methods, including DPO (Rafailov et al., 2023), IPO (Azar et al., 2024a), CPO (Xu et al., 2024), KTO (Ethayarajh et al., 2024), ORPO (Hong et al., 2024), R-DPO (Park et al., 2024), rDPO (Chowdhury et al., 2024), \( \beta \)-DPO (Wu et al., 2024), and SimPO (Meng et al., 2024). We also use SFT model as a baseline. Further information regarding the baselines and training details can be found in Appendix B.

4.2 Experiment Results

\( \gamma \)-PO consistently outperforms the baselines. We apply our method to one of the leading baseline preference optimization approaches, SimPO, resulting in \( \gamma \)-SimPO. The results in Table 2 show that \( \gamma \)-SimPO consistently outperforms other methods in both AlpacaEval2 and Arena-Hard, demonstrating stable performance improvements. For instance, in AlpacaEval2, it boosts the performance of Qwen2.5-Instruct from 34.1% to 39.0% of LC, and from 33.3% to 34.3% of WR. In the Mistral-Instruct setting, \( \gamma \)-SimPO outperforms the baseline SimPO by 6.5% (LC) and 7.1% (WR) in AlpacaEval2, and enhances WR of Arena-Hard from 20.7% to 22.9%. On average, our method leads to a 4.4% improvement. Additionally, we also evaluate our
<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="5">Llama3-Instruct (8B)</th>
    <th colspan="5">Mistral-Instruct (7B)</th>
  </tr>
  <tr>
    <th>AlpacaEval 2</th>
    <th>Arena-Hard</th>
    <th>Average</th>
    <th>AlpacaEval 2</th>
    <th>Arena-Hard</th>
    <th>Average</th>
  </tr>
  <tr>
    <th></th>
    <th>LC(%)</th>
    <th>WR(%)</th>
    <th>err(%)</th>
    <th>WR(%)</th>
    <th>95%CI (%)</th>
    <th>LC(%)</th>
    <th>WR(%)</th>
    <th>err(%)</th>
    <th>WR(%)</th>
    <th>95%CI (%)</th>
  </tr>
  <tr>
    <td>SFT</td>
    <td>28.2</td>
    <td>27.4</td>
    <td>1.38</td>
    <td>24.3</td>
    <td>(-1.7, 1.9)</td>
    <td>26.6</td>
    <td>21.3</td>
    <td>16.0</td>
    <td>1.15</td>
    <td>13.1</td>
    <td>(-1.5, 1.1)</td>
    <td>16.8</td>
  </tr>
  <tr>
    <td>IPO</td>
    <td>38.3</td>
    <td>35.9</td>
    <td>1.48</td>
    <td>31.4</td>
    <td>(-2.3, 2.5)</td>
    <td>35.2</td>
    <td>24.0</td>
    <td>19.3</td>
    <td>1.21</td>
    <td>16.3</td>
    <td>(-1.6, 1.5)</td>
    <td>19.9</td>
  </tr>
  <tr>
    <td>KTO</td>
    <td>34.9</td>
    <td>32.5</td>
    <td>1.45</td>
    <td>27.8</td>
    <td>(-2.8, 3.5)</td>
    <td>31.7</td>
    <td>18.9</td>
    <td>20.2</td>
    <td>1.23</td>
    <td>14.4</td>
    <td>(-1.6, 1.6)</td>
    <td>17.8</td>
  </tr>
  <tr>
    <td>CPO</td>
    <td>27.9</td>
    <td>26.8</td>
    <td>1.36</td>
    <td>25.4</td>
    <td>(-2.3, 2.9)</td>
    <td>26.7</td>
    <td>25.3</td>
    <td>22.6</td>
    <td>1.30</td>
    <td>18.4</td>
    <td>(-1.7, 1.9)</td>
    <td>22.1</td>
  </tr>
  <tr>
    <td>ORPO</td>
    <td>32.0</td>
    <td>30.3</td>
    <td>1.41</td>
    <td>25.9</td>
    <td>(-1.9, 2.4)</td>
    <td>29.4</td>
    <td>24.0</td>
    <td>21.8</td>
    <td>1.27</td>
    <td>20.4</td>
    <td>(-2.1, 2.0)</td>
    <td>22.1</td>
  </tr>
  <tr>
    <td>R-DPO</td>
    <td>47.9</td>
    <td>43.5</td>
    <td>1.48</td>
    <td>32.9</td>
    <td>(-2.5, 2.7)</td>
    <td>41.4</td>
    <td>24.8</td>
    <td>22.0</td>
    <td>1.27</td>
    <td>17.1</td>
    <td>(-1.6, 1.7)</td>
    <td>21.3</td>
  </tr>
  <tr>
    <td>rDPO</td>
    <td>47.8</td>
    <td>43.7</td>
    <td>1.48</td>
    <td>34.4</td>
    <td>(-2.2, 2.4)</td>
    <td>42.0</td>
    <td>23.7</td>
    <td>20.9</td>
    <td>1.25</td>
    <td>16.3</td>
    <td>(-1.5, 1.7)</td>
    <td>20.3</td>
  </tr>
  <tr>
    <td>DPO</td>
    <td>46.8</td>
    <td>42.7</td>
    <td>1.48</td>
    <td>32.7</td>
    <td>(-2.6, 2.7)</td>
    <td>40.7</td>
    <td>22.5</td>
    <td>20.2</td>
    <td>1.20</td>
    <td>17.1</td>
    <td>(-2.0, 1.6)</td>
    <td>19.9</td>
  </tr>
  <tr>
    <td>\( \beta \)-DPO</td>
    <td>48.2</td>
    <td>44.0</td>
    <td>1.48</td>
    <td>36.4</td>
    <td>(-2.0, 2.1)</td>
    <td>42.9</td>
    <td>23.3</td>
    <td>20.6</td>
    <td>1.22</td>
    <td>17.5</td>
    <td>(-1.4, 1.6)</td>
    <td>20.5</td>
  </tr>
  <tr>
    <td>SimPO</td>
    <td>54.3</td>
    <td>46.6</td>
    <td>1.53</td>
    <td>31.5</td>
    <td>(-2.4, 3.0)</td>
    <td>44.1</td>
    <td>29.3</td>
    <td>30.8</td>
    <td>1.38</td>
    <td>20.7</td>
    <td>(-2.0, 1.8)</td>
    <td>26.9</td>
  </tr>
  <tr>
    <td>\( \gamma \)-SimPO</td>
    <td><b>55.4</b></td>
    <td><b>48.7</b></td>
    <td><b>1.48</b></td>
    <td><b>37.3</b></td>
    <td>(-2.6, 2.8)</td>
    <td><b>47.1</b></td>
    <td><b>31.2</b></td>
    <td><b>33.0</b></td>
    <td><b>1.39</b></td>
    <td><b>22.9</b></td>
    <td>(-1.7, 2.2)</td>
    <td><b>29.0</b></td>
  </tr>
  <tr>
    <td>Improve</td>
    <td>+2.0%</td>
    <td>+4.5%</td>
    <td>-</td>
    <td>+6.8%</td>
    <td>+2.5%</td>
    <td>-</td>
    <td>+6.5%</td>
    <td>+7.1%</td>
    <td>-</td>
    <td>+10.6%</td>
    <td>-</td>
    <td>+7.8%</td>
  </tr>
</table>

<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="5">Gemma2-Instruct (9B)</th>
    <th colspan="5">Qwen2.5-Instruct (7B)</th>
  </tr>
  <tr>
    <th>AlpacaEval 2</th>
    <th>Arena-Hard</th>
    <th>Average</th>
    <th>AlpacaEval 2</th>
    <th>Arena-Hard</th>
    <th>Average</th>
  </tr>
  <tr>
    <th></th>
    <th>LC (%)</th>
    <th>WR (%)</th>
    <th>err (%)</th>
    <th>WR (%)</th>
    <th>95%CI (%)</th>
    <th>LC (%)</th>
    <th>WR (%)</th>
    <th>err (%)</th>
    <th>WR (%)</th>
    <th>95%CI (%)</th>
  </tr>
  <tr>
    <td>SFT</td>
    <td>50.2</td>
    <td>38.1</td>
    <td>1.44</td>
    <td>40.8</td>
    <td>(-2.3, 2.1)</td>
    <td>43.0</td>
    <td>23.2</td>
    <td>22.2</td>
    <td>1.29</td>
    <td>35.9</td>
    <td>(-3.1, 3.1)</td>
    <td>27.1</td>
  </tr>
  <tr>
    <td>IPO</td>
    <td>61.3</td>
    <td>57.2</td>
    <td>1.45</td>
    <td>56.7</td>
    <td>(-2.8, 2.6)</td>
    <td>58.4</td>
    <td>25.1</td>
    <td>25.4</td>
    <td>1.34</td>
    <td>43.7</td>
    <td>(-4.4, 2.5)</td>
    <td>31.4</td>
  </tr>
  <tr>
    <td>KTO</td>
    <td>60.7</td>
    <td>54.5</td>
    <td>1.46</td>
    <td>51.8</td>
    <td>(-3.3, 2.8)</td>
    <td>55.7</td>
    <td>32.6</td>
    <td>30.5</td>
    <td>1.38</td>
    <td>46.6</td>
    <td>(-3.6, 3.0)</td>
    <td>36.6</td>
  </tr>
  <tr>
    <td>CPO</td>
    <td>50.3</td>
    <td>38.8</td>
    <td>1.45</td>
    <td>41.8</td>
    <td>(-2.4, 2.1)</td>
    <td>43.6</td>
    <td>23.3</td>
    <td>19.5</td>
    <td>1.29</td>
    <td>38.3</td>
    <td>(-3.1, 3.1)</td>
    <td>27.0</td>
  </tr>
  <tr>
    <td>ORPO</td>
    <td>54.9</td>
    <td>46.4</td>
    <td>1.48</td>
    <td>44.5</td>
    <td>(-2.3, 2.2)</td>
    <td>48.6</td>
    <td>26.4</td>
    <td>23.0</td>
    <td>1.29</td>
    <td>38.3</td>
    <td>(-3.2, 2.7)</td>
    <td>29.2</td>
  </tr>
  <tr>
    <td>R-DPO</td>
    <td>66.9</td>
    <td><b>65.9</b></td>
    <td>1.37</td>
    <td>60.9</td>
    <td>(-2.3, 2.8)</td>
    <td>64.6</td>
    <td>33.2</td>
    <td>31.7</td>
    <td>1.39</td>
    <td>53.1</td>
    <td>(-3.6, 3.5)</td>
    <td>39.3</td>
  </tr>
  <tr>
    <td>rDPO</td>
    <td>68.2</td>
    <td><b>65.9</b></td>
    <td>1.39</td>
    <td>59.1</td>
    <td>(-2.4, 2.9)</td>
    <td>64.4</td>
    <td><b>34.1</b></td>
    <td><b>33.3</b></td>
    <td><b>1.42</b></td>
    <td><b>65.2</b></td>
    <td>(-2.0, 2.6)</td>
    <td><b>44.2</b></td>
  </tr>
  <tr>
    <td>DPO</td>
    <td>67.8</td>
    <td>65.8</td>
    <td>1.39</td>
    <td>60.3</td>
    <td>(-2.0, 2.3)</td>
    <td>64.6</td>
    <td>32.3</td>
    <td>31.8</td>
    <td>1.40</td>
    <td>54.3</td>
    <td>(-3.7, 2.4)</td>
    <td>39.5</td>
  </tr>
  <tr>
    <td>\( \beta \)-DPO</td>
    <td>68.0</td>
    <td><b>65.9</b></td>
    <td>1.37</td>
    <td>61.0</td>
    <td>(-2.2, 2.2)</td>
    <td>65.0</td>
    <td>32.4</td>
    <td>31.4</td>
    <td>1.39</td>
    <td>61.0</td>
    <td>(-2.2, 2.2)</td>
    <td>41.6</td>
  </tr>
  <tr>
    <td>SimPO</td>
    <td>70.9</td>
    <td>64.5</td>
    <td>1.43</td>
    <td>55.3</td>
    <td>(-2.4, 2.0)</td>
    <td>63.6</td>
    <td>33.3</td>
    <td>28.2</td>
    <td>1.37</td>
    <td>59.0</td>
    <td>(-2.9, 3.0)</td>
    <td>40.2</td>
  </tr>
  <tr>
    <td>\( \gamma \)-SimPO</td>
    <td><b>72.0</b></td>
    <td><b>65.9</b></td>
    <td>1.42</td>
    <td><b>62.0</b></td>
    <td>(-2.5, 1.8)</td>
    <td><b>66.6</b></td>
    <td><b>39.0</b></td>
    <td><b>34.3</b></td>
    <td><b>1.46</b></td>
    <td><b>64.8</b></td>
    <td>(-2.1, 2.3)</td>
    <td><b>46.0</b></td>
  </tr>
  <tr>
    <td>Improve</td>
    <td>+1.6%</td>
    <td>+0.0%</td>
    <td>-</td>
    <td>+1.6%</td>
    <td>-</td>
    <td>+2.5%</td>
    <td>+14.4%</td>
    <td>+3.0%</td>
    <td>-</td>
    <td>-0.6%</td>
    <td>-</td>
    <td>+4.1%</td>
  </tr>
</table>

Table 2: The results from AlpacaEval and Arena-Hard under four different settings are shown. “LC” and “WR” represent the Length-Controlled and Raw Win Rates, respectively. Bold indicates the best performance for each metric, while underlined values represent the best performance excluding our methods (our method is \( \gamma \)-SimPO, i.e., our plug-and-play modules applied to SimPO). “err” refers to the standard error, while “95% CI” stands for the 95% confidence interval. “Improve” denotes the percentage improvement of the bold value over the underlined one. Each metric, except for “err” and “95% CI”, is considered better when it has a higher value.

methods on MT-Bench, as shown in Appendix F. The results presented in Table 10 demonstrate that our proposed methods, \( \gamma \)-DPO and \( \gamma \)-SimPO, consistently achieve superior performance compared to baseline methods across all models. These results highlight the effectiveness of our approach in better aligning LLMs with human preferences.

\( \gamma \)-PO outperforms other plug-and-play methods.
We compared \( \gamma \)-PO with other plug-and-play techniques, such as rDPO and \( \beta \)-DPO. Specifically, we applied the \( \beta \)-DPO and rDPO to SimPO, resulting in \( \beta \)-SimPO and rSimPO, respectively. Additionally, we apply the \( \gamma \)-PO approach to both DPO and SimPO, leading to the development of \( \gamma \)-DPO and \( \gamma \)-SimPO. The experimental results in Table 3 show that our method significantly outperforms both rDPO and \( \beta \)-DPO in improving model performance. For instance, in the Qwen2.5-Instruct scenario, \( \gamma \)-SimPO exceeds the baseline SimPO by 6.1 in AlpacaEval2 and 5.8 in Arena-Hard. Moreover, both \( \gamma \)-DPO and \( \gamma \)-SimPO achieved the highest average scores across all metrics and settings, with just one exception. Overall, \( \gamma \)-PO demonstrates a more effective enhancement of alignment performance compared to other plug-and-play methods.

\( \gamma \)-PO handles flipped data better. To evaluate the resilience of the \( \gamma \)-SimPO method in the presence of label uncertainty, we deliberately introduce label noise into the training dataset by flipping labels at random rates of 10% and 20%. Building on the optimal baseline provided by the SimPO method, we conducted comparative experiments us-
<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="3">Llama3-Instruct (8B)</th>
    <th colspan="3">Mistral-Instruct (7B)</th>
  </tr>
  <tr>
    <th>AlpacaEval 2<br>LC(%)<br>WR(%)</th>
    <th>Arena-Hard<br>WR(%)</th>
    <th>Average<br>Avg. (rank)</th>
    <th>AlpacaEval 2<br>LC(%)<br>WR(%)</th>
    <th>Arena-Hard<br>WR(%)</th>
    <th>Average<br>Avg. (rank)</th>
  </tr>
  <tr>
    <td>DPO</td>
    <td>46.8<sup>+0.0</sup> 42.7<sup>+0.0</sup></td>
    <td>32.7<sup>+0.0</sup></td>
    <td>40.7 (4)</td>
    <td>22.5<sup>+0.0</sup> 20.2<sup>+0.0</sup></td>
    <td>17.1<sup>+0.0</sup></td>
    <td>19.9 (4)</td>
  </tr>
  <tr>
    <td>rDPO</td>
    <td>47.8<sup>+1.0</sup> 43.7<sup>+1.0</sup></td>
    <td>34.4<sup>+1.7</sup></td>
    <td>42.0 (2)</td>
    <td>23.7<sup>+1.2</sup> 20.9<sup>+0.7</sup></td>
    <td>17.5<sup>+0.4</sup></td>
    <td>20.7 (2)</td>
  </tr>
  <tr>
    <td>\( \beta \)-DPO</td>
    <td><b>48.2</b><sup>+1.4</sup> <b>44.0</b><sup>+1.3</sup></td>
    <td><b>36.4</b><sup>+3.7</sup></td>
    <td><b>42.9</b> (1)</td>
    <td>23.3<sup>+0.8</sup> 20.6<sup>+0.4</sup></td>
    <td>17.5<sup>+0.4</sup></td>
    <td>20.5 (3)</td>
  </tr>
  <tr>
    <td>\( \gamma \)-DPO</td>
    <td>47.8<sup>+1.0</sup> 43.0<sup>+0.3</sup></td>
    <td>35.1<sup>+2.4</sup></td>
    <td>42.0 (2)</td>
    <td><b>23.8</b><sup>+1.3</sup> <b>21.7</b><sup>+1.5</sup></td>
    <td><b>17.5</b><sup>+0.4</sup></td>
    <td><b>21.0</b> (1)</td>
  </tr>
  <tr>
    <td>SimPO</td>
    <td>54.3<sup>+0.0</sup> 46.6<sup>+0.0</sup></td>
    <td>31.5<sup>+0.0</sup></td>
    <td>44.1 (4)</td>
    <td>29.3<sup>+0.0</sup> 30.8<sup>+0.0</sup></td>
    <td>20.7<sup>+0.0</sup></td>
    <td>26.9 (4)</td>
  </tr>
  <tr>
    <td>rSimPO</td>
    <td>53.0<sup>-1.3</sup> 45.5<sup>-1.1</sup></td>
    <td>36.5<sup>+5.0</sup></td>
    <td>45.0 (3)</td>
    <td>28.9<sup>-0.4</sup> 30.3<sup>-0.5</sup></td>
    <td>22.2<sup>+1.5</sup></td>
    <td>27.1 (3)</td>
  </tr>
  <tr>
    <td>\( \beta \)-SimPO</td>
    <td>54.8<sup>+0.5</sup> 48.1<sup>+1.5</sup></td>
    <td>32.6<sup>+1.1</sup></td>
    <td>45.2 (2)</td>
    <td>29.4<sup>+0.1</sup> 31.1<sup>+0.3</sup></td>
    <td>22.6<sup>+1.9</sup></td>
    <td>27.7 (2)</td>
  </tr>
  <tr>
    <td>\( \gamma \)-SimPO</td>
    <td><b>55.4</b><sup>+1.1</sup> <b>48.7</b><sup>+2.1</sup></td>
    <td><b>37.3</b><sup>+5.8</sup></td>
    <td><b>47.1</b> (1)</td>
    <td><b>31.2</b><sup>+1.9</sup> <b>33.0</b><sup>+2.2</sup></td>
    <td><b>22.9</b><sup>+2.2</sup></td>
    <td><b>29.0</b> (1)</td>
  </tr>
</table>

<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="3">Gemma2-Instruct (9B)</th>
    <th colspan="3">Qwen2.5-Instruct (7B)</th>
  </tr>
  <tr>
    <th>AlpacaEval 2<br>LC(%)<br>WR(%)</th>
    <th>Arena-Hard<br>WR(%)</th>
    <th>Average<br>Avg. (rank)</th>
    <th>AlpacaEval 2<br>LC(%)<br>WR(%)</th>
    <th>Arena-Hard<br>WR(%)</th>
    <th>Average<br>Avg. (rank)</th>
  </tr>
  <tr>
    <td>DPO</td>
    <td>67.8<sup>+0.0</sup> 65.8<sup>+0.0</sup></td>
    <td>60.3<sup>+0.0</sup></td>
    <td>64.6 (3)</td>
    <td>32.3<sup>+0.0</sup> 31.8<sup>+0.0</sup></td>
    <td>54.3<sup>+0.0</sup></td>
    <td>39.5 (4)</td>
  </tr>
  <tr>
    <td>rDPO</td>
    <td>68.2<sup>+0.4</sup> 65.9<sup>+0.1</sup></td>
    <td>59.1<sup>+1.2</sup></td>
    <td>64.4 (4)</td>
    <td>34.1<sup>+1.8</sup> 33.3<sup>+1.5</sup></td>
    <td>65.2<sup>+10.9</sup></td>
    <td>44.2 (2)</td>
  </tr>
  <tr>
    <td>\( \beta \)-DPO</td>
    <td>68.0<sup>+0.2</sup> 65.9<sup>+0.1</sup></td>
    <td>61.0<sup>+0.7</sup></td>
    <td>65.0 (2)</td>
    <td>32.4<sup>+0.1</sup> 31.4<sup>-0.4</sup></td>
    <td>61.0<sup>+6.7</sup></td>
    <td>41.6 (3)</td>
  </tr>
  <tr>
    <td>\( \gamma \)-DPO</td>
    <td><b>70.6</b><sup>+2.8</sup> <b>68.0</b><sup>+2.2</sup></td>
    <td><b>64.1</b><sup>+3.8</sup></td>
    <td><b>67.6</b> (1)</td>
    <td><b>34.4</b><sup>+2.1</sup> <b>34.3</b><sup>+2.5</sup></td>
    <td><b>68.4</b><sup>+14.1</sup></td>
    <td><b>45.7</b> (1)</td>
  </tr>
  <tr>
    <td>SimPO</td>
    <td>70.9<sup>+0.0</sup> 64.5<sup>+0.0</sup></td>
    <td>55.3<sup>+0.0</sup></td>
    <td>63.6 (4)</td>
    <td>33.3<sup>+0.0</sup> 28.2<sup>+0.0</sup></td>
    <td>59.0<sup>+0.0</sup></td>
    <td>40.2 (4)</td>
  </tr>
  <tr>
    <td>rSimPO</td>
    <td>71.6<sup>+0.7</sup> 64.5<sup>+0.0</sup></td>
    <td>58.6<sup>+3.3</sup></td>
    <td>64.9 (2)</td>
    <td>38.8<sup>+5.5</sup> 31.7<sup>+3.5</sup></td>
    <td>61.4<sup>+2.4</sup></td>
    <td>44.0 (2)</td>
  </tr>
  <tr>
    <td>\( \beta \)-SimPO</td>
    <td>71.1<sup>+0.2</sup> 62.8<sup>-1.7</sup></td>
    <td>59.0<sup>+3.7</sup></td>
    <td>64.3 (3)</td>
    <td>34.5<sup>+1.2</sup> 29.5<sup>+1.3</sup></td>
    <td>59.1<sup>+0.1</sup></td>
    <td>41.0 (3)</td>
  </tr>
  <tr>
    <td>\( \gamma \)-SimPO</td>
    <td><b>72.0</b><sup>+1.1</sup> <b>65.9</b><sup>+1.4</sup></td>
    <td><b>62.0</b><sup>+6.7</sup></td>
    <td><b>66.6</b> (1)</td>
    <td><b>39.0</b><sup>+5.7</sup> <b>34.3</b><sup>+6.1</sup></td>
    <td><b>64.8</b><sup>+5.8</sup></td>
    <td><b>46.0</b> (1)</td>
  </tr>
</table>

Table 3: The improvements in results from AlpacaEval2 and Arena-Hard across various plug-and-play methods. ‘LC’ stands for Length-Controlled, while “WR” represents Raw Win Rates. The methods rDPO and rSimPO refer to the application of rDPO to the DPO and SimPO methods, respectively. And the method \( \beta \)-SimPO refer to the application of \( \beta \)-DPO to the SimPO method. The superscript indicates enhancements based on the baselines, such as DPO and SimPO, where red represents improvement and blue signifies degradation. Performance metrics displayed in bold indicate the best results among the plugins. For each metric, a higher value denotes better performance.

<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="3">Label Flip Rate 0.1</th>
    <th colspan="3">Label Flip Rate 0.2</th>
  </tr>
  <tr>
    <th>AlpacaEval 2<br>LC(%)<br>WR(%)</th>
    <th>Arena-Hard<br>WR(%)</th>
    <th>Average<br>Avg. (rank)</th>
    <th>AlpacaEval 2<br>LC(%) / WR(%)</th>
    <th>Arena-Hard<br>WR(%)</th>
    <th>Average<br>Avg. (rank)</th>
  </tr>
  <tr>
    <td>SimPO</td>
    <td>26.2<sup>+0.0</sup> 28.6<sup>+0.0</sup></td>
    <td>20.0<sup>+0.0</sup></td>
    <td>24.9 (4)</td>
    <td>24.5<sup>+0.0</sup> 26.1<sup>+0.0</sup></td>
    <td>18.7<sup>+0.0</sup></td>
    <td>23.1 (4)</td>
  </tr>
  <tr>
    <td>rSimPO</td>
    <td>27.1<sup>+0.9</sup> <b>28.7</b><sup>+0.1</sup></td>
    <td>21.9<sup>+1.9</sup></td>
    <td>25.9 (2)</td>
    <td>25.8<sup>+1.3</sup> 27.0<sup>+0.9</sup></td>
    <td>19.6<sup>+0.9</sup></td>
    <td>24.1 (3)</td>
  </tr>
  <tr>
    <td>\( \beta \)-SimPO</td>
    <td>26.6<sup>+0.4</sup> 28.4<sup>-0.2</sup></td>
    <td>21.6<sup>+1.6</sup></td>
    <td>25.5 (3)</td>
    <td>26.3<sup>+1.8</sup> 27.9<sup>+1.8</sup></td>
    <td>18.9<sup>+0.2</sup></td>
    <td>24.4 (2)</td>
  </tr>
  <tr>
    <td>\( \gamma \)-SimPO</td>
    <td><b>28.2</b><sup>+2.0</sup> <b>28.3</b><sup>-0.3</sup></td>
    <td><b>22.4</b><sup>+2.4</sup></td>
    <td><b>26.3</b> (1)</td>
    <td><b>25.9</b><sup>+1.4</sup> <b>26.3</b><sup>+0.2</sup></td>
    <td><b>21.9</b><sup>+3.2</sup></td>
    <td><b>24.7</b> (1)</td>
  </tr>
</table>

Table 4: The results from AlpacaEval2 and Arena-Hard illustrate the performance of various plug-and-play methods in a noisy label setting. In this context, “LC” stands for Length-Controlled, and “WR” stands for Raw Win Rates. Metrics that are highlighted in bold indicate the best performance among the plugins evaluated. The base model used for this analysis is Mistral-7B-Instruct. For each metric, a higher value indicates better performance.

ing rSimPO, \( \beta \)-SimPO, and our proposed \( \gamma \)-SimPO, all utilizing the Mistral-7B-Instruct as the base model. As shown in Table 4, \( \gamma \)-SimPO outperforms the other methods, achieving the highest average score, which is further illustrated in Figure 4. These results demonstrate that \( \gamma \)-SimPO exhibits
Figure 4: Comparison of average winning rates with randomly flipped labels at different probabilities.

robustness in scenarios involving noise, such as label flipping.

γ-PO requires no additional computing overhead. To evaluate the computational efficiency of γ-PO, we measure the training duration when combining γ-PO with DPO and SimPO across four base models. As shown in Table 5, γ-PO maintains computational efficiency comparable to the baseline methods while also delivering performance gains. Specifically, γ-DPO has only a 0.7% longer training time than DPO, while γ-SimPO demonstrates an even more minor increase of 0.5% over SimPO. These results confirm that our method performs better without significantly increasing GPU resource consumption.

4.3 Visualization of Dynamic Target Margin

Figure 5 depicts the relationship between the target margin \( \gamma_t \) and the reward gap for various LLMs (Llama, Mistral, Gemma, and Qwen, which stand for Llama3-Instruct (8B), Mistral-Instruct (7B), Gemma2-Instruct (9B), and Qwen2.5-Instruct (7B), respectively). The target margin exhibits a sigmoid-like increase with the reward gap, with the rapid upward slope closely corresponding to the range of the dynamic target margin. This behavior suggests that the target margin is increased when the reward gap exceeds the threshold, prioritizing learning from high-gap pairs. In contrast, when the reward gap is smaller than the threshold, corresponding to low-gap pairs that may contain noise, the target margin is reduced to prevent overfitting to these pairs. Furthermore, we observed that the target margin does not exhibit extreme values despite large or small reward gaps; it varies within a controlled, reasonable range. This visualization confirms the effectiveness of our approach in dynamically adjusting the target margin based on the reward gap, illustrating how it can be tailored to enhance learning from high-gap pairs while minimizing the influence of unreliable pairs.

4.4 Hyperparameter

We thoroughly investigate how the hyperparameter \( \tau \) in γ-PO affects the performance of AlpacaEval2 and Arena-Hard. Figure 6 illustrates the performance metrics across different \( \tau \) values for the Gemma base model. We plot three key metrics: Length-Controlled Winning Rate (AE2:LC) and Raw Winning Rate (AE2:WR) from AlpacaEval2, and Raw Winning Rate from Arena Hard (AH:WR). The results demonstrate a clear upward trend for all three metrics as \( \tau \) increases when \( \tau < 20 \), followed by a decline when \( \tau > 20 \). This suggests that the optimal value of \( \tau \) is almost the same across all metrics. The range for hyperparameter search is detailed in Table 6, while the final optimal hyperparameters are presented in Table 7.

5 Related Work

5.1 Reinforcement Learning from Human Feedback

Reinforcement Learning from Human Feedback (RLHF) (Christiano et al., 2017; Ziegler et al., 2019; Azar et al., 2024b) is a crucial post-training step following Supervised Fine-Tuning (SFT). RLHF helps align LLMs outputs with human values such as helpfulness, harmlessness, and honesty (Ouyang et al., 2022b; Bai et al., 2022). It has been widely applied in the revolutionary closed-source LLMs, such as GPT-4 (OpenAI, 2023) and Gemini (Anil et al., 2023), as well as powerful open-source models like LLaMA-3 (Dubey et al., 2024), Gemma-2 (Rivière et al., 2024), and Qwen-2.5 (Yang et al., 2024). The RLHF process generally consists of three main steps. First, a policy \( \pi_{SFT} \) is trained on a large dataset through SFT (Zhou et al., 2023; Ding et al., 2023; Köpf et al., 2023). Next, \( \pi_{SFT} \) generates multiple responses, which are then labeled by humans into preference pairs. These pairs are used to train a reward model, \( r_\theta \), that scores the model’s responses based on their alignment with human preferences (Chen et al., 2024; Gao et al., 2024; Lightman et al., 2024; Havrilla et al., 2024; Luo et al., 2023; Lambert et al., 2024; Dong et al., 2024). Lastly, the reward model updates the policy model via PPO algorithm (Schulman et al., 2017) in RL (Anthony et al., 2017).
<table>
  <tr>
    <th>Base Model</th>
    <th>DPO</th>
    <th>\( \gamma \)-DPO</th>
    <th>SimPO</th>
    <th>\( \gamma \)-SimPO</th>
  </tr>
  <tr>
    <td>Llama</td>
    <td>4A100*4h54min</td>
    <td>4A100*4h55min<sup>+1min</sup></td>
    <td>4A100*3h30min</td>
    <td>4A100*3h33min<sup>+3min</sup></td>
  </tr>
  <tr>
    <td>Mistral</td>
    <td>4A100*4h45min</td>
    <td>4A100*4h53min<sup>+8min</sup></td>
    <td>4A100*3h41min</td>
    <td>4A100*3h42min<sup>+1min</sup></td>
  </tr>
  <tr>
    <td>Gemma</td>
    <td>8A100*4h04min</td>
    <td>8A100*4h03min<sup>+1min</sup></td>
    <td>8A100*3h13min</td>
    <td>8A100*3h12min<sup>-1min</sup></td>
  </tr>
  <tr>
    <td>Qwen</td>
    <td>4A100*5h26min</td>
    <td>4A100*5h29min<sup>+3min</sup></td>
    <td>4A100*3h33min</td>
    <td>4A100*3h36min<sup>+3min</sup></td>
  </tr>
  <tr>
    <td>Avg. GPU time</td>
    <td>23.22h·A100</td>
    <td>23.38h·A100<sup>+0.7%</sup></td>
    <td>17.17h·A100</td>
    <td>17.25h·A100<sup>+0.5%</sup></td>
  </tr>
</table>

Table 5: A100 GPU time consumption for training DPO, \( \gamma \)-DPO, SimPO, and \( \gamma \)-SimPO across all base models.

![Visualization of dynamic target margin (\( \gamma_i \)) with reward gaps. The horizontal line indicates the initial value of target margin (\( \gamma_0 \)).](page_180_370_1087_246.png)

Figure 5: Visualization of dynamic target margin (\( \gamma_i \)) with reward gaps. The horizontal line indicates the initial value of target margin (\( \gamma_0 \)).

![Performance metrics across different \( \tau \) settings for the Gemma model, showing Length-Controlled Winning Rate (AE2:LC) and Raw Winning Rate (AE2:WR) from AlpacaEval2, and Raw Winning Rate (AH:WR) from Arena Hard.](page_180_624_1087_246.png)

Figure 6: Performance metrics across different \( \tau \) settings for the Gemma model, showing Length-Controlled Winning Rate (AE2:LC) and Raw Winning Rate (AE2:WR) from AlpacaEval2, and Raw Winning Rate (AH:WR) from Arena Hard.

5.2 Direct Preference Optimization

Recent studies introduce several enhancements to DPO, including IPO (Azar et al., 2024a), KTO (Ethayarajh et al., 2024), CPO (Xu et al., 2024), ORPO (Hong et al., 2024), R-DPO (Park et al., 2024), \( \beta \)-DPO (Wu et al., 2024), SimPO (Meng et al., 2024), \( \alpha \)-DPO (Wu et al., 2025a), WPO (Zhou et al., 2024), and RainbowPO (Zhao et al., 2024a). In addition, Dr. DPO (Wu et al., 2025b) uses Distributionally Robust Optimization to handle noisy data and align human preferences. Huang et al. (2025) introduce the alignment potential metric \( M_{AP} \) to optimize preference data selection for LLM alignment.

6 Conclusion

In this work, we introduce a plug-and-play approach called \( \gamma \)-PO, which is applicable to DPO and its variants. The key idea behind \( \gamma \)-PO is personalizing the target margin based on the reward margin, increasing them for high-confidence pairs to prioritize learning from these instances while reducing them for low-reward margin pairs to mitigate the impact of uncertain preference, which improves the handling of ambiguous preference data and reduces the risk of noisy data distorting the alignment policy. Furthermore, \( \gamma \)-PO offers a simple and efficient solution with minimal impact on training efficiency, making it a valuable tool for improving alignment in practical applications. Extensive experiments demonstrate that the \( \gamma \)-PO enhances the alignment performance, achieving robust results on AlpacaEval2 and Arena-Hard. Our findings highlight the potential of dynamic margin strategies as a promising direction for improving the alignment of LLMs, particularly in the presence of noisy data. There is still substantial opportunity for deeper exploration of this approach to enhance model alignment in real-world settings.
Limitations

Although the \( \gamma \)-PO method demonstrates significant improvements in LLMs alignment, it introduces several limitations that warrant further investigation and refinement in future work.

Additional Hyperparameter. The \( \gamma \)-PO method, while enhancing performance, introduces an additional hyperparameter, \( \tau \), which significantly impacts model alignment. Future efforts will focus on eliminating this parameter, aiming to compute the adaptive margin automatically using existing parameters.

Limited Evaluation. In our experiments, we used two evaluation benchmarks: AlpacaEval2 (Li and Zhang, 2023) and Arena-Hard (Li et al., 2024). Although these benchmarks are widely used in the field (Wu et al., 2025a; Meng et al., 2024; Wu et al., 2024), we observed that their results are influenced by the choice of parameters (\emph{e.g.}, the version of ‘alpaca\_eval’). In the future, incorporating a broader range of LLMs evaluation metrics and developing benchmarks that are more robust to parameter variations could provide a more comprehensive assessment of model performance.

7 Ethical Considerations

While the UltraFeedback dataset (Cui et al., 2024) provides high-quality preference and textual feedback, covering aspects such as instruction following, truthfulness, honesty, and helpfulness, it still faces challenges in addressing the safety of LLMs. This includes mitigating risks such as toxicity, stereotype bias, adversarial and out-of-distribution robustness, privacy, machine ethics, and fairness (Wang et al., 2023). Future alignment efforts will require the integration of larger and more diverse training datasets to comprehensively address these issues (Ji et al., 2023; Zhao et al., 2024b).

ACKNOWLEDGEMENT

This work was supported by Ant Group Research Fund.

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Chunting Zhou, Pengfei Liu, Puxin Xu, Sriniwasan Iyer, Jiao Sun, Yuning Mao, Xuezhe Ma, Avia Efrat, Ping Yu, Lili Yu, Susan Zhang, Gargi Ghosh, Mike Lewis, Luke Zettlemoyer, and Omer Levy. 2023. LIMA: less is more for alignment. In Advances in Neural Information Processing Systems 36, New Orleans, LA.

Wenxuan Zhou, Ravi Agrawal, Shujian Zhang, Sathish Reddy Indurthi, Sanqiang Zhao, Kaiqiang Song, Silei Xu, and Chenguang Zhu. 2024. WPO: enhancing RLHF with weighted preference optimization. In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pages 11170–11189, Miami, FL.

Daniel M. Ziegler, Nisan Stiennon, Jeffrey Wu, Tom B. Brown, Alec Radford, Dario Amodei, Paul F. Christiano, and Geoffrey Irving. 2019. Fine-tuning language models from human preferences. arXiv preprint, abs/1909.08593.
A  Proof of Theorem 3.1

Theorem 3.1. Consider the rDPO loss function defined as
\[
\mathcal{L}_{rDPO} = -\varepsilon \log \sigma(r_l - r_w - \gamma_0) \\
-(1-\varepsilon) \log \sigma(r_w - r_l - \gamma_0),
\]
where \( \sigma(x) = \frac{1}{1+e^{-x}} \) is the sigmoid function, and \( \varepsilon \) is a smoothing parameter.

If we set \( \mathcal{L}_{rDPO} = \mathcal{L}_{\gamma\text{-PO}} \) for simplicity, and denote \( \delta = \gamma_i - \gamma_0 \) and \( m = r_w - r_l \), then under the condition \( |\delta| \ll |m| \), the approximate solution for \( \varepsilon \) is given by
\[
\varepsilon \approx \frac{\delta [1 - \sigma(m - \gamma_0)]}{\log (1 + e^{m+\gamma_0}) - \log (1 + e^{\gamma_0-m})}.
\]

Proof. Starting with the given equation:
\[
- \log \sigma(m - \gamma_0 - \delta) \\
= -\varepsilon \log \sigma(-m - \gamma_0) - (1-\varepsilon) \log \sigma(m - \gamma_0),
\]
using the definition of the sigmoid function \( \sigma(x) = \frac{1}{1+e^{-x}} \) and \( -\log \frac{1}{x} = \log x \), we rewrite the equation as:
\[
\log \left( 1 + e^{-(m-\gamma_0-\delta)} \right) = \varepsilon \log \left( 1 + e^{-(m-\gamma_0)} \right) \\
+(1-\varepsilon) \log \left( 1 + e^{-(m-\gamma_0)} \right).
\]
Assuming that \( \delta \) is small, we apply a first-order Taylor expansion around \( \delta = 0 \) to approximate the left-hand side:
\[
\log \left( 1 + e^{-(m-\gamma_0)} \right) + \delta \cdot \frac{e^{-(m-\gamma_0)}}{1 + e^{-(m-\gamma_0)}} \\
\approx (1-\varepsilon) \log \left( 1 + e^{\gamma_0-m} \right) + \varepsilon \log \left( 1 + e^{m+\gamma_0} \right).
\]
Rearranging the terms to solve for \( \delta \), we get:
\[
\delta \cdot \frac{e^{\gamma_0-m}}{1 + e^{\gamma_0-m}} \\
\approx \varepsilon \left[ \log \left( 1 + e^{m+\gamma_0} \right) - \log \left( 1 + e^{\gamma_0-m} \right) \right].
\]
Recognizing that \( \frac{e^{\gamma_0-m}}{1+e^{\gamma_0-m}} = \sigma(\gamma_0 - m) = 1 - \sigma(m - \gamma_0) \), we express \( \delta \) as:
\[
\varepsilon \approx \frac{\delta [1 - \sigma(m - \gamma_0)]}{\log (1 + e^{m+\gamma_0}) - \log (1 + e^{\gamma_0-m})}.
\]

In addition, as training approaches convergence, as shown in Figure 5, most of the values of \( \gamma_i \) lie near \( \gamma_0 \), while \( \delta \) remains small under the control of \( \tau \), thereby satisfying the assumption \( |\delta| \ll |m| \).

B  Experiment Details

Training hyperparameters. We observed that the alignment performance of LLMs is influenced by the choice of hyperparameters. To ensure a fair comparison, we conducted a hyperparameter search for all methods, with the search range detailed in Table 6. For the remaining hyperparameters, the learning rate was selected from the range [3e-7, 5e-7, 6e-7, 8e-7, 1e-6]. A consistent batch size of 128 was used across all methods. All models were trained for one epoch using a cosine learning rate schedule, which included a 10% warm-up phase. The Adam optimizer was employed (Kingma and Ba, 2015). Additionally, the maximum sequence length was set to 2048.

Hyperparameter in \( \gamma \)-PO. Table 7 shows the hyperparameters of our method.

Decoding hyperparameters. The decoding hyperparameters are consistent with those used in SimPO*. We would like to express our sincere gratitude to the SimPO team for generously sharing their valuable insights.

Computation environment. All training experiments in this paper were conducted using 8×A100 GPUs, following the procedures outlined in the alignment-handbook repository*.

C  Algorithm for calculating \( \gamma \)

Due to GPU memory constraints, the batch size is relatively small, which may introduce sampling bias, leading to significant variations in reward margins across different batches. To mitigate this issue, we introduce an additional queue to reduce bias. Specifically, we implement a First-In-First-Out (FIFO) reward margin queue, denoted as \( \mathcal{H} \). This mechanism maintains an updated record of recent reward margins, and incorporating more samples from \( \mathcal{H} \) helps better approximate the true distribution of reward margins. The revised pseudocode is provided in Algorithm 2.

For optimizing \( \gamma_i \), we apply Mirror Descent and the Multiplicative Weight Update Method to efficiently compute \( \gamma \). The optimization problem for \( \gamma \) is formulated as follows:
\[
p = \arg \min_p (-\mathbb{E}_\mathcal{D} [\log \sigma(m - \gamma_i) \\
+ \tau \mathbb{D}_{KL}(p||p_0)]),
\]
where \( m \) represents the reward margins, \( \gamma_i = m_n p_i \gamma_0 \), \( \sigma(x) \) is the sigmoid function, \( p \) is the
<table>
  <tr>
    <th>Method</th>
    <th>Objective</th>
    <th>Hyperparameter</th>
  </tr>
  <tr>
    <td>DPO (Rafailov et al., 2023)</td>
    <td>\( -\log \sigma \left( \beta \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} - \beta \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)} \right) \)</td>
    <td>\( \beta \in [0.01, 0.05, 0.1] \)</td>
  </tr>
  <tr>
    <td>IPO (Azar et al., 2024a)</td>
    <td>\( \left( \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} - \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)} \right)^2 \)</td>
    <td>\( \tau \in [0.01, 0.1, 0.5, 1.0] \)</td>
  </tr>
  <tr>
    <td>CPO (Xu et al., 2024)</td>
    <td>\( -\log \sigma \left( \beta \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} - \beta \log \pi_\theta(y_l|x) \right) - \lambda \log \pi_\theta(y_w|x) \)</td>
    <td>\( \alpha = 1.0,\ \beta \in [0.01, 0.05, 0.1] \)</td>
  </tr>
  <tr>
    <td>KTO (Ethayarajh et al., 2024)</td>
    <td>\( -\lambda_0 \sigma \left( \beta \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} - \pi_{\text{ref}} \right) + \lambda_1 \sigma \left( \pi_{\text{ref}} - \beta \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)} \right) \),<br>where \( \pi_{\text{ref}} = \mathbb{E}_{(x,y)\sim D} [\text{KL}(\pi_\theta(y|x)||\pi_{\text{ref}}(y|x))] \)</td>
    <td>\( \lambda_0 = \lambda_1 = 1.0 \)<br>\( \beta \in [0.01, 0.05, 0.1] \)</td>
  </tr>
  <tr>
    <td>ORPO (Hong et al., 2024)</td>
    <td>\( -\log p_\theta(y_w|x) - \lambda \log \sigma \left( \log \frac{p_\theta(y_w|x)}{1-p_\theta(y_w|x)} - \log \frac{p_\theta(y_l|x)}{1-p_\theta(y_l|x)} \right) \),<br>where \( p_\theta(y|x) = \exp \left( \frac{1}{\beta} \log \pi_\theta(y|x) \right) \)</td>
    <td>\( \lambda \in [0.1, 0.5, 1.0, 2.0] \)</td>
  </tr>
  <tr>
    <td>R-DPO (Park et al., 2024)</td>
    <td>\( -\log \sigma \left( \beta \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} - \beta \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)} - (\alpha|y_w| - \alpha|y_l|) \right) \)</td>
    <td>\( \alpha \in [0.05, 0.1, 0.5, 1.0] \)<br>\( \beta \in [0.01, 0.05, 0.1] \)</td>
  </tr>
  <tr>
    <td>rDPO (Chowdhury et al., 2024)</td>
    <td>\( -(1-\varepsilon) \log \sigma \left( r_g - \varepsilon \log \sigma (-r_g) \right) \)</td>
    <td>\( \beta \in [0.01, 0.05, 0.1] \)<br>\( \varepsilon \in [0.01, 0.05, 0.1] \)</td>
  </tr>
  <tr>
    <td>SimPO (Meng et al., 2024)</td>
    <td>\( -\log \sigma \left( \frac{\beta}{|\gamma_0|} \log \pi_\theta(y_w|x) - \frac{\beta}{|\gamma_1|} \log \pi_\theta(y_l|x) - \gamma \right) \)</td>
    <td>\( \beta \in [2.5, 5.0, 10.0] \)<br>\( \gamma \in [0.3, 0.4, 0.5, 1.0] \)</td>
  </tr>
  <tr>
    <td>\( \beta \)-DPO (Wu et al., 2024)</td>
    <td>\( -\log \sigma \left( \beta_i \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} - \beta_i \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)} \right) \)</td>
    <td>\( \beta \in [2.5, 10.0],\ \gamma \in [0.1, 0.3, 0.5] \)<br>\( M = \beta_0 \log \frac{\pi_\theta(y_w|x)}{\pi_{\text{ref}}(y_w|x)} - \beta_0 \log \frac{\pi_\theta(y_l|x)}{\pi_{\text{ref}}(y_l|x)},\ \alpha \in [1e-2, 5e-2, 0.1, 0.2] \)</td>
  </tr>
</table>

Table 6: Various preference optimization objectives and hyperparameter search range.

<table>
  <tr>
    <th>Setting</th>
    <th>\( \beta \)</th>
    <th>\( \gamma \)</th>
    <th>\( \tau \)</th>
    <th>Learning rate</th>
  </tr>
  <tr>
    <td>Llama3-Instruct</td>
    <td>10.0</td>
    <td>0.4</td>
    <td>10.0</td>
    <td>1e-6</td>
  </tr>
  <tr>
    <td>Mistral-Instruct</td>
    <td>2.5</td>
    <td>0.15</td>
    <td>10.0</td>
    <td>6e-7</td>
  </tr>
  <tr>
    <td>Gemma2-Instruct</td>
    <td>10.0</td>
    <td>0.5</td>
    <td>20.0</td>
    <td>8e-7</td>
  </tr>
  <tr>
    <td>Qwen2.5-Instruct</td>
    <td>10.0</td>
    <td>0.3</td>
    <td>3.0</td>
    <td>1e-6</td>
  </tr>
</table>

Table 7: The hyperparameter values in \( \gamma \)-PO used for each training setting.

probability distribution over the reward gaps, \( \gamma_0 \) is the initial margin value, \( p_0 \) is the uniform distribution (used as the target), and \( \tau \) is a regularization parameter controlling the KL divergence term. For simplicity, We denote \( L = -\mathbb{E}_D[\log \sigma(m - \gamma_i) + \tau \mathbb{D}_{\text{KL}}(p||p_0)] \). The gradients of the loss with respect to \( p \) are computed as:

\[
\frac{\partial L}{\partial p_i} = \frac{\gamma_0}{1 + e^{m - m_n p_i \gamma_0}} + \tau (1 + \log(m_n p_i)),
\]

where \( m_n \) is the number of reward margin \( m \). These gradients are then used to iteratively update \( p \) using a multiplicative update rule:

\[
p_i \leftarrow p_i \cdot \frac{\exp(-\eta \cdot \text{grad}(p_i))}{\sum_i \exp(-\eta \cdot \text{grad}(p_i))},
\]

where \( \eta \) is the learning rate. Furthermore, we derive the second-order gradients with respect to \( p \) as:

\[
\frac{\partial^2 L}{\partial p_i^2} = \frac{m_n \gamma_0^2 e^{m - m_n p_i \gamma_0}}{(1 + e^{m - m_n p_i \gamma_0})^2} + \frac{\tau}{p_i} > 0,
\]

which indicates that it is a convex optimization problem and that an optimal solution for \( p \) exists.

Algorithm 2 \( \gamma \)-PO Algorithm

1: Input: SFT model \( \pi_{\text{SFT}} \), dataset \( \mathcal{D} \), total number of iterations \( T \), learning rate \( \alpha \).
2: Initial policy model \( \pi_\theta = \pi_{\text{SFT}} \), empty FIFO queue \( \mathcal{H} \).
3: for \( t = 0 \) to \( T \) do
4:   Sample a batch of tuples \( (x, y_w, y_l) \) from \( \mathcal{D} \).
5:   Sample a reward pair \( (r_w, r_l) \) from \( \mathcal{H} \).
6:   Calculate the rewards pair \( (r_w, r_l) \) via Table 1 and store them in \( \mathcal{H} \).
7:   Compute the adaptive margin \( \gamma_i \) using Equation (9) with reward pairs \( (r_w, r_l) \) both from \( \mathcal{H} \) current batch.
8:   Extract the adaptive margin \( \gamma_i \) from the current batch, calculate the loss \( \mathcal{L}_{\gamma\text{-PO}} \) using Equation (10),
9:   Update the policy parameters \( \theta \) for \( \pi_\theta \) via gradient descent: \( \theta_{t+1} = \theta_t - \alpha \nabla_{\theta_t} \mathcal{L}_{\gamma\text{-PO}} \).
10: end for

In our implementation, we set \( \eta = \min \left( \frac{0.5}{\tau}, 1 \right) \) and \( m_n = 256 \). The optimization process is repeated \( T = 20 \) times, which allows for sufficient convergence.
<table>
  <tr>
    <th>Instruction:</th>
    <td>Hi, I’m trying to solve a crossword puzzle, but I’ve never done one of these before. Can you help me out?</td>
  </tr>
  <tr>
    <th>DPO output:</th>
    <td>I’d love to help you with your crossword puzzle! Here’s how we can break it down to make it easier:\n\n**1. Tell me the specifics:**\n* **Clues:** Share the **across** and **down** clues you’re stuck on. Be as specific as possible about the number of letters each answer should have (e.g., “7 letters across”).\n* **Theme (if any):** Some crosswords have a theme. Knowing it can give big hints.\n* **Intersection Help:** If you have some letters already filled in, let me know where they intersect with the clues you need help with. This can narrow down possibilities.\n* **2. I can offer:**\n* **Definitions and synonyms:** I can provide definitions or related words to help you decipher clues.\n* **Wordplay hints:** Crosswords often use puns, anagrams, or other wordplay. I can help spot those.\n* **Letter suggestions:** If you have partial answers, I might be able to suggest missing letters.\n\nLet’s get started! What’s your first clue?</td>
  </tr>
  <tr>
    <th>γ-SimPO output:</th>
    <td>Absolutely! I can definitely help you with your crossword puzzle. To give you the best assistance, I need a bit more information:* **Can you provide the clues, especially the ones you’re stuck on?** I’m good at understanding wordplay, synonyms, and different clue types.* **What’s the crossword’s theme (if any)?** Themes often give hints about interconnected clues.* **Are there any intersecting letters you already have filled in?** Those can be crucial for narrowing down possibilities. Once you provide these, I’ll do my best to help you crack the clues and solve the puzzle!</td>
  </tr>
  <tr>
    <th>GPT-4-Turbor(1106) output:</th>
    <td>Absolutely! I’d be delighted to help you with your crossword puzzle. Just tell me the clues and the number of letters needed for each answer (and any letters you may have already filled in), and I’ll do my best to help you find the solutions. If you have any specific questions about how to approach solving crossword puzzles in general, feel free to ask those as well!</td>
  </tr>
</table>

Table 8: Output example from Gemma base model.

D Outputs Example

We present sample outputs generated by the Gemma base model, as shown in Table 8.

E The artifacts

Base models. In this manuscript, we use several publicly available resources, including four base models (LLaMA-3-8B-Instruct (Dubey et al., 2024), Mistral-7B-Instruct (Jiang et al., 2023), Gemma-2-9B-Instruct (Rivière et al., 2024), and Qwen-2.5-7B-Instruct (Yang et al., 2024)), which are licensed under their respective terms for research and non-commercial use. Additionally, we utilize the Ultrafeedback Binarized dataset*, which is publicly available under the CC-BY-4.0 license for academic and research purposes. The reward model, RLHFFlow/ArmoRM-Llama3-8B-v0.1 *, is also publicly available on Hugging Face with terms permitting non-commercial use.

Evaluation. We evaluate our methods using two datasets—AlpacaEval2 (Dubois et al., 2024) and Arena-Hard (Li et al., 2024)—both of which are publicly available and licensed for research purposes. All assets used in this work are provided with clear attribution and comply with the licensing terms of the respective sources. Full license details and terms of use for each artifact can be found in the accompanying repository and documentation.

Dataset. The original UltraFeedback dataset contains 64k prompts, each accompanied by four model completions from various open and proprietary models. GPT-4 was then used to assign scores to each completion based on criteria such as usefulness and honesty. To create UltraFeedback Binarized, we selected the highest overall score as the "chosen" completion and randomly selected one of the remaining three as the "rejected" completion. This defines the preference modeling split

*HuggingFaceH/ultrafeedback_binarized
*RLHFFlow/ArmoRM-Llama3-8B-v0.1
<table>
  <tr>
    <th>Name</th>
    <th>Notation</th>
    <th>Description</th>
  </tr>
  <tr>
    <td>Prompt</td>
    <td>x</td>
    <td>Prompt is the input sequence passed to the model.</td>
  </tr>
  <tr>
    <td>Response</td>
    <td>y</td>
    <td>Response is the output sequence of the model.</td>
  </tr>
  <tr>
    <td>Preferred Response</td>
    <td>y_w</td>
    <td>The sample has a higher reward score or is preferred by humans.</td>
  </tr>
  <tr>
    <td>Non-preferred Response</td>
    <td>y_l</td>
    <td>The sample has a lower reward score or non-preferred by humans.</td>
  </tr>
  <tr>
    <td>Reward of Preferred Response</td>
    <td>r_w</td>
    <td>Reward score of \( y_w \).</td>
  </tr>
  <tr>
    <td>Reward of Less Preferred Response</td>
    <td>r_l</td>
    <td>Reward score of \( y_l \).</td>
  </tr>
  <tr>
    <td>Reward Margin</td>
    <td>m</td>
    <td>Reward margin between \( r_w \) and \( r_l \), i.e., \( m = r_w - r_l \).</td>
  </tr>
  <tr>
    <td>Margin Batchsize</td>
    <td>m_n</td>
    <td>The number of preference pairs in one batch.</td>
  </tr>
  <tr>
    <td>Initial Target Margin</td>
    <td>\( \gamma_0 \)</td>
    <td>The initial target margin for preference learning.</td>
  </tr>
  <tr>
    <td>Adaptive Target Margin</td>
    <td>\( \gamma_s \)</td>
    <td>The adaptive target margin for preference learning.</td>
  </tr>
  <tr>
    <td>Uniform-distribution</td>
    <td>p_0</td>
    <td>Uniform distribution of \( \gamma_0 \): \( p_i = 1 / m_n \).</td>
  </tr>
  <tr>
    <td>\( \gamma \)-distribution</td>
    <td>p_i</td>
    <td>The normalized distribution of \( \gamma_i \): \( p_i = \gamma_i^* / \sum_j \gamma_j^* \), where \( \gamma_i^* = \max(0, \gamma_i) \).</td>
  </tr>
  <tr>
    <td>Policy Model</td>
    <td>\( \pi_\theta \)</td>
    <td>The generative model accepts prompts and outputs response.</td>
  </tr>
  <tr>
    <td>Reference Model</td>
    <td>\( \pi_{ref} \)</td>
    <td>The generative model is employed as a reference to ensure minimal deviation from the policy model.</td>
  </tr>
  <tr>
    <td>Preference Dataset</td>
    <td>\( \mathcal{D} \)</td>
    <td>Dataset comprising a set of triples \( (x, y_w, y_l) \).</td>
  </tr>
  <tr>
    <td>Loss Function</td>
    <td>\( \mathcal{L} \)</td>
    <td>Loss function.</td>
  </tr>
  <tr>
    <td>Historical Reward Gap Queue</td>
    <td>\( \mathcal{H} \)</td>
    <td>The First-In-First-Out queue to maintain the recent reward gaps.</td>
  </tr>
  <tr>
    <td>Smoothing Parameter</td>
    <td>\( \varepsilon \)</td>
    <td>The initial margin for preference learning, also considered as the label flip rate.</td>
  </tr>
  <tr>
    <td>Dynamic Smoothing Parameter</td>
    <td>\( \varepsilon_i \)</td>
    <td>The smoothing parameter that adaptive to different samples.</td>
  </tr>
  <tr>
    <td>Regularization Hyper-parameters</td>
    <td>\( \tau \)</td>
    <td>Regularization Hyper-parameters control the deviation from initial margin.</td>
  </tr>
</table>

Table 9: Table of Terminology and Notation.

for techniques like reward modeling or DPO. We also created splits for supervised fine-tuning (SFT), using the "chosen" column as the conversation to be modeled, and for generations (such as rejection sampling or PPO). For detailed information on dataset processing, please refer to the accompanying scripts.

F More Evaluation

We also evaluation our methods on MT-Bench (Zheng et al., 2023a), results are shown in Table 10.
<table>
  <tr>
    <th>Method</th>
    <th>Llama3-Instruct (8B)</th>
    <th>Mistral-Instruct (7B)</th>
    <th>Gemma2-Instruct (9B)</th>
    <th>Qwen2.5-Instruct (7B)</th>
    <th>Avg. (rank)</th>
  </tr>
  <tr>
    <td>SFT</td>
    <td>8.1</td>
    <td>7.5</td>
    <td>8.4</td>
    <td>8.5</td>
    <td>8.13(11)</td>
  </tr>
  <tr>
    <td>IPO</td>
    <td>9.1</td>
    <td>8.4</td>
    <td>9.2</td>
    <td>8.8</td>
    <td>8.88(8)</td>
  </tr>
  <tr>
    <td>KTO</td>
    <td>9.1</td>
    <td>7.9</td>
    <td>9.2</td>
    <td>8.9</td>
    <td>8.78(10)</td>
  </tr>
  <tr>
    <td>CPO</td>
    <td>8.9</td>
    <td>8.8</td>
    <td>8.8</td>
    <td>8.9</td>
    <td>8.85(9)</td>
  </tr>
  <tr>
    <td>ORPO</td>
    <td>8.8</td>
    <td><b>9.1</b></td>
    <td>8.9</td>
    <td>8.9</td>
    <td>8.93(7)</td>
  </tr>
  <tr>
    <td>R-DPO</td>
    <td>9.0</td>
    <td>8.8</td>
    <td>9.2</td>
    <td>9.0</td>
    <td>9.00(5)</td>
  </tr>
  <tr>
    <td>DPO</td>
    <td>9.1</td>
    <td>8.9</td>
    <td>9.2</td>
    <td>9.0</td>
    <td>9.05(3)</td>
  </tr>
  <tr>
    <td>\( \beta \)-DPO</td>
    <td>9.2</td>
    <td>8.7</td>
    <td><b>9.3</b></td>
    <td>9.0</td>
    <td>9.05(3)</td>
  </tr>
  <tr>
    <td>SimPO</td>
    <td>9.0</td>
    <td>8.9</td>
    <td>9.0</td>
    <td>9.0</td>
    <td>8.98(6)</td>
  </tr>
  <tr>
    <td>\( \gamma \)-DPO</td>
    <td>9.2</td>
    <td>8.8</td>
    <td><b>9.3</b></td>
    <td><b>9.1</b></td>
    <td>9.10(2)</td>
  </tr>
  <tr>
    <td>\( \gamma \)-SimPO</td>
    <td><b>9.3</b></td>
    <td><b>9.1</b></td>
    <td>9.2</td>
    <td><b>9.1</b></td>
    <td><b>9.18(1)</b></td>
  </tr>
  <tr>
    <td>Improve</td>
    <td>+1.1%</td>
    <td>+0.0%</td>
    <td>+0.0%</td>
    <td>+1.1%</td>
    <td>+1.4%</td>
  </tr>
</table>

Table 10: The Win Rate from MT-Bench (Zheng et al., 2023a) across different methods. **Bold** indicates the best performance for each metric, while underlined values represent the best performance excluding our methods, *i.e.*, \( \gamma \)-DPO and \( \gamma \)-SimPO. “Improve” denotes the percentage improvement of the bold value over the underlined one. This metric is considered better when it has a higher value.

Reviewer paper summaries: The paper introduces γ-PO to deal with noisy data in DPO. γ-PO dynamically adjusts reward margins for preference pairs, prioritizing high-confidence ones while reducing noise from ambiguous pairs. It is compatible with existing DPO-based methods. And it improves model performance by 4.4% on AlpacaEval2 and Arena-Hard.

The paper introduces γ-PO, a dynamic target margin preference optimization algorithm designed to improve the alignment of Large Language Models with human preferences. Unlike standard Direct Preference Optimization, which uses a fixed margin, γ-PO adaptively adjusts reward margins at the pairwise level, prioritizing high-confidence samples while suppressing noise from ambiguous pairs.
Summary of strengths: 1. Addresses ambiguity in pair-wise samples: γ-PO dynamically adjusts margins to handle noisy or uncertain data, improving alignment.
2. Easy integration: It works with any DPO-based method, making it versatile.
3.Theoretical foundation: this paper connects γ-PO to dynamic label smoothing, supporting its approach with theoretical analysis.

1. The paper is easy to follow.

2. The paper provides a analysis that connects γ-PO to dynamic label smoothing is interesting.

3. Extensive experiments on multiple base models (LLaMA-3, Mistral, Gemma, Qwen) and benchmarks show consistent performance gains, validating γ-PO’s effectiveness.
Summary of weaknesses: 1.  This paper introduces an extra hyperparameter, τ, which impacts performance. Future work could automate this.
2. Limited evaluation: The paper uses only two benchmarks, limiting the assessment of the method's generalizability.
3. The method requires high-quality re-annotated datasets, which may not always be available.

1. The experiments only evaluated on AlpacaEval and Arena, what about including more widely used LLM leaderboards (e.g., Open LLM Leaderboard v1/v2 or MT-Bench)? Moreover, testing on real-world noisy datasets would better validate robustness.

2. May be you should discuss the paper's relation with [1]

[1] α-DPO: Adaptive Reward Margin is What Direct Preference Optimization Needs
Comments, suggestions and typos: 1. More evaluation: testing on more diverse datasets and models would strengthen the claims.
2. Hyperparameter ablation: provide more details on the impact of τ on performance.

I wonder what would happen if a different prior distribution (other than uniform) were chosen. Would the choice of distribution have an impact? Could it affect the connection to dynamic label smoothing?
Overall assessment scores: 3.5

3
Metareview score: 3
Metareview: This paper introduces $\gamma$-PO, a DPO/SimPO based method that learns the target margin in a dynamic manner. During the discussion phase, all reviewers agree with the effectiveness and theoretical foundation of the proposed method.  However, most reviewers identify the lack of sufficient experiments as an important weakness aspect. For instance, reviewer gQiG and AnM5 pointed out limited benchmark evaluations on two benchmarks, where real-world evaluations are expected. Reviewer qkH7 pointed out the lack of sufficient discussion on hyperparameters. Reviewer oRxM is not a domain-expert and is therefore not taken into account. Moreover, the reliance on the high-quality re-annotated data is considered as a limitation of the proposed method (Reviewer AnM5). The meta-review acknowledge the supplemented experiments of MT-bench and more discussion on the hyperparameters and standard deviation during the rebuttal phase. Considering the current stage of the work, the paper is recommended as a Findings paper for publication.
Summary of reasons to publish: 1. Clear and intuitive presentation.
2. Theoretical analysis and experimental validation.
3. Versatile integration with DPO-based methods.