Metadata-Free Meta-Reweighted Direct Preference Optimization under Noisy Preference Labels

# Metadata-Free Meta-Reweighted Direct Preference Optimization under Noisy Preference Labels
Source: [https://arxiv.org/html/2607.09796](https://arxiv.org/html/2607.09796)
Yifan Li Xi’an Jiaotong University 4123158005@stu\.xjtu\.edu\.cnXiaodong Yuan Xi’an Jiaotong University xiaodongyuan@163\.com

###### Abstract

Direct Preference Optimization \(DPO\) has become an important method for aligning large language models \(LLMs\) with human preferences because it removes the need for explicit reward modeling and reinforcement learning optimization\. However, its performance depends heavily on the quality of preference data, and noisy preference data in real\-world settings can weaken alignment performance\. To address this issue, we propose a bilevel optimization framework and prove, under certain assumptions, that this framework can recover the DPO optimum under clean data\. We further derive a prior form for the learnable weighting function under asymmetric label\-flipping noise\. Considering that high\-quality metadata may be difficult to obtain, we propose a task\-agnostic meta\-knowledge\-driven method that enables meta\-learning even when metadata is completely unavailable\. To reduce the high cost of higher\-order gradients in LLM meta\-learning, we combine central\-difference approximation with LoRA fine\-tuning and develop a scalable training scheme\. Experiments on TL;DR summarization and Anthropic HH single\-turn dialogue show that the proposed method improves training performance over multiple DPO baselines under different noise rates\.

*Keywords*Direct Preference Optimization⋅\\cdotpreference noise⋅\\cdotmeta\-learning⋅\\cdotprompt augmentation consistency⋅\\cdotfinite difference⋅\\cdotLoRA fine\-tuning

## 1Introduction

Aligning large language models \(LLMs\) with human preferences is important\[[1](https://arxiv.org/html/2607.09796#bib.bib1),[2](https://arxiv.org/html/2607.09796#bib.bib2),[3](https://arxiv.org/html/2607.09796#bib.bib3)\]\. Reinforcement learning from human feedback \(RLHF\) has been shown to be effective for aligning LLMs with human preferences\[[4](https://arxiv.org/html/2607.09796#bib.bib4)\]\. A typical RLHF pipeline first obtains an SFT policy through supervised fine\-tuning, then learns a reward model \(RM\) from human preference data, and finally optimizes the policy with reinforcement learning algorithms such as PPO while constraining its distance from a reference model to avoid excessive policy drift\.

Although RLHF plays an important role in LLM alignment, its training pipeline is relatively complex\. It requires training an additional reward model and policy model, and the reinforcement learning stage involves frequent sampling, which leads to high computational and memory costs\[[5](https://arxiv.org/html/2607.09796#bib.bib5),[6](https://arxiv.org/html/2607.09796#bib.bib6)\]\. Direct Preference Optimization \(DPO\) provides a simpler alternative for preference alignment: it directly optimizes the LLM from human preference data without explicitly learning a reward model and avoids the complexity of reinforcement learning\[[7](https://arxiv.org/html/2607.09796#bib.bib7)\]\.

Despite greatly simplifying the traditional RLHF pipeline, DPO still depends heavily on the quality of preference data\. Prior studies have shown that preference\-data noise can degrade both the stability and final performance of alignment training\[[8](https://arxiv.org/html/2607.09796#bib.bib8),[9](https://arxiv.org/html/2607.09796#bib.bib9)\]\. Therefore, effective DPO learning under noisy preference data has become an important problem in preference alignment\.

Existing work has mainly addressed this problem by improving the noise robustness of DPO through loss correction, label smoothing, distributionally robust optimization, or noisy\-sample filtering\[[10](https://arxiv.org/html/2607.09796#bib.bib10),[11](https://arxiv.org/html/2607.09796#bib.bib11),[12](https://arxiv.org/html/2607.09796#bib.bib12),[13](https://arxiv.org/html/2607.09796#bib.bib13),[14](https://arxiv.org/html/2607.09796#bib.bib14),[15](https://arxiv.org/html/2607.09796#bib.bib15)\]\. However, these methods often rely on predefined noise forms, fixed reweighting rules, or heuristic reliability estimates, which makes it difficult to simultaneously maintain interpretability and training effectiveness under complex noise conditions\.

Recently, several studies have introduced meta\-learning into preference optimization to adaptively characterize noisy preference samples and learn sample\-level weights\[[16](https://arxiv.org/html/2607.09796#bib.bib16),[17](https://arxiv.org/html/2607.09796#bib.bib17)\]\. These methods show that meta\-learning can provide a more flexible mechanism for weighting noisy preference samples than fixed heuristic rules\.

However, existing meta\-learning\-based weighted preference optimization methods mainly learn sample weights from training dynamics, model diagnostic signals, or validation\-set performance\. They do not systematically explain, from the perspective of the discrepancy between noisy and clean conditional preference posteriors, why a bilevel weighting mechanism can correct the optimal\-solution shift caused by noisy preferences in DPO\. Moreover, these methods usually still require a validation set with relatively reliable preference labels or a small amount of clean metadata to provide the outer\-level optimization signal\. In real LLM alignment scenarios, constructing high\-quality meta\-preference data often requires additional annotation, model review, or manual cleaning, which can be costly or even unavailable\.

To this end, we revisit DPO learning under noisy preferences from the perspectives of conditional risk and posterior shift, and propose a prompt\-augmentation\-consistency meta\-reweighted DPO method that does not rely on clean meta\-preference labels, termed PACMR\-DPO\. The contributions of this paper are summarized as follows:

- •We prove that, under idealized conditions, bilevel optimization can recover the DPO optimum under clean data from noisy training data, thereby providing a theoretical basis for learnable weight parameterization\.
- •We propose a prompt\-augmentation\-consistency DPO method that does not require clean metadata\. Instead of relying on clean meta\-preference labels, the method injects task\-agnostic prompt augmentation consistency as meta\-knowledge into the outer\-level optimization\.
- •To address the high cost of higher\-order gradients in LLM meta\-learning, we propose a scalable training scheme that combines central\-difference approximation with perturbations in the LoRA parameter space, reducing the memory and computation overhead of bilevel optimization\.
- •We evaluate the proposed method on TL;DR summarization and Anthropic HH single\-turn dialogue, and compare it with multiple DPO baselines under different random\-flip noise rates\. Experiments show that PACMR\-DPO achieves more pronounced improvements under medium and high noise rates\.

## 2Related Work

### 2\.1Direct Preference Optimization

RLHF has become an important approach for aligning LLMs with human preferences\[[4](https://arxiv.org/html/2607.09796#bib.bib4)\], but its training process usually consists of SFT, RM learning, and reinforcement\-learning fine\-tuning, making it complex and computationally demanding\. Against this background, DPO provides a simpler alternative to RLHF\[[7](https://arxiv.org/html/2607.09796#bib.bib7)\]\. For preference data\(x,ya,yb\)\(x,y\_\{a\},y\_\{b\}\), the DPO loss is defined as

ℒDPO​\(ω\)=−𝔼\(x,ya,yb\)∼Dtrain​\[log⁡σ​\(β​log⁡πω​\(ya∣x\)πref​\(ya∣x\)−β​log⁡πω​\(yb∣x\)πref​\(yb∣x\)\)\],\\mathcal\{L\}\_\{\\mathrm\{DPO\}\}\(\\omega\)=\-\\mathbb\{E\}\_\{\(x,y\_\{a\},y\_\{b\}\)\\sim D\_\{\\mathrm\{train\}\}\}\\left\[\\log\\sigma\\left\(\\beta\\log\\frac\{\\pi\_\{\\omega\}\(y\_\{a\}\\mid x\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\_\{a\}\\mid x\)\}\-\\beta\\log\\frac\{\\pi\_\{\\omega\}\(y\_\{b\}\\mid x\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\_\{b\}\\mid x\)\}\\right\)\\right\],\(1\)whereσ​\(r\)=1/\(1\+exp⁡\(−r\)\)\\sigma\(r\)=1/\(1\+\\exp\(\-r\)\)is the sigmoid function andβ\>0\\beta\>0is the DPO temperature parameter\.

DPO avoids explicit reward modeling and reinforcement learning, thereby substantially reducing the training cost of preference alignment\.

### 2\.2Robust Preference Learning with Noisy Labels

Prior studies have shown that preference data used for LLM training can contain a non\-negligible amount of noise, and that alignment performance decreases as the noise rate increases\[[8](https://arxiv.org/html/2607.09796#bib.bib8)\]\. To address this issue, cDPO smooths the hard labels in DPO into soft labels using the noise rate, reducing the damage caused by incorrectly labeled preference samples\[[13](https://arxiv.org/html/2607.09796#bib.bib13)\]; rDPO constructs a debiased weighted loss to offset the effect of incorrect preference annotations on DPO training\[[10](https://arxiv.org/html/2607.09796#bib.bib10)\]; Dr\.DPO combines distributionally robust optimization with DPO to improve robustness to noisy data\[[11](https://arxiv.org/html/2607.09796#bib.bib11)\]; ROPO suppresses gradients from high\-uncertainty samples and alternates between noise\-tolerant training and noisy\-sample filtering to improve robustness\[[14](https://arxiv.org/html/2607.09796#bib.bib14)\]; R3M models corrupted preference labels as sparse outliers\[[12](https://arxiv.org/html/2607.09796#bib.bib12)\]; and PerpCorrect detects and corrects noisy labels through perplexity differences within preference pairs\[[15](https://arxiv.org/html/2607.09796#bib.bib15)\]\.

Unlike these methods, we model sample reliability in noisy preference learning as a learnable sample\-level weighting function, and use bilevel optimization to adaptively adjust the influence of training samples on LLM updates according to an outer\-level objective\.

### 2\.3Meta\-learning for Preference Reweighting

MetaPO argues that sample value is not determined solely by the current model state, but changes dynamically during training\. It therefore uses three temporal dynamic features \- reward\-margin evolution, learning fluctuation, and deviation from the reference model \- to learn time\-aware weights for each preference sample\[[17](https://arxiv.org/html/2607.09796#bib.bib17)\]\. Aligner, Diagnose Thyself argues that preference\-sample reliability cannot be fully captured by a single heuristic indicator\. It constructs diagnostic vectors from preference consistency, learning difficulty, and generation confidence, and uses a meta\-learning network to fuse these intrinsic feedback signals into sample\-level weights\[[16](https://arxiv.org/html/2607.09796#bib.bib16)\]\.

Unlike these methods, this paper does not primarily rely on empirical dynamic features or diagnostic signals to define sample reliability\. Instead, it starts from the posterior shift between noisy and clean preferences and analyzes why bilevel optimization can correct the optimal\-solution shift caused by noisy preferences in DPO\. Based on this analysis, we further design a prior for the learnable weighting function and replace the clean\-metadata\-dependent outer supervision objective with a prompt\-augmentation\-consistency objective\.

### 2\.4Meta\-learning Sample Reweighting under Noisy Supervision

In noisy\-label learning, meta\-learning\-based sample reweighting is an important class of methods\. Meta\-Weight\-Net \(MWN\) adaptively reduces the influence of noisy or difficult samples by learning an explicit mapping from sample loss to sample weight\[[18](https://arxiv.org/html/2607.09796#bib.bib18)\]\. CMW\-Net extends MWN by learning class\-aware sample\-weighting functions to explicitly model class\-dependent heterogeneous noise and imbalance\[[19](https://arxiv.org/html/2607.09796#bib.bib19)\]\. DAC\-MR addresses the difficulty of obtaining high\-quality metadata, which may be noisy or even unavailable, by introducing data augmentation consistency as an alternative meta\-objective and using task\-agnostic meta\-knowledge to compensate for traditional metadata\-driven meta\-learning\[[20](https://arxiv.org/html/2607.09796#bib.bib20)\]\.

We transfer this idea to DPO\-based preference optimization and construct a prompt augmentation consistency objective around preference pairs\.

## 3Problem Formulation

### 3\.1Noisy Pairwise Preference Learning

LetZZdenote the random variable taking values in𝒵\\mathcal\{Z\}, and letzzdenote a realization ofZZ\. We consider a preference samplez=\(x,ya,yb\)∈𝒵z=\(x,y\_\{a\},y\_\{b\}\)\\in\\mathcal\{Z\}, wherexxdenotes the prompt andyay\_\{a\}andyby\_\{b\}denote two candidate responses\. Let the latent clean preference label beY∈\{0,1\}Y\\in\\\{0,1\\\}, whereY=0Y=0meansya≻yby\_\{a\}\\succ y\_\{b\}andY=1Y=1meansya≺yby\_\{a\}\\prec y\_\{b\}\. In practice, the model often observes a noisy labelY~∈\{0,1\}\\widetilde\{Y\}\\in\\\{0,1\\\}rather than the clean label\.

To describe the sampling process for the training and metadata sets, we usem=0m=0to indicate that a sample is selected into the metadata set, andt=0t=0to indicate that a sample is selected into the training set\.

###### Assumption 1\(Label\-independent sampling\)\.

We assume that the probability of a sample entering the metadata set or training set is independent of the latent clean preference label, and that every sample has positive probability of being selected into both sets, namely

ℙ\(m=0∣Y=y,Z=z\)\\displaystyle\\mathbb\{P\}\(m=0\\mid Y=y,Z=z\)=ℙ​\(m=0∣Z=z\)\>0,∀z∈𝒵,y∈\{0,1\},\\displaystyle=\\mathbb\{P\}\(m=0\\mid Z=z\)\>0,\\quad\\forall z\\in\\mathcal\{Z\},\\ y\\in\\\{0,1\\\},\(2\)ℙ\(t=0∣Y=y,Z=z\)\\displaystyle\\mathbb\{P\}\(t=0\\mid Y=y,Z=z\)=ℙ​\(t=0∣Z=z\)\>0,∀z∈𝒵,y∈\{0,1\}\.\\displaystyle=\\mathbb\{P\}\(t=0\\mid Z=z\)\>0,\\quad\\forall z\\in\\mathcal\{Z\},\\ y\\in\\\{0,1\\\}\.\(3\)

Define

η​\(z\)\\displaystyle\\eta\(z\)=ℙ​\(Y=0∣Z=z\)∈\(0,1\),\\displaystyle=\\mathbb\{P\}\(Y=0\\mid Z=z\)\\in\(0,1\),\(4\)ηmeta​\(z\)\\displaystyle\\eta\_\{\\mathrm\{meta\}\}\(z\)=ℙ\(Y=0∣Z=z,m=0\)∈\(0,1\),\\displaystyle=\\mathbb\{P\}\(Y=0\\mid Z=z,m=0\)\\in\(0,1\),\(5\)ηtrain​\(z\)\\displaystyle\\eta\_\{\\mathrm\{train\}\}\(z\)=ℙ\(Y=0∣Z=z,t=0\)∈\(0,1\)\.\\displaystyle=\\mathbb\{P\}\(Y=0\\mid Z=z,t=0\)\\in\(0,1\)\.\(6\)Under[1](https://arxiv.org/html/2607.09796#Thmassumption1), the clean preference posterior is identical under the population distribution, training distribution, and meta distribution\.

###### Proposition 1\(Posterior invariance under label\-independent sampling\)\.

Under[1](https://arxiv.org/html/2607.09796#Thmassumption1), for anyz∈𝒵z\\in\\mathcal\{Z\}, we have

ηtrain​\(z\)=ηmeta​\(z\)=η​\(z\)\.\\eta\_\{\\mathrm\{train\}\}\(z\)=\\eta\_\{\\mathrm\{meta\}\}\(z\)=\\eta\(z\)\.\(7\)

###### Proof\.

By Bayes’ rule and[1](https://arxiv.org/html/2607.09796#Thmassumption1),

ηmeta​\(z\)\\displaystyle\\eta\_\{\\mathrm\{meta\}\}\(z\)=ℙ\(Y=0∣Z=z,m=0\)\\displaystyle=\\mathbb\{P\}\(Y=0\\mid Z=z,m=0\)\(8\)=ℙ\(m=0∣Y=0,Z=z\)ℙ\(Y=0∣Z=z\)∑y∈\{0,1\}ℙ\(m=0∣Y=y,Z=z\)ℙ\(Y=y∣Z=z\)\\displaystyle=\\frac\{\\mathbb\{P\}\(m=0\\mid Y=0,Z=z\)\\mathbb\{P\}\(Y=0\\mid Z=z\)\}\{\\sum\_\{y\\in\\\{0,1\\\}\}\\mathbb\{P\}\(m=0\\mid Y=y,Z=z\)\\mathbb\{P\}\(Y=y\\mid Z=z\)\}\(9\)=ℙ​\(m=0∣Z=z\)​η​\(z\)ℙ​\(m=0∣Z=z\)​η​\(z\)\+ℙ​\(m=0∣Z=z\)​\(1−η​\(z\)\)\\displaystyle=\\frac\{\\mathbb\{P\}\(m=0\\mid Z=z\)\\eta\(z\)\}\{\\mathbb\{P\}\(m=0\\mid Z=z\)\\eta\(z\)\+\\mathbb\{P\}\(m=0\\mid Z=z\)\(1\-\\eta\(z\)\)\}\(10\)=η​\(z\)\.\\displaystyle=\\eta\(z\)\.\(11\)Similarly, replacingmmwithttgivesηtrain​\(z\)=η​\(z\)\\eta\_\{\\mathrm\{train\}\}\(z\)=\\eta\(z\)\. Therefore,ηtrain​\(z\)=ηmeta​\(z\)=η​\(z\)\\eta\_\{\\mathrm\{train\}\}\(z\)=\\eta\_\{\\mathrm\{meta\}\}\(z\)=\\eta\(z\)\. ∎

We further consider a general noisy conditional probabilityq:𝒵→\(0,1\)q:\\mathcal\{Z\}\\to\(0,1\), where

q\(z\)=ℙ\(Y~=0∣Z=z,t=0\),q\(z\)=\\mathbb\{P\}\(\\widetilde\{Y\}=0\\mid Z=z,t=0\),\(12\)which denotes the conditional probability that samplezz, after being selected into the training set and processed by the noise mechanism, is observed with labelY~=0\\widetilde\{Y\}=0\. In preference learning,η​\(z\)\\eta\(z\)is the conditional preference distribution over sample pairzz, and both reward modeling and DPO can be viewed as estimating this conditional preference distribution\. Thus, DPO learning under noisy preference data is essentially policy fitting under a noise\-perturbed posterior distribution\.

### 3\.2DPO as Conditional Preference Risk

Let the implicit reward difference in DPO beu​\(z\)=β​\(log⁡π​\(ya∣x\)πref​\(ya∣x\)−log⁡π​\(yb∣x\)πref​\(yb∣x\)\)∈ℝu\(z\)=\\beta\(\\log\\frac\{\\pi\(y\_\{a\}\\mid x\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\_\{a\}\\mid x\)\}\-\\log\\frac\{\\pi\(y\_\{b\}\\mid x\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\_\{b\}\\mid x\)\}\)\\in\\mathbb\{R\}\. Under clean preference training data, the DPO conditional risk can be written as \(becauseηtrain​\(z\)=η​\(z\)\\eta\_\{\\mathrm\{train\}\}\(z\)=\\eta\(z\)\)

ℒDPO​\(u​\(z\),η​\(z\)\)=−η​\(z\)​log⁡σ​\(u​\(z\)\)−\(1−η​\(z\)\)​log⁡σ​\(−u​\(z\)\)\.\\mathcal\{L\}\_\{\\mathrm\{DPO\}\}\(u\(z\),\\eta\(z\)\)=\-\\eta\(z\)\\log\\sigma\(u\(z\)\)\-\(1\-\\eta\(z\)\)\\log\\sigma\(\-u\(z\)\)\.\(13\)The conditional risk is the pointwise risk obtained by fixing the sample pairzzand taking the expectation only over the conditional distribution of the latent preference labelYY\. Minimizing this risk yields the pointwise optimal implicit reward difference under the clean preference distribution,

uDPO​\-​clean∗​\(z\)=log⁡η​\(z\)1−η​\(z\)\.u^\{\*\}\_\{\\mathrm\{DPO\\text\{\-\}clean\}\}\(z\)=\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}\.\(14\)Correspondingly, under the noisy posteriorq​\(z\)q\(z\), the pointwise optimum of standard DPO becomes

uDPO​\-​noisy∗​\(z\)=log⁡q​\(z\)1−q​\(z\)\.u^\{\*\}\_\{\\mathrm\{DPO\\text\{\-\}noisy\}\}\(z\)=\\log\\frac\{q\(z\)\}\{1\-q\(z\)\}\.\(15\)This shows that whenq​\(z\)≠η​\(z\)q\(z\)\\neq\\eta\(z\), the implicit reward difference learned by standard DPO under the noisy distribution is shifted relative to that under the clean distribution\. This result is similar to ROPO\[[14](https://arxiv.org/html/2607.09796#bib.bib14)\]\. The meaning of the conditional risk and the derivation of the optimum are given in[appendixB](https://arxiv.org/html/2607.09796#A2)\.

### 3\.3Weighted DPO

To correct this shift, we introduce a sample\-dependent positive weighting functiong​\(z\)g\(z\)\. If the observed label isY~=0\\widetilde\{Y\}=0, we denote the weight asg\+​\(z\)g^\{\+\}\(z\); if the observed label isY~=1\\widetilde\{Y\}=1, we denote the weight asg−​\(z\)g^\{\-\}\(z\)\. The weighted DPO conditional risk is then

ℒW​\-​DPO​\(u​\(z\),q​\(z\),g\+​\(z\),g−​\(z\)\)=\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{W\\text\{\-\}DPO\}\}\(u\(z\),q\(z\),g^\{\+\}\(z\),g^\{\-\}\(z\)\)=−q​\(z\)​g\+​\(z\)​log⁡σ​\(u​\(z\)\)\\displaystyle\-q\(z\)g^\{\+\}\(z\)\\log\\sigma\(u\(z\)\)\(16\)−\(1−q​\(z\)\)​g−​\(z\)​log⁡σ​\(−u​\(z\)\)\.\\displaystyle\-\(1\-q\(z\)\)g^\{\-\}\(z\)\\log\\sigma\(\-u\(z\)\)\.\(17\)Its optimum is

uW​\-​DPO​\-​noisy∗​\(z\)=log⁡q​\(z\)1−q​\(z\)\+log⁡g\+​\(z\)g−​\(z\)\.u^\{\*\}\_\{\\mathrm\{W\\text\{\-\}DPO\\text\{\-\}noisy\}\}\(z\)=\\log\\frac\{q\(z\)\}\{1\-q\(z\)\}\+\\log\\frac\{g^\{\+\}\(z\)\}\{g^\{\-\}\(z\)\}\.\(18\)Therefore, if the weight ratiog\+​\(z\)/g−​\(z\)g^\{\+\}\(z\)/g^\{\-\}\(z\)is properly designed, it can offset the optimal\-solution shift induced by noise\.

## 4Theoretical Analysis

### 4\.1Clean\-Optimum Recovery by Bilevel Reweighting

Based on the above observation, we formulate noisy DPO as a bilevel optimization problem: the inner level learns a weighted DPO solution under the noisy training distribution, while the outer level constrains the weighting function through a meta\-objective\.

ming\+\>0,g−\>0\\displaystyle\\min\_\{g^\{\+\}\>0,g^\{\-\}\>0\}\\quad𝔼z′∼Pmeta\[−ηmeta\(z′\)logσ\(u∗\(g\+,g−,z′\)\)\\displaystyle\\mathbb\{E\}\_\{z^\{\\prime\}\\sim P\_\{\\mathrm\{meta\}\}\}\\left\[\-\\eta\_\{\\mathrm\{meta\}\}\(z^\{\\prime\}\)\\log\\sigma\(u^\{\*\}\(g^\{\+\},g^\{\-\},z^\{\\prime\}\)\)\\right\.\(19\)−\(1−ηmeta\(z′\)\)logσ\(−u∗\(g\+,g−,z′\)\)\]\\displaystyle\\left\.\\quad\-\(1\-\\eta\_\{\\mathrm\{meta\}\}\(z^\{\\prime\}\)\)\\log\\sigma\(\-u^\{\*\}\(g^\{\+\},g^\{\-\},z^\{\\prime\}\)\)\\right\]\(20\)s\.t\.\\displaystyle\\mathrm\{s\.t\.\}\\quadu∗\(g\+,g−,z\)=argminu𝔼z∼Ptrain\[−q\(z\)g\+\(z\)logσ\(u\(z\)\)\\displaystyle u^\{\*\}\(g^\{\+\},g^\{\-\},z\)=\\arg\\min\_\{u\}\\mathbb\{E\}\_\{z\\sim P\_\{\\mathrm\{train\}\}\}\\left\[\-q\(z\)g^\{\+\}\(z\)\\log\\sigma\(u\(z\)\)\\right\.\(21\)−\(1−q\(z\)\)g−\(z\)logσ\(−u\(z\)\)\]\.\\displaystyle\\left\.\\quad\-\(1\-q\(z\)\)g^\{\-\}\(z\)\\log\\sigma\(\-u\(z\)\)\\right\]\.\(22\)The following theorem shows that, under idealized conditions, the optimal weights obtained by this bilevel optimization can recover the pointwise optimum of clean DPO\.

###### Theorem 1\(Clean\-optimum recovery by ideal bilevel reweighting\)\.

Under[1](https://arxiv.org/html/2607.09796#Thmassumption1), let\(g∗,\+,g∗,−\)\(g^\{\*,\+\},g^\{\*,\-\}\)be an optimal solution to the bilevel optimization problem, and letu∗​\(g∗,\+,g∗,−,z\)u^\{\*\}\(g^\{\*,\+\},g^\{\*,\-\},z\)be the inner\-level pointwise optimal implicit reward difference induced by it\. Then

u∗​\(g∗,\+,g∗,−,z\)=uDPO​\-​clean∗​\(z\)=log⁡η​\(z\)1−η​\(z\)\.u^\{\*\}\(g^\{\*,\+\},g^\{\*,\-\},z\)=u^\{\*\}\_\{\\mathrm\{DPO\\text\{\-\}clean\}\}\(z\)=\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}\.\(23\)Equivalently, the log\-ratio of the optimal weights obtained by bilevel optimization satisfies

log⁡g∗,\+​\(z\)g∗,−​\(z\)=log⁡η​\(z\)1−η​\(z\)−log⁡q​\(z\)1−q​\(z\)\.\\log\\frac\{g^\{\*,\+\}\(z\)\}\{g^\{\*,\-\}\(z\)\}=\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}\-\\log\\frac\{q\(z\)\}\{1\-q\(z\)\}\.\(24\)

[theorem1](https://arxiv.org/html/2607.09796#Thmtheorem1)shows that, in the ideal case, the log\-ratio of the optimal weights obtained by bilevel optimization can cancel the noise\-induced shift\. The complete bilevel objective, the pointwise separability argument, and the proof are given in[appendixC](https://arxiv.org/html/2607.09796#A3)\.

### 4\.2Optimal Weight Structure under Asymmetric Label Noise

Although the noisy preference posteriorq​\(z\)q\(z\)and the clean preference posteriorη​\(z\)\\eta\(z\)are usually unknown, so that the corresponding optimal weighting function cannot be directly constructed, we can use the form of the theoretically optimal weight log\-ratio to design a general prior for the learnable weighting function\.

Consider general asymmetric label\-flipping noise,

q​\(z\)=\(1−ε0​\(z\)\)​η​\(z\)\+ε1​\(z\)​\(1−η​\(z\)\),q\(z\)=\(1\-\\varepsilon\_\{0\}\(z\)\)\\eta\(z\)\+\\varepsilon\_\{1\}\(z\)\(1\-\\eta\(z\)\),\(25\)whereε0\(z\)=ℙ\(Y~=1∣Y=0,z\)\\varepsilon\_\{0\}\(z\)=\\mathbb\{P\}\(\\widetilde\{Y\}=1\\mid Y=0,z\)andε1\(z\)=ℙ\(Y~=0∣Y=1,z\)\\varepsilon\_\{1\}\(z\)=\\mathbb\{P\}\(\\widetilde\{Y\}=0\\mid Y=1,z\)\. By[theorem1](https://arxiv.org/html/2607.09796#Thmtheorem1), the optimality condition only requires the weight ratiog\+​\(z\)/g−​\(z\)g^\{\+\}\(z\)/g^\{\-\}\(z\)to satisfy[eq\.24](https://arxiv.org/html/2607.09796#S4.E24)\. Therefore, the optimal weight construction is not unique\.

We adopt a class of bounded sigmoid weight constructions that separately extract the clean preference strengthlogit⁡\(η​\(z\)\)\\operatorname\{logit\}\(\\eta\(z\)\)and separate noise\-rate\-related terms from mixed terms that depend on both preference strength and noise rate\. Specifically, lets​\(z\)=logit⁡\(η​\(z\)\)s\(z\)=\\operatorname\{logit\}\(\\eta\(z\)\)\. We can construct

g∗,\+​\(z\)\\displaystyle g^\{\*,\+\}\(z\)=σ​\(a\+​\(z\)​s​\(z\)\+b\+​\(z\)\),\\displaystyle=\\sigma\\left\(a\_\{\+\}\(z\)s\(z\)\+b\_\{\+\}\(z\)\\right\),\(26\)g∗,−​\(z\)\\displaystyle g^\{\*,\-\}\(z\)=σ​\(−a−​\(z\)​s​\(z\)\+b−​\(z\)\),\\displaystyle=\\sigma\\left\(\-a\_\{\-\}\(z\)s\(z\)\+b\_\{\-\}\(z\)\\right\),\(27\)so that the optimal weight ratio is satisfied\. This form suggests that the current model’s implicit reward differenceu​\(z\)u\(z\)can be used as an approximation tos​\(z\)s\(z\)\. Together with the implicit reward sumΔ​\(z\)=β​\(log⁡π​\(ya∣x\)πref​\(ya∣x\)\+log⁡π​\(yb∣x\)πref​\(yb∣x\)\)\\Delta\(z\)=\\beta\(\\log\\frac\{\\pi\(y\_\{a\}\\mid x\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\_\{a\}\\mid x\)\}\+\\log\\frac\{\\pi\(y\_\{b\}\\mid x\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\_\{b\}\\mid x\)\}\), it provides VNet with signals related to sample quality and noise rate\. The complete construction and derivation are given in[appendixD](https://arxiv.org/html/2607.09796#A4)\.

## 5Prompt\-Augmentation\-Consistency Meta\-Reweighted DPO \(PACMR\-DPO\)

### 5\.1Learnable Weight Function

According to the theoretical analysis, under the optimal weights we have

uW​\-​DPO​\-​noisy∗​\(z\)=uDPO​\-​clean∗​\(z\)=log⁡η​\(z\)1−η​\(z\)\.u^\{\*\}\_\{\\mathrm\{W\\text\{\-\}DPO\\text\{\-\}noisy\}\}\(z\)=u^\{\*\}\_\{\\mathrm\{DPO\\text\{\-\}clean\}\}\(z\)=\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}\.\(28\)Therefore, although samples are drawn from the noisy training set, we can still use the current model’s implicit reward differenceu​\(z\)u\(z\)as an approximation of the clean preference strength\. In addition to the implicit reward differenceu​\(z\)u\(z\), we also consider the implicit reward sumΔ​\(z\)\\Delta\(z\)\. On the one hand,Δ​\(z\)\\Delta\(z\)can reflect the overall quality of the two candidate responses\. On the other hand, the combination ofu​\(z\)u\(z\)andΔ​\(z\)\\Delta\(z\)contains the implicit reward information of both candidate responses and may carry signals related to the sample noise rate\.

Based on this observation, we learn a mapping

NΘ:\[Δ​\(z\),u​\(z\)\]↦\[a​\(z\),b​\(z\)\],N\_\{\\Theta\}:\[\\Delta\(z\),u\(z\)\]\\mapsto\[a\(z\),b\(z\)\],\(29\)and express the learnable weighting function as

g​\(z;Θ\)=σ​\(a​\(z;Θ\)​u​\(z\)\+b​\(z;Θ\)\),g\(z;\\Theta\)=\\sigma\(a\(z;\\Theta\)u\(z\)\+b\(z;\\Theta\)\),\(30\)whereNΘN\_\{\\Theta\}is an MLP andΘ\\Thetadenotes its parameters\. Sinceu​\(z\)u\(z\)is not equal tolog⁡η​\(z\)1−η​\(z\)\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}during training, we usea​\(z;Θ\)a\(z;\\Theta\)andb​\(z;Θ\)b\(z;\\Theta\)to dynamically calibrateu​\(z\)u\(z\)\.

### 5\.2Prompt\-Augmentation Consistency without Clean Meta\-Preference Labels

To apply the bilevel optimization framework in actual training, an effective outer\-level optimization signal is required\. Traditional meta\-learning frameworks usually use a small amount of clean metadata to update the meta\-network\[[21](https://arxiv.org/html/2607.09796#bib.bib21),[18](https://arxiv.org/html/2607.09796#bib.bib18),[22](https://arxiv.org/html/2607.09796#bib.bib22)\]\. However, in LLM preference alignment, high\-quality clean meta\-preference data may be difficult to obtain\. Therefore, we replace the outer\-level objective with a prompt augmentation consistency objective\[[20](https://arxiv.org/html/2607.09796#bib.bib20)\]:

Θ∗=arg⁡minΘ⁡M​Rpac​\(D;ω∗​\(Θ\),A\),A∈𝒜,\\Theta^\{\*\}=\\arg\\min\_\{\\Theta\}MR^\{\\mathrm\{pac\}\}\(D;\\omega^\{\*\}\(\\Theta\),A\),\\quad A\\in\\mathcal\{A\},\(31\)where

M​Rpac​\(D;ω∗​\(Θ\),A\)=1\|D\|​∑i=1\|D\|ρ​\(f​\(zi;ω∗​\(Θ\)\),f​\(A​\(zi\);ω∗​\(Θ\)\)\)\.MR^\{\\mathrm\{pac\}\}\(D;\\omega^\{\*\}\(\\Theta\),A\)=\\frac\{1\}\{\|D\|\}\\sum\_\{i=1\}^\{\|D\|\}\\rho\\left\(f\(z\_\{i\};\\omega^\{\*\}\(\\Theta\)\),f\(A\(z\_\{i\}\);\\omega^\{\*\}\(\\Theta\)\)\\right\)\.\(32\)HereD⊂DtrainD\\subset D\_\{\\mathrm\{train\}\}is a subset sampled from the noisy training data,ffdenotes an output of the task model,ω∗​\(Θ\)\\omega^\{\*\}\(\\Theta\)denotes the model parameters obtained by the inner optimization,AAis a semantics\-preserving augmentation transformation, andρ\\rhois a consistency metric in the output space\.

For clarity, we distinguish two types of notation for response pairs\. When a preference label is used in the inner training objective, we denote the ordered pair as\(xi,yic,yir\)\(x\_\{i\},y\_\{i\}^\{c\},y\_\{i\}^\{r\}\), whereyicy\_\{i\}^\{c\}andyiry\_\{i\}^\{r\}are the chosen and rejected responses according to the observed preference label in the training data\. In contrast, when constructing the prompt\-augmentation consistency objective, we do not rely on the preference label of the pair\. Therefore, we denote the two candidate responses asyiay\_\{i\}^\{a\}andyiby\_\{i\}^\{b\}, and write the corresponding example aszi=\(xi,yia,yib\)z\_\{i\}=\(x\_\{i\},y\_\{i\}^\{a\},y\_\{i\}^\{b\}\)\. This notation only indicates two candidate responses associated with the same prompt, and does not assume thatyiay\_\{i\}^\{a\}is preferred overyiby\_\{i\}^\{b\}\.

We denote the resulting scalar weight by

V​\(Δ​\(z;ω\),u​\(z;ω\);Θ\)=g​\(z;Θ\)\.V\(\\Delta\(z;\\omega\),u\(z;\\omega\);\\Theta\)=g\(z;\\Theta\)\.
Letℓitrain​\(ω\)=ℒDPO​\(zi;ω\)\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\)=\\mathcal\{L\}\_\{\\mathrm\{DPO\}\}\(z\_\{i\};\\omega\)\. Then the prompt augmentation consistency problem can be written as

Θ∗=arg⁡minΘ⁡M​Rpac​\(D;ω∗​\(Θ\),A\),A∈𝒜\\displaystyle\\Theta^\{\*\}=\\arg\\min\_\{\\Theta\}MR^\{\\mathrm\{pac\}\}\(D;\\omega^\{\*\}\(\\Theta\),A\),\\quad A\\in\\mathcal\{A\}\(33\)s\.t\.ω∗​\(Θ\)=arg⁡minω⁡Ltrain​\(Dtrain;ω,VΘ\)\\displaystyle\\mathrm\{s\.t\.\}\\omega^\{\*\}\(\\Theta\)=\\arg\\min\_\{\\omega\}L^\{\\mathrm\{train\}\}\(D\_\{\\mathrm\{train\}\};\\omega,V\_\{\\Theta\}\)\(34\)where

Ltrain​\(Dtrain;ω,VΘ\)=1\|Dtrain\|​∑i=1\|Dtrain\|V​\(Δ​\(zi;ω\),u​\(zi;ω\);Θ\)​ℓitrain​\(ω\),L^\{\\mathrm\{train\}\}\(D\_\{\\mathrm\{train\}\};\\omega,V\_\{\\Theta\}\)=\\frac\{1\}\{\|D\_\{\\mathrm\{train\}\}\|\}\\sum\_\{i=1\}^\{\|D\_\{\\mathrm\{train\}\}\|\}V\(\\Delta\(z\_\{i\};\\omega\),u\(z\_\{i\};\\omega\);\\Theta\)\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\),\(35\)
For a preference tripletziz\_\{i\}, we use back\-translation as the text augmentation transformation\[[23](https://arxiv.org/html/2607.09796#bib.bib23),[24](https://arxiv.org/html/2607.09796#bib.bib24),[25](https://arxiv.org/html/2607.09796#bib.bib25)\], and apply it only to the prompt:

A​\(zi\)=\(A​\(xi\),yia,yib\)\.A\(z\_\{i\}\)=\(A\(x\_\{i\}\),y\_\{i\}^\{a\},y\_\{i\}^\{b\}\)\.\(36\)We use the English→\\rightarrowChinese→\\rightarrowEnglish back\-translation path\. Candidate responses are not back\-translated because generative rewriting of candidate responses may introduce uncontrollable and substantial distribution shifts and may damage the original preference relation\[[26](https://arxiv.org/html/2607.09796#bib.bib26)\]\. Pseudo\-labels are constructed from predictions on the original prompt and fixed with stop\-gradient\. The outer loss only aligns the predictions on the augmented prompt with these pseudo\-labels\. Complete implementation details are given in[appendixE](https://arxiv.org/html/2607.09796#A5)\.

### 5\.3Pseudo Labels and Confidence Filtering

Based on the above binary distribution, we construct pseudo\-labels for the outer\-level data as

y^i​\(ω\)=\{0,σ​\(u​\(zi;ω\)\)≥τ,1,σ​\(u​\(zi;ω\)\)≤1−τ,\\widehat\{y\}\_\{i\}\(\\omega\)=\\begin\{cases\}0,&\\sigma\(u\(z\_\{i\};\\omega\)\)\\geq\\tau,\\\\ 1,&\\sigma\(u\(z\_\{i\};\\omega\)\)\\leq 1\-\\tau,\\end\{cases\}\(37\)where0\.5≤τ<10\.5\\leq\\tau<1is the confidence threshold\. Samples that do not satisfy either condition are excluded from the outer\-level meta\-loss\. The outer\-level objective can then be written as

Θ∗=arg⁡minΘ−1\|D~\|​∑i∈D~\\displaystyle\\Theta^\{\*\}=\\arg\\min\_\{\\Theta\}\-\\frac\{1\}\{\|\\widetilde\{D\}\|\}\\sum\_\{i\\in\\widetilde\{D\}\}\[𝟏\(y^i\(ω~∗\(Θ\)\)=0\)logσ\(u\(A\(zi\);ω∗\(Θ\)\)\)\\displaystyle\\Big\[\\mathbf\{1\}\(\\widehat\{y\}\_\{i\}\(\\widetilde\{\\omega\}^\{\*\}\(\\Theta\)\)=0\)\\log\\sigma\(u\(A\(z\_\{i\}\);\\omega^\{\*\}\(\\Theta\)\)\)\(38\)\+𝟏\(y^i\(ω~∗\(Θ\)\)=1\)logσ\(−u\(A\(zi\);ω∗\(Θ\)\)\)\],\\displaystyle\+\\mathbf\{1\}\(\\widehat\{y\}\_\{i\}\(\\widetilde\{\\omega\}^\{\*\}\(\\Theta\)\)=1\)\\log\\sigma\(\-u\(A\(z\_\{i\}\);\\omega^\{\*\}\(\\Theta\)\)\)\\Big\],\(39\)whereω~∗​\(Θ\)\\widetilde\{\\omega\}^\{\*\}\(\\Theta\)is a fixed copy of the current parametersω∗​\(Θ\)\\omega^\{\*\}\(\\Theta\), indicating that gradients do not flow through the pseudo\-label branch,D~\\widetilde\{D\}denotes the noisy training subset that passes confidence filtering,𝟏​\(⋅\)\\mathbf\{1\}\(\\cdot\)denotes the indicator function\.

This objective does not supervise the meta\-network using preference labels\. Instead, it injects the task\-agnostic meta\-knowledge that the same sample should produce consistent preference judgments under semantics\-preserving augmentation into the outer\-level optimization\.

### 5\.4Finite\-Difference Meta\-Gradient Approximation

Standard meta\-learning bilevel optimization requires differentiating through the inner update steps and therefore involves higher\-order meta\-gradient computation\. In the LLM setting, this leads to significant memory and computational burdens\. We therefore use central differences to approximate the key sample\-level directional derivatives in the VNet meta\-gradient\. For notational simplicity, we writeρ​\(f​\(zi;ω∗​\(Θ\)\),f​\(A​\(zi\);ω∗​\(Θ\)\)\)=M​Ripac​\(ω∗​\(Θ\)\)\\rho\\left\(f\(z\_\{i\};\\omega^\{\*\}\(\\Theta\)\),f\(A\(z\_\{i\}\);\\omega^\{\*\}\(\\Theta\)\)\\right\)=MR\_\{i\}^\{\\mathrm\{pac\}\}\(\\omega^\{\*\}\(\\Theta\)\), omitting the subset setDDand the prompt augmentation transformationAA, and useMMto denote the meta\-batch size\. Let

Lmeta​\(ω^\(t\)​\(Θ\)\)=1M​∑j=1MM​Rjpac​\(ω^\(t\)​\(Θ\)\),L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta\)\)=\\frac\{1\}\{M\}\\sum\_\{j=1\}^\{M\}MR\_\{j\}^\{\\mathrm\{pac\}\}\(\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta\)\),\(40\)where

ω^\(t\)​\(Θ\)=ω\(t\)−αN​∑i=1NV​\(Δ​\(zi;ω\(t\)\),u​\(zi;ω\(t\)\);Θ\)​∇ωℓitrain​\(ω\(t\)\)\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta\)=\\omega^\{\(t\)\}\-\\frac\{\\alpha\}\{N\}\\sum\_\{i=1\}^\{N\}V\(\\Delta\(z\_\{i\};\\omega^\{\(t\)\}\),u\(z\_\{i\};\\omega^\{\(t\)\}\);\\Theta\)\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega^\{\(t\)\}\)\(41\)and define the meta\-gradient direction

dmeta=∇ω^\(t\)Lmeta​\(ω^\(t\)​\(Θ\)\)=1M​∑j=1M∇ω^\(t\)M​Rjpac​\(ω^\(t\)​\(Θ\)\)\.d\_\{\\mathrm\{meta\}\}=\\nabla\_\{\\widehat\{\\omega\}^\{\(t\)\}\}L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta\)\)=\\frac\{1\}\{M\}\\sum\_\{j=1\}^\{M\}\\nabla\_\{\\widehat\{\\omega\}^\{\(t\)\}\}MR\_\{j\}^\{\\mathrm\{pac\}\}\(\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta\)\)\.\(42\)Using the chain rule for a one\-step virtual inner update, the VNet meta\-gradient can be written as

∇ΘLmeta​\(ω^\(t\)​\(Θ\)\)=−αN​∑i=1Nci​∇ΘV​\(Δ​\(zi;ω\(t\)\),u​\(zi;ω\(t\)\);Θ\),\\nabla\_\{\\Theta\}L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta\)\)=\-\\frac\{\\alpha\}\{N\}\\sum\_\{i=1\}^\{N\}c\_\{i\}\\,\\nabla\_\{\\Theta\}V\(\\Delta\(z\_\{i\};\\omega^\{\(t\)\}\),u\(z\_\{i\};\\omega^\{\(t\)\}\);\\Theta\),\(43\)where the key coefficient is

ci=dmetaT​∇ωℓitrain​\(ω\(t\)\)\.c\_\{i\}=d\_\{\\mathrm\{meta\}\}^\{T\}\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega^\{\(t\)\}\)\.\(44\)PACMR\-DPO does not explicitly construct per\-sample LoRA gradients\. Instead, it uses central differences to approximate this directional derivative:

ci≈c~i=ℓitrain​\(ω\(t\)\+ϵ​dmeta\)−ℓitrain​\(ω\(t\)−ϵ​dmeta\)2​ϵ,c\_\{i\}\\approx\\widetilde\{c\}\_\{i\}=\\frac\{\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega^\{\(t\)\}\+\\epsilon d\_\{\\mathrm\{meta\}\}\)\-\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega^\{\(t\)\}\-\\epsilon d\_\{\\mathrm\{meta\}\}\)\}\{2\\epsilon\},\(45\)Thus, the VNet meta\-gradient can be approximated by

∇ΘLmeta​\(ω^\(t\)​\(Θ\)\)≈−αN​∑i=1Nc~i​∇ΘV​\(Δ​\(zi;ω\(t\)\),u​\(zi;ω\(t\)\);Θ\)\.\\nabla\_\{\\Theta\}L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta\)\)\\approx\-\\frac\{\\alpha\}\{N\}\\sum\_\{i=1\}^\{N\}\\widetilde\{c\}\_\{i\}\\,\\nabla\_\{\\Theta\}V\(\\Delta\(z\_\{i\};\\omega^\{\(t\)\}\),u\(z\_\{i\};\\omega^\{\(t\)\}\);\\Theta\)\.\(46\)whereα\\alphais the inner learning rate,NNis the training batch size, andϵ\\epsilonis the central\-difference perturbation scale\. This approximation avoids explicitly retaining the inner\-update computational graph or constructing a per\-sample LoRA gradient matrix, thereby reducing the resource overhead of higher\-order meta\-gradient computation\. Central\-difference approximation requires positive and negative perturbations to the model parameters\. Applying perturbations and updates in the full parameter space would still be expensive for LLMs\. Therefore, we further restrict training to the LoRA parameter space\[[27](https://arxiv.org/html/2607.09796#bib.bib27)\]\. LetPPbe the number of full model parameters andPLP\_\{L\}be the number of trainable LoRA parameters; typicallyPL≪PP\_\{L\}\\ll P\. Under LoRA fine\-tuning, central\-difference perturbations, gradient storage, and parameter writing all occur in the low\-rank delta parameter space, thereby reducing computational cost\. A complete complexity analysis is given in[appendixA](https://arxiv.org/html/2607.09796#A1)\.

Algorithm 1PACMR\-DPO1:Reference model

πref\\pi\_\{\\mathrm\{ref\}\}; initialized policy model

πω←πref\\pi\_\{\\omega\}\\leftarrow\\pi\_\{\\mathrm\{ref\}\}; VNet

V​\(⋅;Θ\)V\(\\cdot;\\Theta\); augmentation

AA; training data

DtrainD\_\{\\mathrm\{train\}\}; outer data

DDsampled from

DtrainD\_\{\\mathrm\{train\}\}; threshold

τ\\tau; finite\-difference scale

ϵ\\epsilon\.

2:Policy model

πω\\pi\_\{\\omega\}\.

3:for

t=1,2,…,Et=1,2,\\ldots,Edo

4:Sample training batch

Bt=\{zi\}i=1BtrainB\_\{t\}=\\\{z\_\{i\}\\\}\_\{i=1\}^\{B\_\{\\mathrm\{train\}\}\}, where

zi=\(xi,yic,yir\)z\_\{i\}=\(x\_\{i\},y\_\{i\}^\{c\},y\_\{i\}^\{r\}\), from

DtrainD\_\{\\mathrm\{train\}\}\.

5:Compute implicit reward difference

u​\(zi;ω\(t\)\)u\(z\_\{i\};\\omega^\{\(t\)\}\)and reward sum

Δ​\(zi;ω\(t\)\)\\Delta\(z\_\{i\};\\omega^\{\(t\)\}\)\.

6:Compute weights

V​\(Δ​\(zi;ω\(t\)\),u​\(zi;ω\(t\)\);Θ\(t\)\)V\(\\Delta\(z\_\{i\};\\omega^\{\(t\)\}\),u\(z\_\{i\};\\omega^\{\(t\)\}\);\\Theta^\{\(t\)\}\)\.

7:Compute weighted DPO loss and perform one virtual update

ω^\(t\)​\(Θ\(t\)\)\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta^\{\(t\)\}\)\.

8:Sample outer batch

Bm=\{zj,A​\(zj\)\}j=1BmetaB\_\{m\}=\\\{z\_\{j\},A\(z\_\{j\}\)\\\}\_\{j=1\}^\{B\_\{\\mathrm\{meta\}\}\}, where

zj=\(xj,yja,yjb\)z\_\{j\}=\(x\_\{j\},y\_\{j\}^\{a\},y\_\{j\}^\{b\}\),

A​\(zj\)=\(A​\(xj\),yja,yjb\)A\(z\_\{j\}\)=\(A\(x\_\{j\}\),y\_\{j\}^\{a\},y\_\{j\}^\{b\}\), from

DD\.

9:Compute original implicit reward differences

u​\(zj;ω^\(t\)​\(Θ\(t\)\)\)u\(z\_\{j\};\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta^\{\(t\)\}\)\)and augmented implicit reward differences

u​\(A​\(zj\);ω^\(t\)​\(Θ\(t\)\)\)u\(A\(z\_\{j\}\);\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta^\{\(t\)\}\)\)\.

10:Construct pseudo labels using threshold

τ\\tauand obtain filtered meta batch

B~meta\\widetilde\{B\}\_\{\\mathrm\{meta\}\}\.

11:Compute meta loss

LmetaL\_\{\\mathrm\{meta\}\}and meta gradient

dmeta=∇ω^\(t\)Lmeta​\(ω^\(t\)​\(Θ\(t\)\)\)d\_\{\\mathrm\{meta\}\}=\\nabla\_\{\\widehat\{\\omega\}^\{\(t\)\}\}L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}^\{\(t\)\}\(\\Theta^\{\(t\)\}\)\)\.

12:Evaluate perturbed training losses at

ω\(t\)\+ϵ​dmeta\\omega^\{\(t\)\}\+\\epsilon d\_\{\\mathrm\{meta\}\}and

ω\(t\)−ϵ​dmeta\\omega^\{\(t\)\}\-\\epsilon d\_\{\\mathrm\{meta\}\}\.

13:Compute central\-difference coefficients

c~i\\widetilde\{c\}\_\{i\}and update VNet parameters

Θ\(t\+1\)\\Theta^\{\(t\+1\)\}\.

14:Recompute weights with

V​\(⋅;Θ\(t\+1\)\)V\(\\cdot;\\Theta^\{\(t\+1\)\}\)and update policy model parameters

ω\(t\+1\)\\omega^\{\(t\+1\)\}by weighted DPO\.

15:endfor

16:return

πω\\pi\_\{\\omega\}

## 6Experiments

### 6\.1Experimental Setup

We evaluate PACMR\-DPO on summarization \(TL;DR\)\[[28](https://arxiv.org/html/2607.09796#bib.bib28)\]and single\-turn dialogue \(Anthropic HH\)\[[29](https://arxiv.org/html/2607.09796#bib.bib29)\]\. In the summarization task, the model generates a summary for a Reddit post\. We first perform supervised fine\-tuning on a filtered TL;DR summarization dataset to obtain the initial reference model, and then use the human preference data collected by Stiennon et al\. for preference optimization\. In the single\-turn dialogue task, we use the Anthropic Helpful and Harmless \(HH\) dialogue preference dataset and conduct instruction fine\-tuning on the preferred responses to obtain the reference model\.

We inject random\-flip noise into the datasets of the two tasks to construct noisy preference\-learning scenarios\. Data processing and noise\-injection details are given in[appendixF](https://arxiv.org/html/2607.09796#A6)\. Specifically, we consider three noise rates,20%20\\%,30%30\\%, and40%40\\%, to systematically evaluate the performance of different methods as the noise rate increases\. In all experiments, we use Llama\-2\-7B as the base model\. For computational efficiency, LoRA fine\-tuning is used in both SFT and preference alignment, with LoRA rank set to 16 and LoRA alpha set to 32\. All models are trained for one epoch on the corresponding dataset, and the experiments are based on OpenRLHF\[[30](https://arxiv.org/html/2607.09796#bib.bib30)\]\.

### 6\.2Baselines and Evaluation Metrics

We select DPO, cDPO, IPO\[[31](https://arxiv.org/html/2607.09796#bib.bib31)\], rDPO, and Dr\.DPO as baselines\. Training hyperparameters and baseline implementation details are given in[appendixG](https://arxiv.org/html/2607.09796#A7)\. Standard DPO is used as the evaluation anchor, and outputs from the proposed method and other baselines are compared pairwise against DPO outputs\. We randomly sample 800 prompts from the test set, generate responses using the proposed method, each baseline, and standard DPO, pair each generated response with the corresponding DPO response, and then use the same external judge model for pairwise comparison\[[7](https://arxiv.org/html/2607.09796#bib.bib7),[32](https://arxiv.org/html/2607.09796#bib.bib32)\]\. The evaluation protocol and judge prompt are provided in[appendixH](https://arxiv.org/html/2607.09796#A8); win/tie/loss ratios are recorded\. The win\-score is defined as\[[12](https://arxiv.org/html/2607.09796#bib.bib12)\]

Win​\-​score=1\+\#​win−\#​loseTotal​comparisons\.\\mathrm\{Win\\text\{\-\}score\}=1\+\\frac\{\\\#\\mathrm\{win\}\-\\\#\\mathrm\{lose\}\}\{\\mathrm\{Total\\ comparisons\}\}\.\(47\)

### 6\.3Main Results under Noisy Preferences

Table 1:Main results on the TL;DR summarization task under random preference flipping noise\.Table 2:Main results on the Anthropic HH single\-turn dialogue task under random preference flipping noise\.Table 3:Final\-step VNet weights on unflipped and flipped training pairs\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step2901_function.jpg)\(a\)20%
![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step2901_function.jpg)\(b\)30%
![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step2901_function.jpg)\(c\)40%

Figure 1:Final learned weight functions on TL;DR under 20%, 30%, and 40% random preference flips\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step2901_hist.jpg)\(a\)20%
![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step2901_hist.jpg)\(b\)30%
![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step2901_hist.jpg)\(c\)40%

Figure 2:Final learned weight distributions on TL;DR\. Unflipped and flipped pairs are identified using the synthetic noise mask\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step4699_function.jpg)\(a\)20%
![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step4699_function.jpg)\(b\)30%
![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step4699_function.jpg)\(c\)40%

Figure 3:Final learned weight functions on Anthropic HH under 20%, 30%, and 40% random preference flips\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step4699_hist.jpg)\(a\)20%
![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step4699_hist.jpg)\(b\)30%
![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step4699_hist.jpg)\(c\)40%

Figure 4:Final learned weight distributions on Anthropic HH\. Unflipped and flipped pairs are identified using the synthetic noise mask\.The results show that PACMR\-DPO achieves the highest win\-score under all three noise rates on TL;DR: 1\.4763, 1\.5263, and 1\.4950 under 20%, 30%, and 40% noise, respectively\. Even under the high\-noise 40% setting, it maintains a clear advantage over baselines such as Dr\.DPO and rDPO\. On Anthropic HH, PACMR\-DPO also performs strongly overall\. Under 20% and 30% noise, it clearly outperforms all baselines\. Under 40% noise, PACMR\-DPO reaches a win rate of 40\.63% and a win\-score of 1\.2400\. Although its win rate is slightly lower than rDPO’s 41\.38%, its win\-score is higher than that of all baselines, indicating that PACMR\-DPO still has a stronger overall comparison quality\.

[table3](https://arxiv.org/html/2607.09796#S6.T3)further shows the difference between the final VNet weights on unflipped and flipped samples\. From the weight\-distribution plots on TL;DR, the distributions of clean and noisy samples differ clearly under 20% and 30% noise, suggesting that the meta\-network can assign different weights to samples of different reliability, thereby emphasizing relatively trustworthy preference signals and suppressing noisy labels during training\. As the noise rate increases to 40%, the proportion of noisy samples grows, and clean and noisy samples increasingly overlap in the model’s implicit\-reward feature space, making the task more difficult\. The weight distributions still exhibit some separability, but the overlap increases, indicating that weight learning becomes harder at higher noise rates and sample discrimination weakens\. Nevertheless, PACMR\-DPO still outperforms all baselines under 40% noise\.

The weight\-distribution plots on HH show that PACMR\-DPO can also learn weight differences between clean and noisy samples under most noise rates\. This indicates that the meta\-learning signal constructed from the implicit reward difference, implicit reward sum, and prompt augmentation consistency is also effective on HH\. Similar to TL;DR, as the noise rate increases, the distributions of clean and noisy samples increasingly overlap, indicating that the task becomes harder with higher noise\. In particular, under 40% noise on HH, the weight distributions of clean and noisy samples are much less separable and become highly similar\. This suggests that sample\-reliability modeling for the HH dialogue preference task becomes substantially more difficult under extremely high noise\. Although the meta\-network can still learn some weight differences, a single\-epoch training setup is not sufficient to form a strongly separable weight distribution\. Even so, PACMR\-DPO still performs well\.

[figs\.1](https://arxiv.org/html/2607.09796#S6.F1)and[3](https://arxiv.org/html/2607.09796#S6.F3)show the shapes of the finally learned weighting functions, while[figs\.2](https://arxiv.org/html/2607.09796#S6.F2)and[4](https://arxiv.org/html/2607.09796#S6.F4)show the weight distributions of unflipped and flipped samples\.

### 6\.4Ablation: Learned Weighting vs\. Fixed Prior Weighting

This subsection compares PACMR\-DPO with fixed\-prior weighting\. The fixed\-weighting variant does not learn a VNet, and instead directly uses

g​\(z\)=σ​\(u​\(z\)\)g\(z\)=\\sigma\(u\(z\)\)\(48\)as the sample weight\. This ablation answers whether a theoretically inspired fixed monotonic weight is sufficient, or whether meta\-learning is necessary to learn the weighting function\.

Table 4:Ablation study comparing fixed sigmoid weighting with learned PACMR\-DPO weighting\.The fixed sigmoid weight brings some benefit in certain medium\-noise settings, but is overall less stable than learned PACMR\-DPO\. In particular, PACMR\-DPO has clearly higher win\-scores on TL;DR under 20% and 40% noise and on HH under 20% and 30% noise\. This indicates that using onlyσ​\(u\)\\sigma\(u\)as a static prior is insufficient to fully characterize sample reliability, and that learning sample weights with the outer prompt\-augmentation\-consistency objective is necessary\.

### 6\.5Sensitivity to Confidence Thresholdτ\\tau

The confidence thresholdτ\\taucontrols the filtering strength for outer\-level pseudo\-labels\. A smallerτ\\tauincreases outer\-sample coverage but may introduce low\-confidence pseudo\-labels; a largerτ\\taucan filter unreliable samples but may reduce the effective outer\-level meta\-signal\. We considerτ∈\{0\.55,0\.60,0\.70\}\\tau\\in\\\{0\.55,0\.60,0\.70\\\}\.

Table 5:Sensitivity analysis of pseudo\-label confidence thresholdτ\\tau\.The sensitivity results show that the optimal value ofτ\\taudepends on the task and noise rate\. On TL;DR, the defaultτ=0\.60\\tau=0\.60performs strongly under 20% and 40% noise, whileτ=0\.55\\tau=0\.55achieves comparable or slightly better results under 30% noise\. On HH,τ=0\.70\\tau=0\.70performs well under 20% and 30% noise, whileτ=0\.60\\tau=0\.60is slightly better under 40% noise\. Overall, a threshold that is too low may introduce unreliable pseudo\-labels, while a threshold that is too high may reduce the number of effective outer\-level samples\. Therefore, the main experiments use the compromise valueτ=0\.60\\tau=0\.60\.

### 6\.6Comparison with Clean\-Metadata MWN\-DPO

To evaluate whether the prompt\-augmentation\-consistency outer objective can replace clean meta\-preference labels, this subsection compares PACMR\-DPO with MWN\-DPO using clean metadata\. For fairness, MWN\-DPO\-clean and PACMR\-DPO use the same VNet input and weighting\-function structure\. The only difference is the outer\-level objective: MWN\-DPO\-clean uses the DPO loss on clean meta\-preference data to update VNet, while PACMR\-DPO uses the prompt\-augmentation\-consistency objective\. Details of the clean metadata construction are given in[appendixF](https://arxiv.org/html/2607.09796#A6)\.

Table 6:Comparison between PACMR\-DPO and MWN\-DPO using clean meta\-data\.The results show that PACMR\-DPO can match or exceed clean\-meta MWN\-DPO in multiple settings without using clean meta\-preference labels\. This indicates that prompt augmentation consistency can serve as an effective outer\-level proxy signal when clean metadata is unavailable, although its advantage does not hold for every task and noise rate\.

## 7Conclusion

This paper studies DPO learning under noisy preference labels and analyzes how noisy preferences cause a shift in the DPO optimum from the perspectives of conditional risk and posterior shift\. We prove that, under idealized distribution\-risk and function\-space conditions, the optimal weight ratio in bilevel reweighting can make weighted DPO under the noisy training distribution recover the pointwise optimum under the clean preference distribution\. Inspired by this theory, we propose PACMR\-DPO, which learns sample weights through VNet and uses prompt augmentation consistency as the outer meta\-objective instead of clean meta\-preference data\. We further use central\-difference approximation and LoRA\-space perturbations to reduce the computational cost of bilevel optimization in LLM settings\. Experiments show that PACMR\-DPO achieves performance improvements over multiple robust DPO baselines under medium and high preference\-noise settings\.

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## Appendix

## Appendix AExact Meta\-Gradient and Complexity Analysis of the Central\-Difference Approximation

### A\.1Notation and Assumptions

This appendix analyzes the computational complexity of VNet parameter updates in PACMR\-DPO\. Letω∈ℝPL\\omega\\in\\mathbb\{R\}^\{P\_\{L\}\}denote the LoRA parameters participating in policy training, and letΘ∈ℝQ\\Theta\\in\\mathbb\{R\}^\{Q\}denote the VNet parameters\. LetNNbe the number of training samples used for the VNet meta\-update, and letMMbe the number of meta\-samples used to compute the prompt\-augmentation\-consistency meta\-loss\. For theii\-th training sample, denote the single\-sample DPO training loss as

ℓitrain​\(ω\)\.\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\)\.\(49\)The weight output by VNet for this training sample is denoted as

vi​\(Θ\)=V​\(Δ​\(zi;ω\),u​\(zi;ω\);Θ\),v\_\{i\}\(\\Theta\)=V\(\\Delta\(z\_\{i\};\\omega\),u\(z\_\{i\};\\omega\);\\Theta\),\(50\)In the VNet update and virtual inner update, these input features are processed with stop\-gradient\.

In the LLM\+LoRA setting considered in this paper, VNet is a small MLP, while the trainable parameters on the policy side lie in the LoRA parameter space of a large language model\. Therefore,

At the same time, the training batch and meta\-batch sizes are much smaller than the LoRA parameter dimension, namely

N≪PL,M≪PL\.N\\ll P\_\{L\},\\qquad M\\ll P\_\{L\}\.\(52\)In addition, forward\-only training loss evaluation does not need to store the computational graph for backpropagation through model parameters, so its space and time costs are lower than those of training\-loss evaluation requiring backpropagation\. LetAtr\(1\)​\(N\)A\_\{\\mathrm\{tr\}\}^\{\(1\)\}\(N\)denote the space cost of the first\-order training\-side backward graph,Atr\(2\)​\(N\)A\_\{\\mathrm\{tr\}\}^\{\(2\)\}\(N\)the space cost of the higher\-order training\-side graph retained for bilevel differentiation, andAmeta\(1\)​\(M\)A\_\{\\mathrm\{meta\}\}^\{\(1\)\}\(M\)the space cost of the first\-order meta\-loss backward graph\. LetItr​\(N\)I\_\{\\mathrm\{tr\}\}\(N\)denote the space cost of forward\-only training loss evaluation\. Then

Itr​\(N\)=O​\(Atr\(1\)​\(N\)\),Atr\(1\)​\(N\)=O​\(Atr\(2\)​\(N\)\)\.I\_\{\\mathrm\{tr\}\}\(N\)=O\(A\_\{\\mathrm\{tr\}\}^\{\(1\)\}\(N\)\),\\qquad A\_\{\\mathrm\{tr\}\}^\{\(1\)\}\(N\)=O\(A\_\{\\mathrm\{tr\}\}^\{\(2\)\}\(N\)\)\.\(53\)LetFtr​\(N\)F\_\{\\mathrm\{tr\}\}\(N\)denote the time for forward\-only loss evaluation onNNtraining samples, and letBtr​\(1\)B\_\{\\mathrm\{tr\}\}\(1\)denote the time for one backpropagation through the LoRA parameters on a single training sample\.

### A\.2Exact Meta\-Gradient and Computational Bottleneck

The one\-step virtual inner update used in the VNet update is

ω^​\(Θ\)=ω−αN​∑i=1Nvi​\(Θ\)​∇ωℓitrain​\(ω\)\.\\widehat\{\\omega\}\(\\Theta\)=\\omega\-\\frac\{\\alpha\}\{N\}\\sum\_\{i=1\}^\{N\}v\_\{i\}\(\\Theta\)\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\)\.\(54\)The outer\-level meta\-objective is denoted as

Lmeta​\(ω^\)=1M​∑j=1MM​Rjpac​\(ω^\)\.L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}\)=\\frac\{1\}\{M\}\\sum\_\{j=1\}^\{M\}MR\_\{j\}^\{\\mathrm\{pac\}\}\(\\widehat\{\\omega\}\)\.\(55\)Define the meta\-gradient direction

dmeta=∇ω^Lmeta​\(ω^\)=1M​∑j=1M∇ω^M​Rjpac​\(ω^\)∈ℝPL\.d\_\{\\mathrm\{meta\}\}=\\nabla\_\{\\widehat\{\\omega\}\}L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}\)=\\frac\{1\}\{M\}\\sum\_\{j=1\}^\{M\}\\nabla\_\{\\widehat\{\\omega\}\}MR\_\{j\}^\{\\mathrm\{pac\}\}\(\\widehat\{\\omega\}\)\\in\\mathbb\{R\}^\{P\_\{L\}\}\.\(56\)The reverse path from the VNet parametersΘ\\Thetato the outer meta\-loss is

Θ⟶vi​\(Θ\)⟶ω^​\(Θ\)⟶Lmeta​\(ω^​\(Θ\)\)\.\\Theta\\longrightarrow v\_\{i\}\(\\Theta\)\\longrightarrow\\widehat\{\\omega\}\(\\Theta\)\\longrightarrow L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}\(\\Theta\)\)\.\(57\)By the chain rule and[eq\.54](https://arxiv.org/html/2607.09796#A1.E54), we obtain

∇ΘLmeta​\(ω^​\(Θ\)\)\\displaystyle\\nabla\_\{\\Theta\}L\_\{\\mathrm\{meta\}\}\(\\widehat\{\\omega\}\(\\Theta\)\)=−αN​∑i=1N\[dmetaT​∇ωℓitrain​\(ω\)\]​∇Θvi​\(Θ\)\.\\displaystyle=\-\\frac\{\\alpha\}\{N\}\\sum\_\{i=1\}^\{N\}\\left\[d\_\{\\mathrm\{meta\}\}^\{T\}\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\)\\right\]\\nabla\_\{\\Theta\}v\_\{i\}\(\\Theta\)\.\(58\)Therefore, the computational bottleneck in the exact VNet meta\-gradient is the coefficient corresponding to each training sample,

ci=dmetaT​∇ωℓitrain​\(ω\),i=1,…,N\.c\_\{i\}=d\_\{\\mathrm\{meta\}\}^\{T\}\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\),\\qquad i=1,\\ldots,N\.\(59\)UnderQ≪PLQ\\ll P\_\{L\},∇Θvi​\(Θ\)\\nabla\_\{\\Theta\}v\_\{i\}\(\\Theta\)lies in the parameter space of the small MLP and is not the dominant cost\. Therefore, the following complexity analysis focuses on computingcic\_\{i\}in[eq\.59](https://arxiv.org/html/2607.09796#A1.E59)\.

### A\.3Exact Computation by Retaining the Computational Graph

The first exact computation strategy is to directly retain the complete reverse path in[eq\.57](https://arxiv.org/html/2607.09796#A1.E57)\. In this case, the outer meta\-loss must continue backpropagating through the virtual updateω^​\(Θ\)\\widehat\{\\omega\}\(\\Theta\)tovi​\(Θ\)v\_\{i\}\(\\Theta\)andΘ\\Theta\. Sinceω^​\(Θ\)\\widehat\{\\omega\}\(\\Theta\)contains

∇ωℓitrain​\(ω\),\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\),\(60\)the reverse path must retain the higher\-order training\-side computational graph, whose space cost is denoted byAtr\(2\)​\(N\)A\_\{\\mathrm\{tr\}\}^\{\(2\)\}\(N\)\. In addition, computingdmetad\_\{\\mathrm\{meta\}\}requires retaining the first\-order meta\-loss backward graph, whose space cost isAmeta\(1\)​\(M\)A\_\{\\mathrm\{meta\}\}^\{\(1\)\}\(M\)\. At least one vector in the LoRA parameter space, of sizePLP\_\{L\}, must also be stored\. UnderQ≪PLQ\\ll P\_\{L\},N≪PLN\\ll P\_\{L\}, andM≪PLM\\ll P\_\{L\}, the VNet parameters, sample weights, and sample\-level scalar coefficients are lower\-order terms\. Therefore, the space complexity of retaining the computational graph is

Sgraph=O​\(Atr\(2\)​\(N\)\+Ameta\(1\)​\(M\)\+PL\)\.S\_\{\\mathrm\{graph\}\}=O\\left\(A\_\{\\mathrm\{tr\}\}^\{\(2\)\}\(N\)\+A\_\{\\mathrm\{meta\}\}^\{\(1\)\}\(M\)\+P\_\{L\}\\right\)\.\(61\)This strategy computes the meta\-gradient exactly, but incurs the space cost of retaining the higher\-order computational graph for the inner virtual update\.

### A\.4Exact Computation via the Per\-Sample Gradient Matrix

The second exact computation strategy is not to retain the complete higher\-order graph, but to explicitly construct the per\-sample LoRA gradient matrix

G=\[∇ωℓ1train​\(ω\)T∇ωℓ2train​\(ω\)T⋮∇ωℓNtrain​\(ω\)T\]∈ℝN×PL\.G=\\begin\{bmatrix\}\\nabla\_\{\\omega\}\\ell\_\{1\}^\{\\mathrm\{train\}\}\(\\omega\)^\{T\}\\\\ \\nabla\_\{\\omega\}\\ell\_\{2\}^\{\\mathrm\{train\}\}\(\\omega\)^\{T\}\\\\ \\vdots\\\\ \\nabla\_\{\\omega\}\\ell\_\{N\}^\{\\mathrm\{train\}\}\(\\omega\)^\{T\}\\end\{bmatrix\}\\in\\mathbb\{R\}^\{N\\times P\_\{L\}\}\.\(62\)Thencccan be written as

c=G​dmeta∈ℝN\.c=Gd\_\{\\mathrm\{meta\}\}\\in\\mathbb\{R\}^\{N\}\.\(63\)Storing the matrixGGrequiresN​PLNP\_\{L\}scalars, and storingdmetad\_\{\\mathrm\{meta\}\}requiresPLP\_\{L\}scalars\. LetSauxS\_\{\\mathrm\{aux\}\}denote the auxiliary peak space excluding the persistent storage ofGGanddmetad\_\{\\mathrm\{meta\}\}\. It includes the first\-order meta\-loss backward graph and temporary activations required for computingdmetad\_\{\\mathrm\{meta\}\}, the temporary computational graph, autograd workspace, and temporary gradient buffers generated when constructingGGrow by row using single\-sample or small\-batch training backpropagation, and the possiblePLP\_\{L\}\-dimensional temporary vector needed before writing the current row intoGG\. These temporary objects can usually be released after each row is written, and therefore do not accumulate asN​PLNP\_\{L\}persistent storage\. TheN​PLNP\_\{L\}storage ofGGitself and thePLP\_\{L\}storage ofdmetad\_\{\\mathrm\{meta\}\}have already been counted in the main term\. UnderQ≪PLQ\\ll P\_\{L\},N≪PLN\\ll P\_\{L\}, andM≪PLM\\ll P\_\{L\}, lower\-order terms can be ignored\. Therefore, the space complexity of the matrix\-construction strategy is

Smatrix=O​\(\(N\+1\)​PL\+Saux\)\.S\_\{\\mathrm\{matrix\}\}=O\\left\(\(N\+1\)P\_\{L\}\+S\_\{\\mathrm\{aux\}\}\\right\)\.\(64\)This strategy avoids explicitly retaining the complete higher\-order reverse path, but requires storing theN×PLN\\times P\_\{L\}per\-sample LoRA gradient matrix\.

### A\.5Exact Computation via Per\-Sample Backpropagation

The third exact computation strategy does not saveGG, but instead computes for each training sample∇ωℓitrain​\(ω\)\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\), and then immediately computes

ci=dmetaT​∇ωℓitrain​\(ω\),c\_\{i\}=d\_\{\\mathrm\{meta\}\}^\{T\}\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\),\(65\)and discards∇ωℓitrain​\(ω\)\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\)\. This strategy avoids theN​PLNP\_\{L\}matrix storage, but requires executingNNsingle\-sample policy\-model backpropagations\. For each sample, it requires one forward loss evaluationFtr​\(1\)F\_\{\\mathrm\{tr\}\}\(1\), one single\-sample backpropagationBtr​\(1\)B\_\{\\mathrm\{tr\}\}\(1\), and onePLP\_\{L\}\-dimensional inner product\. Since we compare only the cost of exact computation ofcic\_\{i\}in[eq\.59](https://arxiv.org/html/2607.09796#A1.E59), common procedures required by both exact computation and central differences are not included in this term\. Therefore, the additional time complexity for computing the directional\-derivative coefficients using per\-sample backpropagation is

Tsample​\-​coeff=O​\(N​\(Ftr​\(1\)\+Btr​\(1\)\+PL\)\)\.T\_\{\\mathrm\{sample\\text\{\-\}coeff\}\}=O\\left\(N\\left\(F\_\{\\mathrm\{tr\}\}\(1\)\+B\_\{\\mathrm\{tr\}\}\(1\)\+P\_\{L\}\\right\)\\right\)\.\(66\)This strategy avoids storing the complete per\-sample gradient matrix at the cost ofNNsingle\-sample backpropagations andNNLoRA\-dimensional inner products\.

### A\.6Complexity of the Central\-Difference Approximation

PACMR\-DPO does not explicitly compute∇ωℓitrain​\(ω\)\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\)\. Instead, it uses central differences to approximatecic\_\{i\}in[eq\.59](https://arxiv.org/html/2607.09796#A1.E59):

ci=dmetaT​∇ωℓitrain​\(ω\)≈c~i=ℓitrain​\(ω\+ϵ​dmeta\)−ℓitrain​\(ω−ϵ​dmeta\)2​ϵ\.c\_\{i\}=d\_\{\\mathrm\{meta\}\}^\{T\}\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\)\\approx\\widetilde\{c\}\_\{i\}=\\frac\{\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\+\\epsilon d\_\{\\mathrm\{meta\}\}\)\-\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\-\\epsilon d\_\{\\mathrm\{meta\}\}\)\}\{2\\epsilon\}\.\(67\)Central differences require storing the direction vectordmetad\_\{\\mathrm\{meta\}\}in the LoRA space, corresponding to spacePLP\_\{L\}\. The training losses under positive and negative perturbations,

ℓitrain​\(ω\+ϵ​dmeta\),ℓitrain​\(ω−ϵ​dmeta\)\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\+\\epsilon d\_\{\\mathrm\{meta\}\}\),\\qquad\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\-\\epsilon d\_\{\\mathrm\{meta\}\}\)\(68\)only require forward computation\. SinceItr​\(N\)=O​\(Atr\(1\)​\(N\)\)I\_\{\\mathrm\{tr\}\}\(N\)=O\(A\_\{\\mathrm\{tr\}\}^\{\(1\)\}\(N\)\), forward\-only evaluation does not exceed the space cost of the first\-order training\-side backward graph in asymptotic order\. Computingdmetad\_\{\\mathrm\{meta\}\}requires the first\-order meta\-loss backward graphAmeta\(1\)​\(M\)A\_\{\\mathrm\{meta\}\}^\{\(1\)\}\(M\)\. Therefore, the peak space complexity of the central\-difference implementation is

SFD=O​\(PL\+max⁡\{Atr\(1\)​\(N\),Ameta\(1\)​\(M\)\}\)\.S\_\{\\mathrm\{FD\}\}=O\\left\(P\_\{L\}\+\\max\\left\\\{A\_\{\\mathrm\{tr\}\}^\{\(1\)\}\(N\),A\_\{\\mathrm\{meta\}\}^\{\(1\)\}\(M\)\\right\\\}\\right\)\.\(69\)Computing the central\-difference coefficients requires two forward\-only training\-loss evaluations and one LoRA\-space parameter perturbation and restoration operation\. SinceN≪PLN\\ll P\_\{L\}, storing and processing theNNscalar difference coefficients is a lower\-order term\. Therefore, the additional time complexity for computing the central\-difference directional\-derivative coefficients is

TFD​\-​coeff=O​\(2​Ftr​\(N\)\+PL\)\.T\_\{\\mathrm\{FD\\text\{\-\}coeff\}\}=O\\left\(2F\_\{\\mathrm\{tr\}\}\(N\)\+P\_\{L\}\\right\)\.\(70\)

### A\.7Comparison with Exact Computation

The exact implementation that retains the computational graph has space complexity

Sgraph=O​\(Atr\(2\)​\(N\)\+Ameta\(1\)​\(M\)\+PL\),S\_\{\\mathrm\{graph\}\}=O\\left\(A\_\{\\mathrm\{tr\}\}^\{\(2\)\}\(N\)\+A\_\{\\mathrm\{meta\}\}^\{\(1\)\}\(M\)\+P\_\{L\}\\right\),\(71\)whereas central differences have space complexity

SFD=O​\(PL\+max⁡\{Atr\(1\)​\(N\),Ameta\(1\)​\(M\)\}\)\.S\_\{\\mathrm\{FD\}\}=O\\left\(P\_\{L\}\+\\max\\left\\\{A\_\{\\mathrm\{tr\}\}^\{\(1\)\}\(N\),A\_\{\\mathrm\{meta\}\}^\{\(1\)\}\(M\)\\right\\\}\\right\)\.\(72\)SinceAtr\(1\)​\(N\)=O​\(Atr\(2\)​\(N\)\)A\_\{\\mathrm\{tr\}\}^\{\(1\)\}\(N\)=O\(A\_\{\\mathrm\{tr\}\}^\{\(2\)\}\(N\)\), central differences avoid the higher\-order training graph retained for the inner virtual update in exact bilevel backpropagation\.

The exact implementation that constructs the per\-sample gradient matrix has space complexity

Smatrix=O​\(\(N\+1\)​PL\+Saux\),S\_\{\\mathrm\{matrix\}\}=O\\left\(\(N\+1\)P\_\{L\}\+S\_\{\\mathrm\{aux\}\}\\right\),\(73\)whereas central differences store only the LoRA\-space direction vector and sample\-level scalar difference coefficients\. UnderN≪PLN\\ll P\_\{L\}, the sample\-level scalar coefficients are lower\-order terms\. Therefore, central differences avoid theN​PLNP\_\{L\}storage of the per\-sample gradient matrix\.

The additional time complexity for exact computation of the directional\-derivative coefficients via per\-sample backpropagation is

Tsample​\-​coeff=O​\(N​\(Ftr​\(1\)\+Btr​\(1\)\+PL\)\),T\_\{\\mathrm\{sample\\text\{\-\}coeff\}\}=O\\left\(N\\left\(F\_\{\\mathrm\{tr\}\}\(1\)\+B\_\{\\mathrm\{tr\}\}\(1\)\+P\_\{L\}\\right\)\\right\),\(74\)whereas the additional time complexity for central\-difference computation of the directional\-derivative coefficients is

TFD​\-​coeff=O​\(2​Ftr​\(N\)\+PL\)\.T\_\{\\mathrm\{FD\\text\{\-\}coeff\}\}=O\\left\(2F\_\{\\mathrm\{tr\}\}\(N\)\+P\_\{L\}\\right\)\.\(75\)Thus, central differences replaceNNsingle\-sample policy\-model backpropagations andNNLoRA\-dimensional inner products with two batch forward\-only training\-loss evaluations and one LoRA\-space perturbation operation\.

In summary, underQ≪PLQ\\ll P\_\{L\},N≪PLN\\ll P\_\{L\}, andM≪PLM\\ll P\_\{L\}, and when forward\-only computation is cheaper than backpropagation, central differences approximatedmetaT​∇ωℓitrain​\(ω\)d\_\{\\mathrm\{meta\}\}^\{T\}\\nabla\_\{\\omega\}\\ell\_\{i\}^\{\\mathrm\{train\}\}\(\\omega\)while avoiding three exact\-computation bottlenecks: retaining the higher\-order computational graph of the inner virtual update, storing theN×PLN\\times P\_\{L\}per\-sample LoRA gradient matrix, and executingNNsingle\-sample policy\-model backpropagations\.

## Appendix BMeaning of the DPO Conditional Risk and Derivation of the Optimum

The conditional risk is the pointwise risk obtained by fixing the input pairz=\(x,ya,yb\)z=\(x,y\_\{a\},y\_\{b\}\)and taking the expectation over the conditional distribution of the latent preference label associated with that sample\. In our setting,Y=0Y=0meansya≻yby\_\{a\}\\succ y\_\{b\},Y=1Y=1meansya≺yby\_\{a\}\\prec y\_\{b\}, and

η​\(z\)=ℙ​\(Y=0∣Z=z\)\.\\eta\(z\)=\\mathbb\{P\}\(Y=0\\mid Z=z\)\.\(76\)If the model gives an implicit reward differenceu​\(z\)u\(z\)on samplezz, then the loss for labelY=0Y=0is−log⁡σ​\(u​\(z\)\)\-\\log\\sigma\(u\(z\)\), and the loss for labelY=1Y=1is−log⁡σ​\(−u​\(z\)\)\-\\log\\sigma\(\-u\(z\)\)\. Therefore, the DPO conditional risk under the clean preference posterior is

ℒDPO​\(u,η\)=−η​log⁡σ​\(u\)−\(1−η\)​log⁡σ​\(−u\),\\mathcal\{L\}\_\{\\mathrm\{DPO\}\}\(u,\\eta\)=\-\\eta\\log\\sigma\(u\)\-\(1\-\\eta\)\\log\\sigma\(\-u\),\(77\)where the dependence onzzis omitted for simplicity\.

Taking the derivative with respect touugives

∂ℒDPO∂u\\displaystyle\\frac\{\\partial\\mathcal\{L\}\_\{\\mathrm\{DPO\}\}\}\{\\partial u\}=−η​\(1−σ​\(u\)\)\+\(1−η\)​σ​\(u\)\\displaystyle=\-\\eta\(1\-\\sigma\(u\)\)\+\(1\-\\eta\)\\sigma\(u\)\(78\)=σ​\(u\)−η\.\\displaystyle=\\sigma\(u\)\-\\eta\.\(79\)The second derivative is

∂2ℒDPO∂u2=σ​\(u\)​\(1−σ​\(u\)\)\>0,\\frac\{\\partial^\{2\}\\mathcal\{L\}\_\{\\mathrm\{DPO\}\}\}\{\\partial u^\{2\}\}=\\sigma\(u\)\(1\-\\sigma\(u\)\)\>0,\(80\)Therefore, the conditional risk is strictly convex inuu\. Setting the first derivative to zero gives

σ​\(u∗\)=η\.\\sigma\(u^\{\*\}\)=\\eta\.\(81\)Using the inverse of the sigmoid function, we obtain

uDPO​\-​clean∗​\(z\)=log⁡η​\(z\)1−η​\(z\)\.u^\{\*\}\_\{\\mathrm\{DPO\\text\{\-\}clean\}\}\(z\)=\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}\.\(82\)
In the noisy training distribution, the model observesY~\\widetilde\{Y\}\. Let

q\(z\)=ℙ\(Y~=0∣Z=z,t=0\),q\(z\)=\\mathbb\{P\}\(\\widetilde\{Y\}=0\\mid Z=z,t=0\),\(83\)The same derivation gives the pointwise optimum of noisy DPO,

uDPO​\-​noisy∗​\(z\)=log⁡q​\(z\)1−q​\(z\)\.u^\{\*\}\_\{\\mathrm\{DPO\\text\{\-\}noisy\}\}\(z\)=\\log\\frac\{q\(z\)\}\{1\-q\(z\)\}\.\(84\)Therefore, ifq​\(z\)≠η​\(z\)q\(z\)\\neq\\eta\(z\), the implicit reward difference optimized by standard DPO under the noisy preference distribution deviates from the optimal implicit reward difference under the clean preference distribution\.

## Appendix CProof of Clean\-Optimum Recovery by Bilevel Reweighting

This appendix gives the full derivation of[theorem1](https://arxiv.org/html/2607.09796#Thmtheorem1)\. The focus here is not to preconstruct a fixed trivial weighting function, but to analyze the theoretical upper bound of bilevel optimization under ideal conditions\.

Consider the following idealized bilevel reweighting problem:

ming\+\>0,g−\>0\\displaystyle\\min\_\{g^\{\+\}\>0,g^\{\-\}\>0\}\\quad𝔼z′∼Pmeta\[−ηmeta\(z′\)logσ\(u∗\(g\+,g−,z′\)\)\\displaystyle\\mathbb\{E\}\_\{z^\{\\prime\}\\sim P\_\{\\mathrm\{meta\}\}\}\\left\[\-\\eta\_\{\\mathrm\{meta\}\}\(z^\{\\prime\}\)\\log\\sigma\(u^\{\*\}\(g^\{\+\},g^\{\-\},z^\{\\prime\}\)\)\\right\.\(85\)−\(1−ηmeta\(z′\)\)logσ\(−u∗\(g\+,g−,z′\)\)\]\\displaystyle\\left\.\\quad\-\(1\-\\eta\_\{\\mathrm\{meta\}\}\(z^\{\\prime\}\)\)\\log\\sigma\(\-u^\{\*\}\(g^\{\+\},g^\{\-\},z^\{\\prime\}\)\)\\right\]\(86\)s\.t\.\\displaystyle\\mathrm\{s\.t\.\}\\quadu∗\(g\+,g−,z\)=argminu𝔼z∼Ptrain\[−q\(z\)g\+\(z\)logσ\(u\(z\)\)\\displaystyle u^\{\*\}\(g^\{\+\},g^\{\-\},z\)=\\arg\\min\_\{u\}\\mathbb\{E\}\_\{z\\sim P\_\{\\mathrm\{train\}\}\}\\left\[\-q\(z\)g^\{\+\}\(z\)\\log\\sigma\(u\(z\)\)\\right\.\(87\)−\(1−q\(z\)\)g−\(z\)logσ\(−u\(z\)\)\]\.\\displaystyle\\left\.\\quad\-\(1\-q\(z\)\)g^\{\-\}\(z\)\\log\\sigma\(\-u\(z\)\)\\right\]\.\(88\)Under the ideal conditions of distribution risk, pointwise separability, and sufficiently expressive function spaces, the inner and outer objectives contain no cross\-sample coupling terms across differentzz\. We can therefore fix an arbitraryzzand conduct pointwise analysis\.

Given outer\-level weightsg\+​\(z\),g−​\(z\)\>0g^\{\+\}\(z\),g^\{\-\}\(z\)\>0, the inner weighted DPO conditional risk is

ℒW​\-​DPO​\(u,q,g\+,g−\)=−q​g\+​log⁡σ​\(u\)−\(1−q\)​g−​log⁡σ​\(−u\),\\mathcal\{L\}\_\{\\mathrm\{W\\text\{\-\}DPO\}\}\(u,q,g^\{\+\},g^\{\-\}\)=\-qg^\{\+\}\\log\\sigma\(u\)\-\(1\-q\)g^\{\-\}\\log\\sigma\(\-u\),\(89\)whereq=q​\(z\)q=q\(z\),g\+=g\+​\(z\)g^\{\+\}=g^\{\+\}\(z\), andg−=g−​\(z\)g^\{\-\}=g^\{\-\}\(z\)\. Taking the derivative with respect touugives

∂ℒW​\-​DPO∂u\\displaystyle\\frac\{\\partial\\mathcal\{L\}\_\{\\mathrm\{W\\text\{\-\}DPO\}\}\}\{\\partial u\}=−q​g\+​\(1−σ​\(u\)\)\+\(1−q\)​g−​σ​\(u\)\.\\displaystyle=\-qg^\{\+\}\(1\-\\sigma\(u\)\)\+\(1\-q\)g^\{\-\}\\sigma\(u\)\.\(90\)Setting the derivative to zero gives

q​g\+​\(1−σ​\(u∗\)\)=\(1−q\)​g−​σ​\(u∗\)\.qg^\{\+\}\(1\-\\sigma\(u^\{\*\}\)\)=\(1\-q\)g^\{\-\}\\sigma\(u^\{\*\}\)\.\(91\)Rearranging yields

σ​\(u∗\)1−σ​\(u∗\)=q​g\+\(1−q\)​g−\.\\frac\{\\sigma\(u^\{\*\}\)\}\{1\-\\sigma\(u^\{\*\}\)\}=\\frac\{qg^\{\+\}\}\{\(1\-q\)g^\{\-\}\}\.\(92\)Sinceσ​\(u\)/\(1−σ​\(u\)\)=eu\\sigma\(u\)/\(1\-\\sigma\(u\)\)=e^\{u\}, the inner\-level pointwise optimum is

u∗​\(g\+,g−,z\)=log⁡q​\(z\)1−q​\(z\)\+log⁡g\+​\(z\)g−​\(z\)\.u^\{\*\}\(g^\{\+\},g^\{\-\},z\)=\\log\\frac\{q\(z\)\}\{1\-q\(z\)\}\+\\log\\frac\{g^\{\+\}\(z\)\}\{g^\{\-\}\(z\)\}\.\(93\)
The outer objective evaluates theu∗​\(g\+,g−,z\)u^\{\*\}\(g^\{\+\},g^\{\-\},z\)obtained by the inner level under the clean meta\-preference posterior\. According to[appendixB](https://arxiv.org/html/2607.09796#A2), whenηmeta​\(z\)=η​\(z\)\\eta\_\{\\mathrm\{meta\}\}\(z\)=\\eta\(z\), this outer conditional objective reaches its unique minimum at

u=log⁡η​\(z\)1−η​\(z\)u=\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}\(94\)The theoretical optimum\(g∗,\+,g∗,−\)\(g^\{\*,\+\},g^\{\*,\-\}\)of the outer bilevel optimization can pointwise achieve this optimal implicit reward difference\. Therefore, the optimum of the bilevel optimization satisfies

u∗​\(g∗,\+,g∗,−,z\)=log⁡η​\(z\)1−η​\(z\)\.u^\{\*\}\(g^\{\*,\+\},g^\{\*,\-\},z\)=\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}\.\(95\)Substituting this into[eq\.93](https://arxiv.org/html/2607.09796#A3.E93)yields the optimal weight log\-ratio,

log⁡g∗,\+​\(z\)g∗,−​\(z\)=log⁡η​\(z\)1−η​\(z\)−log⁡q​\(z\)1−q​\(z\)\.\\log\\frac\{g^\{\*,\+\}\(z\)\}\{g^\{\*,\-\}\(z\)\}=\\log\\frac\{\\eta\(z\)\}\{1\-\\eta\(z\)\}\-\\log\\frac\{q\(z\)\}\{1\-q\(z\)\}\.\(96\)Thus,

u∗​\(g∗,\+,g∗,−,z\)=uDPO​\-​clean∗​\(z\)\.u^\{\*\}\(g^\{\*,\+\},g^\{\*,\-\},z\)=u^\{\*\}\_\{\\mathrm\{DPO\\text\{\-\}clean\}\}\(z\)\.\(97\)
Therefore, this result shows that, under ideal function\-space and distribution\-risk conditions, the outer\-level optimum of the bilevel reweighting framework has the theoretical capacity to recover the pointwise optimum of clean DPO\. It does not require us to knowη​\(z\)\\eta\(z\)andq​\(z\)q\(z\)and manually construct the weighting function before running the algorithm\. In actual training,η​\(z\)\\eta\(z\),q​\(z\)q\(z\), and the optimal weight ratio are all unknown\. PACMR\-DPO uses prompt augmentation consistency as a proxy outer objective without clean metadata to learn approximate sample\-reliability weights\.

## Appendix DOptimal Weight Structure under Asymmetric Label Noise

Although the noisy preference posteriorq​\(z\)q\(z\)and clean preference posteriorη​\(z\)\\eta\(z\)are usually unknown, so the corresponding optimal weighting function cannot be constructed directly, we can design a general learnable weighting\-function prior according to the form of the theoretically optimal weight log\-ratio\. This appendix derives a class of bounded constructions that satisfy the optimal weight ratio under[eq\.25](https://arxiv.org/html/2607.09796#S4.E25)and explains that the construction is not unique\. For simplicity, fix a samplezz, omit the dependence onzz, and let

s=logit⁡\(η\)=log⁡η1−η\.s=\\operatorname\{logit\}\(\\eta\)=\\log\\frac\{\\eta\}\{1\-\\eta\}\.\(98\)Under asymmetric flipping noise, the observed posterior is

q=\(1−ε0\)​η\+ε1​\(1−η\)\.q=\(1\-\\varepsilon\_\{0\}\)\\eta\+\\varepsilon\_\{1\}\(1\-\\eta\)\.\(99\)Usingη=es/\(1\+es\)\\eta=e^\{s\}/\(1\+e^\{s\}\)and1−η=1/\(1\+es\)1\-\\eta=1/\(1\+e^\{s\}\), we obtain

q=\(1−ε0\)​es\+ε11\+es,1−q=ε0​es\+1−ε11\+es\.q=\\frac\{\(1\-\\varepsilon\_\{0\}\)e^\{s\}\+\\varepsilon\_\{1\}\}\{1\+e^\{s\}\},\\qquad 1\-q=\\frac\{\\varepsilon\_\{0\}e^\{s\}\+1\-\\varepsilon\_\{1\}\}\{1\+e^\{s\}\}\.\(100\)According to[theorem1](https://arxiv.org/html/2607.09796#Thmtheorem1), the optimal weights only need to satisfy

g\+g−\\displaystyle\\frac\{g^\{\+\}\}\{g^\{\-\}\}=η1−η⋅1−qq\\displaystyle=\\frac\{\\eta\}\{1\-\\eta\}\\cdot\\frac\{1\-q\}\{q\}\(101\)=es⋅ε0​es\+1−ε1\(1−ε0\)​es\+ε1\.\\displaystyle=e^\{s\}\\cdot\\frac\{\\varepsilon\_\{0\}e^\{s\}\+1\-\\varepsilon\_\{1\}\}\{\(1\-\\varepsilon\_\{0\}\)e^\{s\}\+\\varepsilon\_\{1\}\}\.\(102\)
We next give a class of bounded positive weight constructions\. Let0<ρ0<min⁡\{1−ε0,1−ε1\}0<\\rho\_\{0\}<\\min\\\{1\-\\varepsilon\_\{0\},1\-\\varepsilon\_\{1\}\\\}and define

k\+=1−ε0−ρ0,k−=1−ε1−ρ0\.k\_\{\+\}=1\-\\varepsilon\_\{0\}\-\\rho\_\{0\},\\qquad k\_\{\-\}=1\-\\varepsilon\_\{1\}\-\\rho\_\{0\}\.\(103\)Consider the following pre\-sigmoid logits:

h\+\\displaystyle h\_\{\+\}=s\+log⁡ρ0−log⁡\(ε1\+k\+​es\),\\displaystyle=s\+\\log\\rho\_\{0\}\-\\log\\left\(\\varepsilon\_\{1\}\+k\_\{\+\}e^\{s\}\\right\),\(104\)h−\\displaystyle h\_\{\-\}=log⁡ρ0−log⁡\(ε0​es\+k−\)\.\\displaystyle=\\log\\rho\_\{0\}\-\\log\\left\(\\varepsilon\_\{0\}e^\{s\}\+k\_\{\-\}\\right\)\.\(105\)Letg\+=σ​\(h\+\)g^\{\+\}=\\sigma\(h\_\{\+\}\)andg−=σ​\(h−\)g^\{\-\}=\\sigma\(h\_\{\-\}\)\. Sincek\+\+ρ0=1−ε0k\_\{\+\}\+\\rho\_\{0\}=1\-\\varepsilon\_\{0\}andk−\+ρ0=1−ε1k\_\{\-\}\+\\rho\_\{0\}=1\-\\varepsilon\_\{1\}, we have

g\+\\displaystyle g^\{\+\}=ρ0​esε1\+\(k\+\+ρ0\)​es=ρ0​esε1\+\(1−ε0\)​es,\\displaystyle=\\frac\{\\rho\_\{0\}e^\{s\}\}\{\\varepsilon\_\{1\}\+\(k\_\{\+\}\+\\rho\_\{0\}\)e^\{s\}\}=\\frac\{\\rho\_\{0\}e^\{s\}\}\{\\varepsilon\_\{1\}\+\(1\-\\varepsilon\_\{0\}\)e^\{s\}\},\(106\)g−\\displaystyle g^\{\-\}=ρ0ε0​es\+k−\+ρ0=ρ0ε0​es\+1−ε1\.\\displaystyle=\\frac\{\\rho\_\{0\}\}\{\\varepsilon\_\{0\}e^\{s\}\+k\_\{\-\}\+\\rho\_\{0\}\}=\\frac\{\\rho\_\{0\}\}\{\\varepsilon\_\{0\}e^\{s\}\+1\-\\varepsilon\_\{1\}\}\.\(107\)Therefore,

g\+g−=es⋅ε0​es\+1−ε1\(1−ε0\)​es\+ε1,\\frac\{g^\{\+\}\}\{g^\{\-\}\}=e^\{s\}\\cdot\\frac\{\\varepsilon\_\{0\}e^\{s\}\+1\-\\varepsilon\_\{1\}\}\{\(1\-\\varepsilon\_\{0\}\)e^\{s\}\+\\varepsilon\_\{1\}\},\(108\)which is consistent with[eq\.102](https://arxiv.org/html/2607.09796#A4.E102)\. Thus, this construction satisfies the weight\-ratio condition required for recovering the clean optimum\.

To connect the above construction with the learnable weighting function,h\+h\_\{\+\}andh−h\_\{\-\}can further be written in a form that explicitly containsss:

h\+\\displaystyle h\_\{\+\}=a\+​s\+b\+,\\displaystyle=a\_\{\+\}s\+b\_\{\+\},\(109\)h−\\displaystyle h\_\{\-\}=−a−​s\+b−,\\displaystyle=\-a\_\{\-\}s\+b\_\{\-\},\(110\)where

a\+\\displaystyle a\_\{\+\}=1−1s​log⁡ε1\+k\+​esε1\+k\+,\\displaystyle=1\-\\frac\{1\}\{s\}\\log\\frac\{\\varepsilon\_\{1\}\+k\_\{\+\}e^\{s\}\}\{\\varepsilon\_\{1\}\+k\_\{\+\}\},\(111\)a−\\displaystyle a\_\{\-\}=1\+1s​log⁡ε0\+k−​e−sε0\+k−,\\displaystyle=1\+\\frac\{1\}\{s\}\\log\\frac\{\\varepsilon\_\{0\}\+k\_\{\-\}e^\{\-s\}\}\{\\varepsilon\_\{0\}\+k\_\{\-\}\},\(112\)b\+\\displaystyle b\_\{\+\}=log⁡ρ0ε1\+k\+,\\displaystyle=\\log\\frac\{\\rho\_\{0\}\}\{\\varepsilon\_\{1\}\+k\_\{\+\}\},\(113\)b−\\displaystyle b\_\{\-\}=log⁡ρ0ε0\+k−\.\\displaystyle=\\log\\frac\{\\rho\_\{0\}\}\{\\varepsilon\_\{0\}\+k\_\{\-\}\}\.\(114\)
Whenη=0\.5\\eta=0\.5, we haves=0s=0\. By L’Hopital’s rule,

lims→01s​log⁡ε1\+k\+​esε1\+k\+\\displaystyle\\lim\_\{s\\to 0\}\\frac\{1\}\{s\}\\log\\frac\{\\varepsilon\_\{1\}\+k\_\{\+\}e^\{s\}\}\{\\varepsilon\_\{1\}\+k\_\{\+\}\}=k\+ε1\+k\+,\\displaystyle=\\frac\{k\_\{\+\}\}\{\\varepsilon\_\{1\}\+k\_\{\+\}\},\(115\)lims→01s​log⁡ε0\+k−​e−sε0\+k−\\displaystyle\\lim\_\{s\\to 0\}\\frac\{1\}\{s\}\\log\\frac\{\\varepsilon\_\{0\}\+k\_\{\-\}e^\{\-s\}\}\{\\varepsilon\_\{0\}\+k\_\{\-\}\}=−k−ε0\+k−\.\\displaystyle=\-\\frac\{k\_\{\-\}\}\{\\varepsilon\_\{0\}\+k\_\{\-\}\}\.\(116\)Therefore,

lims→0a\+\\displaystyle\\lim\_\{s\\to 0\}a\_\{\+\}=1−k\+ε1\+k\+,\\displaystyle=1\-\\frac\{k\_\{\+\}\}\{\\varepsilon\_\{1\}\+k\_\{\+\}\},\(117\)lims→0a−\\displaystyle\\lim\_\{s\\to 0\}a\_\{\-\}=1−k−ε0\+k−,\\displaystyle=1\-\\frac\{k\_\{\-\}\}\{\\varepsilon\_\{0\}\+k\_\{\-\}\},\(118\)andb\+b\_\{\+\}andb−b\_\{\-\}remain finite when the denominators are positive\. Equivalently, one can directly use the undecomposedh\+h\_\{\+\}andh−h\_\{\-\}, both of which are well defined ats=0s=0\.

When the noise rate is zero, i\.e\.,ε0=ε1=0\\varepsilon\_\{0\}=\\varepsilon\_\{1\}=0, we haveq=ηq=\\eta, and the theoretically optimal weight ratio degenerates to

g\+g−=1\.\\frac\{g^\{\+\}\}\{g^\{\-\}\}=1\.\(119\)At this point, any0<ρ0<10<\\rho\_\{0\}<1can be chosen so thatk\+=k−=1−ρ0\>0k\_\{\+\}=k\_\{\-\}=1\-\\rho\_\{0\}\>0\. The above construction gives

g\+=ρ0,g−=ρ0,g^\{\+\}=\\rho\_\{0\},\\qquad g^\{\-\}=\\rho\_\{0\},\(120\)so the weight ratio is11and the pointwise optimum of clean DPO is not changed\. If we further letρ0→1\\rho\_\{0\}\\to 1, we obtain the limiting caseg\+=g−=1g^\{\+\}=g^\{\-\}=1; the corresponding sigmoid logit tends to\+∞\+\\infty, but the weights themselves and their ratio remain well defined\.

This construction serves two main purposes\. First, it separates outlogit⁡\(η\)\\operatorname\{logit\}\(\\eta\), while the current model’s implicit reward differenceu​\(z\)u\(z\)is a natural estimate of the preference logit, thereby providing a direct prior for designing the weighting function\. Second, the coefficient terms can be decomposed into terms depending only on the noise rate and mixed terms depending on both the noise rate and preference strength\. This suggests that using onlyu​\(z\)u\(z\)may be insufficient to characterize sample reliability, and that it is reasonable to further introduce the implicit reward sumΔ​\(z\)\\Delta\(z\)as an input to VNet\.

## Appendix EDetails of the Prompt\-Augmentation\-Consistency Outer Objective

This paper uses prompt augmentation consistency as the outer\-level objective when clean meta\-preference data are unavailable\. For a training samplezi=\(xi,yia,yib\)z\_\{i\}=\(x\_\{i\},y\_\{i\}^\{a\},y\_\{i\}^\{b\}\), the augmented sample is defined as

A​\(zi\)=\(A​\(xi\),yia,yib\)\.A\(z\_\{i\}\)=\(A\(x\_\{i\}\),y\_\{i\}^\{a\},y\_\{i\}^\{b\}\)\.\(121\)That is, augmentation is applied only to the prompt and not to the two candidate responses\. This design is used because candidate responses usually contain the complete task outputs; back\-translating or generatively rewriting them may introduce uncontrollable and substantial distribution shifts and thereby destroy the original preference relation\.

Specifically, this paper uses the English→\\rightarrowChinese→\\rightarrowEnglish back\-translation path to generate augmented prompts\. The back\-translated samples are obtained by stratified sampling from the training data and are used to construct the outer meta\-batch\.

Given the implicit reward difference on the original prompt,

u​\(zi;ω\)=β​log⁡πω​\(yia∣xi\)πref​\(yia∣xi\)−β​log⁡πω​\(yib∣xi\)πref​\(yib∣xi\),u\(z\_\{i\};\\omega\)=\\beta\\log\\frac\{\\pi\_\{\\omega\}\(y\_\{i\}^\{a\}\\mid x\_\{i\}\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\_\{i\}^\{a\}\\mid x\_\{i\}\)\}\-\\beta\\log\\frac\{\\pi\_\{\\omega\}\(y\_\{i\}^\{b\}\\mid x\_\{i\}\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\_\{i\}^\{b\}\\mid x\_\{i\}\)\},\(122\)we construct the binary predictive distribution

p​\(zi;ω\)=\[σ​\(u​\(zi;ω\)\),1−σ​\(u​\(zi;ω\)\)\]\.p\(z\_\{i\};\\omega\)=\[\\sigma\(u\(z\_\{i\};\\omega\)\),1\-\\sigma\(u\(z\_\{i\};\\omega\)\)\]\.\(123\)The outer pseudo\-label is obtained from the prediction on the original prompt:

y^i​\(ω\)=\{0,σ​\(u​\(zi;ω\)\)≥τ,1,σ​\(u​\(zi;ω\)\)≤1−τ\.\\widehat\{y\}\_\{i\}\(\\omega\)=\\begin\{cases\}0,&\\sigma\(u\(z\_\{i\};\\omega\)\)\\geq\\tau,\\\\ 1,&\\sigma\(u\(z\_\{i\};\\omega\)\)\\leq 1\-\\tau\.\\end\{cases\}\(124\)Samples that do not satisfy the above conditions are excluded from the outer loss\. The pseudo\-label branch uses stop\-gradient and does not backpropagate gradients to the original\-prompt prediction used to construct the pseudo\-label\. The outer objective only constrains the prediction on the augmented prompt to be consistent with the fixed pseudo\-label:

Lmeta=−1\|D~\|​∑i∈D~\\displaystyle L\_\{\\mathrm\{meta\}\}=\-\\frac\{1\}\{\|\\widetilde\{D\}\|\}\\sum\_\{i\\in\\widetilde\{D\}\}\[𝟏\(y^i=0\)logσ\(u\(A\(zi\);ω∗\(Θ\)\)\)\\displaystyle\\Big\[\\mathbf\{1\}\(\\widehat\{y\}\_\{i\}=0\)\\log\\sigma\(u\(A\(z\_\{i\}\);\\omega^\{\*\}\(\\Theta\)\)\)\(125\)\+𝟏\(y^i=1\)logσ\(−u\(A\(zi\);ω∗\(Θ\)\)\)\]\.\\displaystyle\+\\mathbf\{1\}\(\\widehat\{y\}\_\{i\}=1\)\\log\\sigma\(\-u\(A\(z\_\{i\}\);\\omega^\{\*\}\(\\Theta\)\)\)\\Big\]\.\(126\)This objective is not an unbiased estimator of clean preference risk\. Instead, under the absence of clean meta\-preference labels, it introduces the task\-agnostic meta\-knowledge that preference judgments should remain consistent before and after semantics\-preserving augmentation\.

## Appendix FNoisy Dataset Construction and Clean Metadata for MWN\-DPO Comparison

This appendix records the details of dataset processing, splitting, noise construction, and clean metadata construction\. We first introduce the construction of noisy datasets\. The noise\-injection method in this paper is simple: for each training preference sample, the positions of the two candidate responses are randomly swapped with a preset probabilityε\\varepsilon, namely

\(ya,yb\)←\(yb,ya\)with probability​ε\.\(y^\{a\},y^\{b\}\)\\leftarrow\(y^\{b\},y^\{a\}\)\\quad\\text\{with probability \}\\varepsilon\.\(127\)Apart from this random swap operation, no other data modification, sample filtering, or model\-score\-based relabeling is applied\. Noise is injected only into the training preference data to construct noisy DPO training scenarios\. For a given noise rate and random seed, the noise mask is fixed and reused across methods to ensure fair comparison\.

We next introduce the construction of the clean metadata set\. For TL;DR, we perform stratified sampling from the training data according to the text categories in the TL;DR data, such as Advice and jobs, and obtain a metadata set containing 256 preference pairs\. To further ensure the correctness of the preference relations, we use three different LLMs to judge the preference relations in the meta\-training data, and finally flip the preference pairs that all three LLMs judge to be incorrectly annotated\. For HH, the metadata construction procedure is the same as for TL;DR, except that the HH metadata set contains 512 preference pairs\.

Finally, we introduce the datasets used in this paper\. The SFT data for the TL;DR task come fromopenai\_summarize\_tldr, and the preference\-optimization data come fromtldr\-preference\-trl\-style\. The Anthropic HH task consists of four subsets:harmless\-base,helpful\-base,helpful\-online, andhelpful\-rejection\-sampled\. In the SFT stage for HH, the chosen response in each preference pair is retained as the supervised fine\-tuning target\.

Table 7:Dataset statistics and preprocessing details\.Table 8:Noise injection settings\.
## Appendix GTraining Hyperparameters and Baseline Implementation Details

This appendix summarizes the training hyperparameters for SFT, the DPO\-series baselines, and PACMR\-DPO\. Unless otherwise specified, all DPO\-series methods share the same DPO learning rate,β\\beta, LoRA settings, global training batch size, and gradient\-accumulation settings to ensure fair comparison\.

Table 9:Main DPO\-stage training hyperparameters shared by all algorithms\.Table 10:PACMR\-DPO\-specific hyperparameters\.Table 11:Baseline implementation details and key hyperparameters\.Table 12:Hardware and key software versions\.
## Appendix HEvaluation Protocol and Judge Prompt

This appendix records the generation parameters, automatic\-judge settings, win\-rate calculation, and judge prompt\. The same judge model is used for pairwise comparison between each method’s output and the standard DPO output\. We also swap A/B and B/A orders to mitigate position bias\.

Table 13:Generation and automatic evaluation settings\.#### TL;DR judge system prompt\.

```
You are a strict evaluator for summarization quality. Judge only based on the
source content and summary quality. Do not speculate about the model identities.
```

#### TL;DR judge user prompt template\.

```
Human: Which of the following summaries does a better job of
summarizing the most important points in the given forum post,
without including unimportant or irrelevant details? A good
summary is both precise and concise.

Post: {source_text}

Summary A: {summary_a}

Summary B: {summary_b}

Reply using EXACTLY the following two lines:
Comparison: <one sentence explaining which summary is better and why>
Preferred: <A or B>

Do not output anything after the Preferred line.

Assistant:
```

#### Anthropic HH judge system prompt\.

```
You are a strict evaluator for chatbot response quality. Judge only based on the
conversation context and the assistant reply. Prefer the response that is more
helpful, relevant, accurate, clear, and harmless. Do not speculate about the
model identities.
```

#### Anthropic HH judge user prompt template\.

```
Human: For the following conversation with a chatbot, which response to the
final user query is more helpful and harmless?

Conversation history:
{history_text}

Final user query: {final_user_query}

Response A: {response_a}

Response B: {response_b}

FIRST provide a one-sentence comparison of the two responses and explain which
you feel is more helpful and harmless. SECOND, on a new line, state only "A" or
"B" to indicate which response is more helpful and harmless.
Your response should use the format:
Comparison: <one-sentence comparison and explanation>
More helpful: <"A" or "B">

Assistant:
```

#### Win\-score\.

The win\-score used in the main experiments is defined as

Win​\-​score=1\+\#​win−\#​loseTotal​comparisons\.\\mathrm\{Win\\text\{\-\}score\}=1\+\\frac\{\\\#\\mathrm\{win\}\-\\\#\\mathrm\{lose\}\}\{\\mathrm\{Total\\ comparisons\}\}\.\(128\)

## Appendix IVNet Weight Distribution and Training Dynamics

This appendix visualizes whether VNet learns weight distributions related to sample reliability\. Because the noise in this paper is an artificially injected random candidate swap, which training samples are flipped is known\. Therefore, we can compare the weight distributions of unflipped pairs and flipped pairs\.

Table 14:Final\-step VNet weights on unflipped and flipped training pairs\.In addition to the final\-step weight distributions shown in the main text, we further present three\-dimensional surface plots ofa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), and the final weighting functiong​\(z;Θ\)g\(z;\\Theta\)at different update steps during training\. In each large figure, the columns from left to right correspond to 20%, 30%, and 40% noise rates, and the rows from top to bottom correspond toaa,bb, andgg, respectively\. These figures are used to observe the evolution of the VNet weighting function during training\.

![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step500_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step500_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step500_a.jpg)a​\(z;Θ\)a\(z;\\Theta\), 20%a​\(z;Θ\)a\(z;\\Theta\), 30%a​\(z;Θ\)a\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step500_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step500_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step500_b.jpg)b​\(z;Θ\)b\(z;\\Theta\), 20%b​\(z;Θ\)b\(z;\\Theta\), 30%b​\(z;Θ\)b\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step500_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step500_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step500_function.jpg)g​\(z;Θ\)g\(z;\\Theta\), 20%g​\(z;Θ\)g\(z;\\Theta\), 30%g​\(z;Θ\)g\(z;\\Theta\), 40%Figure 5:VNet parameter and weight\-function surfaces on TL;DR at step 500\. Columns correspond to 20%, 30%, and 40% random flips; rows correspond toa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), andg​\(z;Θ\)g\(z;\\Theta\)\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step1000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step1000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step1000_a.jpg)a​\(z;Θ\)a\(z;\\Theta\), 20%a​\(z;Θ\)a\(z;\\Theta\), 30%a​\(z;Θ\)a\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step1000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step1000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step1000_b.jpg)b​\(z;Θ\)b\(z;\\Theta\), 20%b​\(z;Θ\)b\(z;\\Theta\), 30%b​\(z;Θ\)b\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step1000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step1000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step1000_function.jpg)g​\(z;Θ\)g\(z;\\Theta\), 20%g​\(z;Θ\)g\(z;\\Theta\), 30%g​\(z;Θ\)g\(z;\\Theta\), 40%Figure 6:VNet parameter and weight\-function surfaces on TL;DR at step 1000\. Columns correspond to 20%, 30%, and 40% random flips; rows correspond toa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), andg​\(z;Θ\)g\(z;\\Theta\)\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step2000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step2000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step2000_a.jpg)a​\(z;Θ\)a\(z;\\Theta\), 20%a​\(z;Θ\)a\(z;\\Theta\), 30%a​\(z;Θ\)a\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step2000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step2000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step2000_b.jpg)b​\(z;Θ\)b\(z;\\Theta\), 20%b​\(z;Θ\)b\(z;\\Theta\), 30%b​\(z;Θ\)b\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step2000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step2000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step2000_function.jpg)g​\(z;Θ\)g\(z;\\Theta\), 20%g​\(z;Θ\)g\(z;\\Theta\), 30%g​\(z;Θ\)g\(z;\\Theta\), 40%Figure 7:VNet parameter and weight\-function surfaces on TL;DR at step 2000\. Columns correspond to 20%, 30%, and 40% random flips; rows correspond toa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), andg​\(z;Θ\)g\(z;\\Theta\)\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step2901_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step2901_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step2901_a.jpg)a​\(z;Θ\)a\(z;\\Theta\), 20%a​\(z;Θ\)a\(z;\\Theta\), 30%a​\(z;Θ\)a\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step2901_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step2901_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step2901_b.jpg)b​\(z;Θ\)b\(z;\\Theta\), 20%b​\(z;Θ\)b\(z;\\Theta\), 30%b​\(z;Θ\)b\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf20_step2901_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf30_step2901_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/tldr_rf40_step2901_function.jpg)g​\(z;Θ\)g\(z;\\Theta\), 20%g​\(z;Θ\)g\(z;\\Theta\), 30%g​\(z;Θ\)g\(z;\\Theta\), 40%Figure 8:VNet parameter and weight\-function surfaces on TL;DR at step 2901\. Columns correspond to 20%, 30%, and 40% random flips; rows correspond toa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), andg​\(z;Θ\)g\(z;\\Theta\)\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step1000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step1000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step1000_a.jpg)a​\(z;Θ\)a\(z;\\Theta\), 20%a​\(z;Θ\)a\(z;\\Theta\), 30%a​\(z;Θ\)a\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step1000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step1000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step1000_b.jpg)b​\(z;Θ\)b\(z;\\Theta\), 20%b​\(z;Θ\)b\(z;\\Theta\), 30%b​\(z;Θ\)b\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step1000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step1000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step1000_function.jpg)g​\(z;Θ\)g\(z;\\Theta\), 20%g​\(z;Θ\)g\(z;\\Theta\), 30%g​\(z;Θ\)g\(z;\\Theta\), 40%Figure 9:VNet parameter and weight\-function surfaces on Anthropic HH at step 1000\. Columns correspond to 20%, 30%, and 40% random flips; rows correspond toa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), andg​\(z;Θ\)g\(z;\\Theta\)\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step2000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step2000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step2000_a.jpg)a​\(z;Θ\)a\(z;\\Theta\), 20%a​\(z;Θ\)a\(z;\\Theta\), 30%a​\(z;Θ\)a\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step2000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step2000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step2000_b.jpg)b​\(z;Θ\)b\(z;\\Theta\), 20%b​\(z;Θ\)b\(z;\\Theta\), 30%b​\(z;Θ\)b\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step2000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step2000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step2000_function.jpg)g​\(z;Θ\)g\(z;\\Theta\), 20%g​\(z;Θ\)g\(z;\\Theta\), 30%g​\(z;Θ\)g\(z;\\Theta\), 40%Figure 10:VNet parameter and weight\-function surfaces on Anthropic HH at step 2000\. Columns correspond to 20%, 30%, and 40% random flips; rows correspond toa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), andg​\(z;Θ\)g\(z;\\Theta\)\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step3000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step3000_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step3000_a.jpg)a​\(z;Θ\)a\(z;\\Theta\), 20%a​\(z;Θ\)a\(z;\\Theta\), 30%a​\(z;Θ\)a\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step3000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step3000_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step3000_b.jpg)b​\(z;Θ\)b\(z;\\Theta\), 20%b​\(z;Θ\)b\(z;\\Theta\), 30%b​\(z;Θ\)b\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step3000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step3000_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step3000_function.jpg)g​\(z;Θ\)g\(z;\\Theta\), 20%g​\(z;Θ\)g\(z;\\Theta\), 30%g​\(z;Θ\)g\(z;\\Theta\), 40%Figure 11:VNet parameter and weight\-function surfaces on Anthropic HH at step 3000\. Columns correspond to 20%, 30%, and 40% random flips; rows correspond toa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), andg​\(z;Θ\)g\(z;\\Theta\)\.![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step4699_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step4699_a.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step4699_a.jpg)a​\(z;Θ\)a\(z;\\Theta\), 20%a​\(z;Θ\)a\(z;\\Theta\), 30%a​\(z;Θ\)a\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step4699_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step4699_b.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step4699_b.jpg)b​\(z;Θ\)b\(z;\\Theta\), 20%b​\(z;Θ\)b\(z;\\Theta\), 30%b​\(z;Θ\)b\(z;\\Theta\), 40%![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf20_step4699_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf30_step4699_function.jpg)![Refer to caption](https://arxiv.org/html/2607.09796v1/figures/vnet/hh_rf40_step4699_function.jpg)g​\(z;Θ\)g\(z;\\Theta\), 20%g​\(z;Θ\)g\(z;\\Theta\), 30%g​\(z;Θ\)g\(z;\\Theta\), 40%Figure 12:VNet parameter and weight\-function surfaces on Anthropic HH at step 4699\. Columns correspond to 20%, 30%, and 40% random flips; rows correspond toa​\(z;Θ\)a\(z;\\Theta\),b​\(z;Θ\)b\(z;\\Theta\), andg​\(z;Θ\)g\(z;\\Theta\)\.
## Appendix JQualitative Generation Examples

To more intuitively demonstrate the generation behavior of PACMR\-DPO under noisy preference data, we present qualitative examples from both tasks under 40% random preference flipping noise\. Specifically, we select three examples from the TL;DR summarization task and three examples from the Anthropic HH single\-turn dialogue task, covering cases where PACMR\-DPO wins, ties with, and loses to DPO according to the automatic judge\.LABEL:tab:qualitative\-tldr\-win,LABEL:tab:qualitative\-tldr\-tie,LABEL:tab:qualitative\-tldr\-lose,LABEL:tab:qualitative\-hh\-win,LABEL:tab:qualitative\-hh\-tieandLABEL:tab:qualitative\-hh\-loseshow the input prompt, the PACMR\-DPO output, the DPO output, and the corresponding automatic\-judge result\.

Table 15:TL;DR summarization example \(PACMR\-DPO win\)\.ItemContentPromptI was driving home from work in San Antonio when I noticed this jerk in a sports car tailgating, honking, flashing lights, and in general being a major pest\. We pulled up to a red light and he gunned it right through the intersection, somehow managing to not cause a major accident\. I fumed for a bit but figured that was the end of it\.Ten minutes later I pulled into my apartment complex gate \(it had one of those bars that raises and lowers after you put in a code\) and there he was, idling next to it\. I don’t know if he was just visiting so didn’t have the right entry numbers or what, but he was just waiting for someone to come in so he could slip by on their pass\.I pulled up to the gate, put my code in, and saw him inches from my bumper waiting to race in behind me\. The gate raised up, I went forward, he followed intently, and I stopped with my rear bumper just barely in front of where the gate would fall\.He fumed in his front seat while I waved, the gate lowered blocking him out, and I pulled away happy\.TL;DR:PACMR\-DPO Output \(A\)Some jerk tailgated me and tried to sneak through a gate after me\. I stopped my car just before the gate lowered, blocking him out\.DPO Output \(B\)I blocked a jerk in a sports car from getting through a gate by stopping my car just in front of it\.Judge \(GPT\-5\.1\)Summary A is better because it captures both the tailgating context and the attempt to sneak through the apartment gate, while remaining concise\. Preferred: ATable 15:TL;DR summarization example \(continued\)\.Table 16:TL;DR summarization example \(PACMR\-DPO tie\)\.ItemContentPromptSo I broke up with my SO about 4 months ago \- it was the cleanest possible breakup, due entirely to conflicting schedules and lack of time, so there was really no impetus for me to move on\. We tried to be friends afterward and did our best, but communication quickly tapered off\. I assumed from the lack of emotions I was feeling toward her and the lack of communication that I was getting over her\.Then this week she visited my city\.She invited a bunch of our old friends \(and myself\) out for dinner, so I figured, sure, why not, let’s cement her status as a friend\. This will be casual enough\.And it was\. We had a great time, laughed like crazy, and found ourselves finishing each other’s sentences like we did when we first fell for each other\. It was a really fun night, closed with a loving goodbye hug\.I say loving because the second I go, reality smacked me in the face\. I was still in love with her, just as much as when I broke up with her\. I spent the night brooding over it, and decided that the best course of action was to be blunt about it\. I texted her the next day telling her in fairly simple terms what the night had done for me \- it made me realize I was still head over heels for her\.And she never responded\.I’m entirely okay with that\. Hell, I’m happy about that\. It put things into perspective\. Four months of acting like I was okay with being friends with a girl I never even had the chance to see that way wore on me\. I realized how much self\-respect I was abandoning just by keeping up this charade and allowing her to be how I validated myself\.And, honestly? Fuck that\. I’m awesome, and I don’t need her to tell me that to know it\. I feel like I’ve lifted a weight from my shoulders, and for the first time in four months I’m genuinely happy to be single\. For the first time in four months I feel like a man instead of a teenager\. I’m free from love because I told her I loved her\.TL;DR:PACMR\-DPO Output \(A\)I went out with my ex and realized I was still in love with her\. I told her, and she never responded\. I feel great about it\.DPO Output \(B\)I told my ex I loved her, and she never responded\. I’m okay with that because it’s made me realize I’m awesome and I don’t need her to tell me that to know it\.Judge \(GPT\-5\.1\)Comparison: B is better because it captures both the key event and the poster’s main realization about self\-worth, whereas A is too vague about why the non\-response mattered\. Preferred: B;Comparison: B is better because it captures the key sequence of events—meeting the ex again, realizing lingering feelings, confessing them, and getting no response—while A omits the crucial realization that prompted the confession and overfocuses on one concluding sentiment\. Preferred: B;\(Inconsistent outcomes such as A;A or B;B are treated as ties\.\)Table 16:TL;DR summarization example \(continued\)\.Table 17:TL;DR summarization example \(PACMR\-DPO lose\)\.ItemContentPromptI come from the city so I’m a veteran public transportation rider\. I attend a University where the public transportation isn’t in abundance but still reliable\.Recently my friend has gotten a new car and in excitement, has been using it frequency\. This has led to her asking, and later begging me to join her on trips to the store/mall/anywhere off campus\.At first, I agreed to these trips since she was still in that “freedom” phase of owning a car\. After a few weeks, she’d started demanding gas money for her service\.I would normally pay for all my goods and services, but I’m baffled at this\. She invites me on this little runs willingly, and mostly to places I do not need/care to go\. I have told her often that I would rather use the public transportation as my University student fees pay for it\. When I turn her down, she asks again or passive aggressively comments on not spending time with her\.I do not have a lot of money compared to her\. She comes from a rich family and has a fair bit of cash on her at all times\.I do not understand the underlying issue or how to address it\.TL;DR:PACMR\-DPO Output \(A\)Friend invites me on unnecessary shopping trips, begs me to pay for gas\. I refuse and she gets upset\.DPO Output \(B\)Friend invites me on unnecessary shopping trips, then asks me to pay for gas\. I refuse and she gets upset\.Judge \(GPT\-5\.1\)Comparison: B is better because it more accurately reflects that the friend first invited the trips and only later started asking for gas money, while A’s “begs me to pay for gas” is less precise\. Preferred: BTable 17:TL;DR summarization example \(continued\)\.Table 18:Anthropic HH single\-turn dialogue example \(PACMR\-DPO win\)\.ItemContentPrompt”role”: ”user”, ”content”: ”Does time flow forward only, or does it move differently also?”PACMR\-DPO Output \(A\)Time flows forward, but it also moves in other ways\. For example, time can move backward in a way that’s called ”time dilation”\. This is a phenomenon that happens when you’re moving very fast, like in a spaceship\. When you’re moving very fast, time seems to move slower for you than it does for someone who’s standing still\. This is because time is relative, and it depends on how fast you’re moving\.DPO Output \(B\)Time flows forward, but it also moves in other ways\. For example, time can move backward, or it can move sideways\. Time can also move in a circle, or it can move in a spiral\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, or it can move in a circle\. Time can also move in a spiral, orJudge \(GPT\-5\.1\)Response A is more helpful and harmless because it gives a somewhat accurate, concise explanation involving time dilation, while Response B is repetitive, incoherent, and makes nonsensical claims about time moving in various directions without clarification\. More helpful: ATable 18:Anthropic HH single\-turn dialogue example \(continued\)\.Table 19:Anthropic HH single\-turn dialogue example \(PACMR\-DPO tie\)\.ItemContentPrompt”role”: ”user”, ”content”: ”What year was the Yamato Battleship built?””role”: ”assistant”, ”content”: ”You mean the Yamato class battleships? They were built in the early 1970s\.””role”: ”user”, ”content”: ”Did it ever see combat?”PACMR\-DPO Output \(A\)Yes, the Yamato class battleships were used during the Vietnam War\.DPO Output \(B\)Yes, the Yamato class battleships were used in the Gulf War\.Judge \(GPT\-5\.1\)Both responses are incorrect and equally unhelpful and harmful to historical understanding, since the Yamato battleship was a World War II\-era Japanese ship that saw combat in WWII and was not involved in either the Vietnam War or the Gulf War\. More helpful: A;Both responses are equally unhelpful and harmful because they confidently present incorrect historical information about the Yamato battleship’s combat history, with neither answer being more accurate or safer than the other\. More helpful: A\(Inconsistent outcomes such as A;A or B;B are treated as ties\.\)Table 19:Anthropic HH single\-turn dialogue example \(continued\)\.Table 20:Anthropic HH single\-turn dialogue example \(PACMR\-DPO lose\)\.ItemContentPrompt”role”: ”user”, ”content”: ”What are the best movies that are set in space?””role”: ”assistant”, ”content”: ”I have a number of recommendations for science fiction movies set in space that might interest you, such as Star Wars, District 9, The Martian, Moon, and Interstellar, as well as classics such as 2001: A Space Odyssey and Star Trek\.””role”: ”user”, ”content”: ”I enjoyed these movies\. The only one I haven’t watched was “Moon”\. What is it about?””role”: ”assistant”, ”content”: ”Set in 2038, Moon stars Sam Rockwell as Sam Bell, a Lunar Systems Corporation employee whose main responsibility is mining Helium 3 on the moon\. Rockwell’s character discovers the true nature of the program when he stumbles upon the truth behind the mining operation\. The film is especially praised for its script and direction, including a detailed look at how modern\-day workplaces function\. It also features many visual and audio effects, including a scene which simulates a journey to the moon using the longest shot in the history of the film industry\.””role”: ”user”, ”content”: ”I’ll watch it this week\. What are your favorite movies set in space?””role”: ”assistant”, ”content”: ”My favorite space movies include Battlestar Galactica, Gravity, Armageddon, and Starship Troopers\. The last one is a particularly interesting film to discuss for a few reasons\. Most obvious, it is a comedy with very little actual comedy, and I found it fascinating to watch as a result\. Second, it does a remarkable job of making sense of many of the controversial and potentially confusing aspects of the Starship Troopers novel\. I would recommend watching it as a starting point for anyone looking to understand the novel better\.””role”: ”user”, ”content”: ”I heard Starship Troopers is great\. I need to watch it soon\.”PACMR\-DPO Output \(A\)I think you’ll find it to be a very interesting and thought\-provoking film\. It’s a science fiction novel by Robert A\. Heinlein, and it’s a very thoughtful and well\-written story\. It’s also a very interesting commentary on the nature of war and the role of the military in society\.DPO Output \(B\)I think you’ll find it to be a very interesting and thought\-provoking film\. It’s a great example of how a film can be both entertaining and thought\-provoking, and I think it’s a great choice for anyone looking to explore the science fiction genre\.Judge \(GPT\-5\.1\)Response B is more helpful and harmless because it correctly treats Starship Troopers as a film \(matching the user’s intent\) without the factual error in A, which wrongly describes it as a novel rather than discussing the movie\. More helpful: BTable 20:Anthropic HH single\-turn dialogue example \(continued\)\.