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PAPER:


# Distributionally Robust Graph Out-of-Distribution

# Recommendation via Diffusion Model

## Anonymous Author(s)∗

```
Abstract
The distributionally robust optimization (DRO)-based graph neu-
ral network methods improve recommendation systems’ out-of-
distribution (OOD) generalization by optimizing the model’s worst-
case performance. However, these studies fail to consider the impact
of noisy samples in the training data, which results in diminished
generalization capabilities and lower accuracy. Through experi-
mental and theoretical analysis, this paper reveals that current
DRO-based graph recommendation methods assign greater weight
to noise distribution, leading to model parameter learning being
dominated by it. When the model overly focuses on fitting noise
samples in the training data, it may learn irrelevant or meaning-
less features that cannot be generalized to OOD data. To address
this challenge, we design aDistributionallyRobustGraph model
forOOD recommendation (DRGO). Specifically, our method first
employs a simple and effective diffusion paradigm to alleviate the
noisy effect in the latent space. Additionally, an entropy regular-
ization term is introduced in the DRO objective function to avoid
extreme sample weights in the worst-case distribution. Finally, we
provide a theoretical proof of the generalization error bound of
DRGO as well as a theoretical analysis of how our approach mit-
igates noisy sample effects, which helps to better understand the
proposed framework from a theoretical perspective. We conduct
extensive experiments on four datasets to evaluate the effective-
ness of our framework against three typical distribution shifts, and
the results demonstrate its superiority in both independently and
identically distributed distributions (IID) and OOD. Our code is
available at https://anonymous.4open.science/r/DRGO-FED2.
```
CCS Concepts

- Information systems→Recommender systems.

```
Keywords
Graph Recommendation, Distributionally Robust Optimization,
Out-of-Distribution
```
```
ACM Reference Format:
Anonymous Author(s). 2018. Distributionally Robust Graph Out-of-Distribution
Recommendation via Diffusion Model. InProceedings of Make sure to en-
ter the correct conference title from your rights confirmation emai (Confer-
ence acronym ’XX).ACM, New York, NY, USA, 13 pages. https://doi.org/
XXXXXXX.XXXXXXX
```
```
Permission to make digital or hard copies of all or part of this work for personal or
classroom use is granted without fee provided that copies are not made or distributed
for profit or commercial advantage and that copies bear this notice and the full citation
on the first page. Copyrights for components of this work owned by others than the
author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or
republish, to post on servers or to redistribute to lists, requires prior specific permission
and/or a fee. Request permissions from permissions@acm.org.
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
©2018 Copyright held by the owner/author(s). Publication rights licensed to ACM.
ACM ISBN 978-1-4503-XXXX-X/18/
https://doi.org/XXXXXXX.XXXXXXX
```
```
1 Introduction
Recently, Graph Neural Network (GNN)-based recommendation
methods have gained extensive attention in various recommenda-
tion tasks and scenarios [ 3 , 9 , 12 , 41 , 46 ]. Generally, GNN-based
methods learn embeddings of users or items by capturing high-
order collaborative signals from the user-item interaction graph.
Although these GNN-based methods have achieved state-of-the-art
performance in the collaborative recommendation, some assume
that the training and test sets follow an independent and identical
distribution (IID). Unfortunately, this assumption may not hold
when faced with out-of-distribution (OOD) data (i.e., distribution
shift) in real-world recommendation scenarios. Such data distribu-
tion shifts may arise from various factors, such as changes in user
consumption habits influenced by seasons, holidays, policies, etc.
Additionally, various biases (e.g., popularity bias and exposure bias)
in the recommendation system can lead to inconsistencies between
training and test data distributions.
Existing works have attempted to address the challenge of distri-
bution shift using various techniques to enhance the generalization
of recommendation models. For instance, some works [ 5 , 24 , 42 ]
use causal learning and approximate inference methods to infer
environment labels and further learn representations that are in-
sensitive to the environment. Some works [ 30 , 38 ] achieve OOD
generalization by directly decoupling user-variant and invariant
representations. In addition, other works employ data augmenta-
tion or self-supervised learning [ 31 , 37 ] methods to enhance the
generalization performance of GNN-based methods on OOD data.
However, the methods mentioned above also have some limitations
in addressing distribution shifts: (1) The performance of these meth-
ods, such as [ 5 , 24 , 42 ], relies on explicit environments as labels. If
the environmental factors cannot be inferred, achieving satisfac-
tory generalization will be difficult. And these methods [ 30 , 38 ] to
disentangle variations from invariant preferences lack theoretical
guarantees. (2) These methods [ 31 , 37 , 37 ] target a specific type of
bias, such as popularity bias, and cannot address scenarios with
multiple complex data distribution shifts.
Recently, to address the limitations of these methods, researchers
are leveraging Distributionally Robust Optimization (DRO) [ 20 ]
to enhance recommendation models’ ability to generalize beyond
their training data [ 23 , 35 , 43 ]. DRO focuses on improving model
robustness by considering a set of potential distributions and opti-
mizing performance for the worst-case scenario among them. This
approach helps ensure the model performs well even when facing
distribution shifts or operating in new environments. Despite the
success, our experimental and theoretical analyses suggest that ex-
isting DRO-based recommendation methods still face the following
two challenges:
```
- Challenge 1: Existing DRO-based recommendation methods [ 23 ,
    43 ] assign greater weight to noise distribution when searching for
    the worst-case scenario. This causes model parameter learning


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DRO-LightGCN DRGO
Methods
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LightGCN DRO-LightGCN DR-GNN DRGO
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(b) Impact of Noisy
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```
Figure 1: (a) Divide the user-item interaction data in the
Yelp2018 dataset into a major group (90%) and a minor group
(10%) based on user activity levels. Inject an additional 10% of
the data as a noise group. Compare the performance of DRO-
LightGCN and our DRGO, observing the weight changes for
each group across multiple iterations. (b) The model’s perfor-
mance under different noise levels on the Yelp2018 dataset.
```
```
to be dominated by noise, leading to overfitting on irrelevant
features and further reducing generalization on OOD data.
```
- Challenge 2: Certain DRO methods based on the Kullback-
    Leibler (KL) divergence [ 23 , 35 ] assume that there is an overlap
    between the distributions of prior interactions and the testing
    phase. However, due to the randomness and unpredictability
    of user behavior, the overlap between the test distribution and
    the training distribution may be minimal or even nonexistent,
    which makes this assumption limited. This situation hinders the
    effective computation of the KL divergence, thereby restricting
    the robustness and performance of the model.
For the first challenge, as illustrated in Figure 1, we conduct ex-
periments to verify how noise samples affect the generalization
performance of the DRO model. Figure 1(a) illustrates the weight
changes across different groups after several training iterations, re-
vealing that the performance of DRO-LightGCN [ 23 , 43 ] is affected
by the noise distribution. Figure 1(b) shows performance variations
under different noise ratios, indicating that when the noisy ratio
exceeds 20%, the performance of DRO-LightGCN falls behind that
of LightGCN. Although DR-GNN is also affected by noisy data, its
data augmentation component can alleviate the impact of noise
to some extent. Section 3.1 provides a more detailed theoretical
analysis of the challenges mentioned above.
    Considering the limitations and challenges of current solutions,
we introduce a novel Distributionally Robust Graph framework,
termed DRGO, aimed at improving the robustness and generaliza-
tion of recommendation systems when dealing with OOD data.
To address the first challenge, our DRGO utilizes a graph varia-
tional autoencoder to encode the feature structure of the graph
into a fixed distribution. Then, a diffusion model is employed in a
low-dimensional embedding space to alleviate the impact of noise
samples in the interaction data. We also introduce an entropy regu-
larization term into the objective function of DRGO to prevent the
model from optimizing over extreme distributions. For the second
challenge, we attempt to utilize Sinkhorn DRO as a replacement
for KL-based DRO. Sinkhorn DRO maintains model robustness in

```
non-overlapping or shifted distributions, ensuring minimal perfor-
mance degradation in worst-case scenarios. We rigorously provide
the theoretical proof of the generalization bound of our DRGO on
OOD data, as well as the theoretical analysis of how our DRGO
mitigates the impact of noisy samples.
The main contributions of our paper are summarized as follows:
```
- Motivation. Through experimental and theoretical analysis, we
    reveal the limitations and shortcomings of DRO-based graph
    recommendation methods in addressing data distribution shifts.
- Methodology. This work proposes a novel Distributionally
    Robust Graph-based method for OOD recommendation called
    DRGO. It integrates diffusion models and an entropy regular-
    ization term to remove noise while reducing focus on extreme
    distributions (such as noise distribution). Additionally, we pro-
    vide rigorous theoretical guarantees for DRGO.
- Experiment. We conduct extensive experiments to validate the
    superiority of our proposed method under three types of distri-
    bution shifts: popularity, temporal, and exposure. The results
    confirm the model’s exceptional performance on both OOD and
    IID data.

```
2 Preliminary
In this section, we first define the notations used in this paper and
then introduce the definition of DRO. Due to page limitations, we
have provided an introduction to Diffusion model in Appendix E.
```
```
2.1 Problem Definition
Recommendation on Graphs. In GNN-based recommendation
methods, we consider a given user-item interaction matrixR∈
R|U|×|V|, whereU={𝑢 1 ,𝑢 2 ,.. .,𝑢𝑚}denotes the user set and
I={𝑖 1 ,𝑖 2 ,.. .,𝑖𝑛}represents the item set. GNN-based methods
first convert the interaction matrix into graph-structured data
G={U,I,E}and theE=(𝑢,𝑖)|𝑟𝑢,𝑖= 1 is the edge set. GNN-
based methods useGas input to capture higher-order collaborative
signals between users and items, thereby inferring user preferences
towards items.
Distribution shift on Recommendation. Due to environmen-
tal factors, training and testing data distributions may differ in
real-world scenarios. The environment is a key concept, as it is the
direct cause of data-generating distributions. For example, in rec-
ommendation scenarios, weather, policy changes, and user mood
can be considered environments. These factors can lead to changes
in user behavior, resulting in distribution shifts in the data. Most tra-
ditional recommendation models trained under the IID assumption
fail to generalize OOD data well.
```
```
2.2 Distributionally Robust Optimization
Distributionally Robust Optimization (DRO) differs from other OOD
recommendation methods [ 5 , 24 , 42 ] that require environmental
labels. Its core idea is to identify and optimize for the worst-case
data distribution within a predefined range of distributions. This
approach ensures that the model maintains good performance when
facing distribution changes or shifts, thereby improving the model’s
generalization performance on OOD data. DRO is typically solved
through a bi-level optimization problem: the outer optimization
determines the model parameters that perform best under the worst
```

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Distributionally Robust Graph Out-of-Distribution Recommendation via Diffusion Model Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
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```
```
Graph
Encoder ...
```
```
Graph
Decoder
```
```
q E E(|t0)
Steps t
E 0 EtEˆt Eˆ 0
```
```
qE(ˆ 0 | Eˆt-1)
```
```
N( μ , σ )
```
```
Nominal distribution
```
```
Sinkhorn DRO
```
```
Bipartite Graph VGAE Diffusion Module
```
```
Denoising
Embeddings
```
```
σ (s, t | v)
σ (s, t)
Betweenness centrality
```
```
Sinkhorn DRO DRGO BPR
```
```
Joint Optimizing
```
```
Ptrain
```
```
Q = KMeans E()ˆ 0
```
```
Figure 2: The proposed DRGO model schematic. An bipar-
tite graph is first processed using VGAE diffusion to obtain
denoised embeddings. Then, betweenness centrality and𝑘-
means clustering are applied to construct the nominal dis-
tribution and uncertainty set. Finally, a joint optimization
strategy is employed for optimization.
```
```
distribution, while the inner optimization adjusts for all possible
distributions. Its mathematical formulation is as follows:
```
```
min
𝜃∈Θ
```
### (

```
𝑅(𝜃):= sup
𝑄∈P(𝑃𝑡𝑟𝑎𝑖𝑛)
```
### E(𝑥,𝑦)∼𝑄[L(𝑓𝜃(𝑥),𝑦)]

### )

```
s.t.𝐷(𝑄,𝑃𝑡𝑟𝑎𝑖𝑛) ≤𝜌,
```
### (1)

```
where𝜃represents the optimal model parameters in DRO, and
L(·)denotes the loss function.supis the supremum, which means
finding the maximum value under the given conditions. The uncer-
tainty set𝑄approximates potential test distributions. It is typically
defined as a divergence ball with radius𝜌centered around the nom-
inal distribution𝑃𝑡𝑟𝑎𝑖𝑛, expressed as𝑄={𝑄:𝐷(𝑄,𝑃𝑡𝑟𝑎𝑖𝑛) ≤𝜌},
where𝐷(·,·)represents a distance metric such as Wasserstein dis-
tance or Kullback-Leibler (KL) divergence. Generally, constructing
the uncertainty set and the nominal distribution is crucial to DRO.
```
```
3 Methodology
In this section, we first discuss the limitations of existing DRO-
based methods, then propose a novel DRO approach and present
the complete model structure, followed by a theoretical analysis.
Figure 2 and Algorithm 1 (in Appendix) show the overall framework
of the model and the algorithm pseudocode.
```
```
3.1 Limitations of DRO-based Recommendation
Method
Drawback of KL divergence. Some recent works [ 23 , 35 ] assume
an overlap between the distributions of the training and testing
sets and use KL divergence as a constraint for the uncertainty set.
However, in practical scenarios, user behavior may exhibit high lev-
els of uncertainty and significant randomness, leading to situations
```
```
where the distribution of the testing set shares no standard support
with that of the training set. KL divergence becomes extremely
large in such cases, rendering the model optimization meaningless
and preventing effective OOD generalization. We give a detailed
theoretical derivation in the Appendix A.1.
Impact of noise samples. Without loss of generality, we use
the Bayesian Personalized Ranking (BPR) [ 21 ] loss, a commonly
used ranking function, to analyze the impact of noise samples on
recommendation performance. Specifically, DRO can be viewed as
optimizing over a weighted empirical distribution. Following the
method proposed in [ 14 ], we demonstrate from the perspective of
model parameter variance how noise influences the learning of
DRO model parameters through weighted BPR loss. In the BPR loss
function, the variance of the model parameters reflects the model’s
sensitivity to various samples, especially when the training data
contains noise. This sensitivity can lead to increased variance and
instability of the model. Due to page limitations, please refer to
Appendix A.2 for more detailed theoretical derivation. Based on the
weighted BPR loss function and the derived variance of the model
parameters, we can get the following conclusions:
```
- If the weight of noisy samples𝑤𝑜(𝑗)is greater than the weight
    of clean samples𝑤𝑐(𝑖), then noisy samples will have a more
    significant impact on the estimate of𝜃. This can lead to an in-
    crease in the variance of parameter estimates, causing the model
    to become unstable. This instability can potentially reduce the
    model’s generalization ability on OOD data.
- Even if the proportion of noisy samples is small, the more signifi-
    cant variance𝜎𝑜^2 of noisy samples can still significantly affect the
    parameter estimates, leading to overfitting to the noisy samples.
Overall, we can understand that in DRO-based methods, the impact
of noise distribution cannot be ignored when improving the OOD
generalization performance of recommendations. Otherwise, the
model’s performance will be limited.

```
3.2 Sinkhorn DRO
To address the challenge of limited OOD performance in KL diver-
gence based DRO methods, this paper introduces Sinkhorn Distribu-
tionally Robust Optimization (i.e., Sinkhorn DRO). Sinkhorn DRO
combines distributionally robust optimization with the Sinkhorn
distance to tackle robustness optimization problems in distribu-
tional shifts. Sinkhorn distance is a variant of the Wasserstein dis-
tance, smoothed by entropy regularization. It is defined as follows:
```
```
𝑊𝑐,𝜆(𝑃,𝑄)= inf
𝜋∈Π(𝑃,𝑄)
```
### E(𝑥,𝑦)∼𝜋[𝑐(𝑥,𝑦)]+𝜆·𝐻(𝜋), (2)

```
whereΠ(𝑃,𝑄)denotes the set of all joint distributions whose mar-
ginal distributions are𝑃and𝑄, i.e., all possible transportation plans
from distribution𝑃to distribution𝑄.infrepresents infimum.𝜋is
the entropy of the joint distribution𝜋, and𝜆> 0 is the regulariza-
tion parameter. We further provide the mathematical definition of
Sinkhorn DRO as follows:
```
```
min
𝜃∈Θ
```
### (

```
𝑅(𝜃):= sup
𝑄:𝑊𝑐,𝜆(𝑃𝑡𝑟𝑎𝑖𝑛, 𝑄)≤𝜌
```
### E𝑃𝑡𝑟𝑎𝑖𝑛(𝑥,𝑦)∼𝑄[L(𝑓𝜃(𝑥),𝑦)]

### )

### (3)


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Conference acronym ’XX, June 03–05, 2018, Woodstock, NY Anon.
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```
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where𝑊𝑐,𝜆(𝑃𝑡𝑟𝑎𝑖𝑛,𝑄)is the Sinkhorn distance between the nom-
inal distribution𝑃𝑡𝑟𝑎𝑖𝑛and the test distribution𝑄, and𝜌is a ra-
dius that controls the range of robust optimization. By introducing
entropy regularization, Sinkhorn distance avoids the issue of KL
divergence becoming infinite when distributions do not overlap,
allowing the model to perform reasonable optimization through
geometric relationships even when no data in the training set is
similar to the test set.
```
```
3.3 Model Instantiations
In this section, we detail how our proposed DRGO method leverages
diffusion models and DRO to mitigate the limited model general-
ization caused by noisy samples in training data.
```
```
3.3.1 Denoising Diffusion Module.We are motivated by the
effectiveness of diffusion models in producing clean data across
diverse areas, including images [ 47 ] and text [ 17 ]. Our proposed
DRGO method incorporates diffusion models to generate denoised
user-item embeddings. Considering the discrete nature of user-
item interaction graphs, they are unsuitable for direct input into
diffusion models. Therefore, we design a method that integrates
Variational Graph AutoEncoders (VGAE) [ 8 ] with diffusion models
to efficiently denoise graph-structure data. In detail, consider a
user-item interaction graphG𝑜={U,V,E,X}, andXdenotes
the features of the nodes, such as a user’s gender and age or an
item’s category. Existing work [ 13 ] suggests that leveraging stable
features related to the target distribution can effectively improve
a model’s generalization performance on OOD data. We useG𝑜
as the input to the VGAE model, mapping it to a low-dimensional
latent vectorE 0 ∼N(𝜇 0 ,𝜎 0 )to facilitate both forward and reverse
diffusion processes in the latent space. The encoding process is
given by:
𝑞𝜓(E 0 |A,X)=N(E 0 |𝜇 0 ,𝜎 0 ), (4)
whereArepresents the adjacency matrix ofG𝑜,𝜇 0 =𝐺𝐶𝑁𝜇(A,X)
is the matrix of mean vectors, and𝜎 0 =𝐺𝐶𝑁𝜎(A,X)denotes the
matrix of standard deviations. Here,𝐺𝐶𝑁(·)refers to the graph
convolutional network in the graph variational autoencoder. Using
the reparameterization trick,E 0 is calculated as:
E 0 =𝜇 0 +𝜎 0 ⊙𝜖, (5)
where𝜖∼N( 0 ,𝐼)and⊙denotes element-wise multiplication.E 0
will be used as the input to the diffusion model for denoising, and
the entire process can be represented as:
```
```
E 0
```
```
𝜙
−→···E𝑡
```
```
𝜓
−→Eˆ𝑡− 1 ···
```
```
𝜓
−→Eˆ 0 , (6)
where𝜙represents injecting noise into the input embedding, while
𝜓removes the noise through optimization. After obtaining the
denoised embedding, the decoder of the VGAE takesEˆ 0 as input to
reconstruct𝐺 0 , which can be represented as: The entire process is
illustrated as follows:
Aˆ=𝜎(Eˆ 0 Eˆ⊤ 0 ), (7)
where𝜎(·)is the sigmoid function. The variational lower bound
optimizes the object of VGAE is shown as:
```
```
L𝑉𝐺𝐴𝐸=E𝑞(E 0 |A,X)
```
### 

```
log𝑝𝜃(A|X)ˆ
```
### 

### −𝐷𝐾𝐿(𝑞(E 0 |A,X) ∥𝑝(E 0 )),

### (8)

```
where𝑞represents the approximate posterior distribution, while𝑝
represents the reconstruction probability distribution. Additionally,
the optimization objective of the diffusion model is shown in Eq.
(57). We will elaborate on the optimization objective of DRGO in
detail in Section 3.3.4.
```
```
3.3.2 Nominal Distribution.Typically, when using DRO to op-
timize on OOD data, the nominal distribution𝑃𝑡𝑟𝑎𝑖𝑛should ideally
cover the testing distribution within a radius𝜌. However, in prac-
tical recommendation scenarios, due to the unpredictability and
randomness of user behavior, the distribution of the test set is often
unknown, posing challenges for constructing the nominal distribu-
tion. In reality, popular items or users in recommendation scenarios
often influence the future behavior of other users. Considering
the connectivity of graph-structured data, this study leverages be-
tweenness centrality to construct the nominal distribution.The
basic assumption is that groups with high centrality during
training strongly influence unseen distribution groups[ 2 , 18 ].
Betweenness centrality measures the frequency at which an entity
appears on the shortest paths between other entities in a topology.
Entities with higher betweenness centrality have greater control
over the topology. Its mathematical definition is as follows:
```
```
𝑐𝑑𝑎𝑡𝑎𝑣 =
```
### ∑︁

```
𝑠,𝑡∈𝐸train
```
### 𝜎(𝑠,𝑡|𝑣)

### 𝜎(𝑠,𝑡)

### , (9)

```
where𝜎(𝑠,𝑡)is the number of shortest paths between groups𝑠and
𝑡in the graph (i.e.,(𝑠,𝑡)-paths), and𝜎(𝑠,𝑡|𝑣)is the number of
(𝑠,𝑡)-paths that pass through node𝑣in graphG 0. After obtaining
the betweenness centrality values for each node, we sort them in
descending order and use the embeddings of the top-𝑛%nodes to
construct our nominal distribution. In the hyperparameter analysis
experiments, we will discuss in detail the impact of the value of𝑛
on model performance.
3.3.3 Uncertainty Set.When using DRO-based methods, it is
necessary to assign weights to different groups to optimize the
worst-case scenario. However, an excessive number of groups can
make DRO optimization challenging. Drawing on previous work
[ 45 ], we can directly cluster the denoised user embeddings𝐸ˆ𝑢,
thereby reducing the number of samples. Additionally, clustering
algorithms can more accurately group users with similar behaviors
by leveraging user feature information. Formally, the uncertainty
set𝑄can be obtained as follows:
𝑄={𝑞 1 ,𝑞 2 ,.. .,𝑞𝑘}=KMeans(𝐸ˆ𝑢), (10)
where the number of clusters𝑘will be discussed in detail as a
hyperparameter.
3.3.4 Joint Optimization.We propose the DRGO method, which
uses LightGCN as the backbone and employs Bayesian Personalized
Ranking as the optimization objective for the recommendation task:
L𝑟𝑒𝑐=
```
### ∑︁

```
(𝑢,𝑖+,𝑖−)
```
```
−log𝜎(𝑟ˆ𝑢,𝑖+−𝑟ˆ𝑢,𝑖−), (11)
```
```
where(𝑢,𝑖+,𝑖−)is a triplet sample for pairwise recommendation
training.𝑖+represents positive samples from which the user has
interacted, and𝑖−are the negative samples randomly drawn from
the items with which the user has not interacted, respectively.𝑟ˆ𝑢,𝑖+
represents the positive prediction score and𝑟ˆ𝑢,𝑖−is the negative
```

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Distributionally Robust Graph Out-of-Distribution Recommendation via Diffusion Model Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
```
```
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```
```
prediction score. Therefore, the overall optimization objective of
our DRGO is as follows:
```
```
min
𝜃
```
### 

```
𝑅(𝜃,𝑤𝑞𝑖):= sup
𝑄:𝑊𝑐,𝜆(𝑃𝑡𝑟𝑎𝑖𝑛, 𝑄)≤𝜌
```
### ∑︁𝑘

```
𝑖= 1
```
### 𝑤𝑞𝑖L𝑟𝑒𝑐(𝑓(𝑥),𝑦)

### −𝛽

### ∑︁𝑘

```
𝑖= 1
```
```
𝑤𝑞𝑖log𝑤𝑞𝑖+L𝑑𝑒𝑛𝑜𝑖𝑠𝑖𝑛𝑔(G 0 )
```
### 

### ,

### (12)

```
where𝑤𝑞𝑖represents the weights of different groups𝑞𝑖, which
need to be dynamically adjusted during training. The parameter𝛽
serves as the penalty factor coefficient. The term
```
### Í𝑛

```
𝑖= 1 𝑤𝑞𝑖log𝑤𝑞𝑖
represents an entropy regularization component that encourages
the distribution𝑞to approach uniformity and𝑘is the number
of groups (i.e, the number of clusters𝑘). This promotes balanced
weighting of all samples during optimization and helps prevent the
model from disproportionately focusing on outliers, such as noisy
samples. The termLdenoisingrepresents the loss function of the
denoising module, primarily comprising the loss from the VGAE,
as defined in Eq. (8), and the diffusion model, as defined in Eq. (57),
which is displayed as:
```
```
Ldenoising=L𝑉𝐺𝐴𝐸+Lsample (13)
```
```
3.4 Theoretical Analysis
In this subsection, we primarily provide two theoretical analyses:
(1) the generalization bound of DRGO on OOD data and (2) how
DRGO mitigates the impact of random noise.
```
```
Theorem 3.1.Assume the loss function𝑅(𝜃,𝑊𝑞𝑖)is bounded by
a constant𝐵. For𝛿> 0 , with probability at least 1 −𝛿, the following
inequality holds for all𝑓∈𝐹:
```
### 𝑅ˆ(𝜃,𝑤𝑞𝑖) ≤𝑅(𝜃,𝑤𝑞𝑖)+ 2 𝑅ˆ𝑛(𝐹)+𝐵𝑊(𝑃𝑡𝑟𝑎𝑖𝑛,𝑄)+𝐵

### √︂

### 𝑙𝑛( 1 /𝜎)

### 2 𝑛

### .

### (14)

```
The upper bound of the generalization risk on𝑄is primarily
influenced by its distance to𝑃𝑡𝑟𝑎𝑖𝑛, denoted as𝑊(𝑃𝑡𝑟𝑎𝑖𝑛,𝑄). By
incorporating the topological prior, the risk on𝑄can be tightly con-
strained by minimizing𝑊(𝑃𝑡𝑟𝑎𝑖𝑛,𝑄), provided that𝑄lies within
the convex hull of the training groups. We experimentally validate
the effectiveness of the topological prior in reducing generalization
risks across unseen distributions. We provide a complete theoretical
proof in Appendix A.3.
We analyze and discuss from the perspective of DRGO’s gradient
updates how DRGO mitigates the impact of random noise, focusing
primarily on the gradients of the second and third terms in the
DRGO optimization objective.
The introduction of the negative entropy term is aimed at reg-
ularizing the weights𝑤𝑞𝑖, preventing certain samples (especially
noisy) from receiving excessively high weights during optimization.
The gradient of the negative entropy term is:
```
### ∇𝑤𝑞𝑖

### −

### ∑︁𝑘

```
𝑖= 1
```
```
𝑤𝑞𝑖log𝑤𝑞𝑖
```
### !

### =

###