GD$^2$: Robust Graph Learning under Label Noise via Dual-View Prediction Discrepancy

Response to Q1

“My main reservation with this paper is that discrepancies in labels between the two views could be an outcome of heterophily in the graph, rather than label noise... Wouldn't the model expect different labels between any ego node and its structural neighborhood?”

We appreciate the reviewer’s thoughtful concern. We would like to clarify that our method does not rely on the homophily assumption, and is in fact designed to function under both homophilous and heterophilous graphs.

The core idea of GD2 is to compare the semantic consistency between two information sources:

• The ego-view reflects the class implied by the node’s own features.
• The structure-view reflects the class pattern inferred from its neighbors via GNN aggregation—not the actual neighbor labels themselves.

Even in heterophilous graphs, nodes of the same class often share similar neighborhood distributions, even if their neighbors belong to different classes. This observation underpins many heterophily-aware GNNs and is supported by prior works [1,2,3]. Therefore, a clean-labeled node should still exhibit alignment between its ego- and structure-view predictions, while noisy nodes often show disagreement across views due to contradictory evidence.

Importantly, GD2 does not penalize cross-view label disagreement per se. Instead, we use the magnitude of the prediction discrepancy as a soft indicator of semantic inconsistency. This formulation allows us to remain effective even when structural labels differ from the ego node’s label—as is typical in heterophilous graphs.

To empirically verify this, we include results on two heterophilous benchmarks: Roman-Empirical and Amazon-Ratings. As shown in Table 1, GD2 achieves consistent improvements over baselines under various noise settings, confirming that our method is robust to heterophily.

[1] Simple and Asymmetric Graph Contrastive Learning without Augmentations, NeurIPS 2023

[2] Is Homophily a Necessity for Graph Neural Networks?, ICLR 2022

[3] Decoupled Self-supervised Learning for Graphs, NeurIPS 2022

Response to Q2

“There is quite some sensitivity to tau selection. Optimal value will also depend on the dataset. Is there some adaptive selection technique authors would recommend that might be useful on top of grid search?”

Thank you for the insightful question. We would like to clarify that GD2 exhibits stable and reliable performance when $\tau \leq 0.7$, with only minor fluctuations. A clear performance drop is observed only at $\tau = 0.9$, where overly aggressive filtering removes too much supervision. These results indicate that the method is not overly sensitive to $\tau$ as long as it is chosen within a reasonable range.

To go beyond grid search, we recommend selecting $\tau$ using a small clean validation set, by maximizing validation accuracy or minimizing label disagreement. Additionally, this setting is closely related to pseudo-label filtering in semi-supervised learning, where model confidence is used to select reliable training samples. A similar principle can be applied here: the cross-view agreement serves as a proxy for prediction confidence, and $\tau$ can be interpreted as a confidence threshold. This aligns with recent techniques such as SoftMatch [1] and FreeMatch [2], which dynamically select samples based on confidence-aware criteria. Inspired by these works, we believe designing adaptive or learnable thresholding mechanisms is a promising future direction.

[1] SoftMatch: Addressing the Quantity-Quality Tradeoff in Semi-supervised Learning, ICLR’23

[2] FreeMatch: Self-adaptive Thresholding for Semi-supervised Learning, ICLR’23

Response to Q3

“How would GD² perform when there is class imbalance? Presumably the GMM learns noise due to a healthy distribution of labels. Would it incorrectly think minority class samples are noisy?”

We thank the reviewer for raising this important question. We clarify that GD2 does not assume any balanced label distribution, and its noise detection mechanism is inherently class-agnostic. Specifically, the GMM is fitted over cross-view prediction discrepancy scores, which measure disagreement between ego- and structure-view predictions, rather than over class frequencies or node labels.

This design ensures that the model focuses on semantic inconsistency rather than minority status. A node from a minority class is not more likely to be marked as noisy unless its two views yield inconsistent predictions.

Moreover, we note that the datasets used in our experiments—such as Computers, Photo, and WikiCS—are naturally imbalanced. The table below shows the number of samples per class in three datasets(for brevity, we display only a subset of classes):

Response to W1 & Q1

“The method's core premise is that a discrepancy between the ego-view and structure-view indicates a noisy label, which is based on the homophily assumption. In heterophily graphs, where connected nodes are often of different classes, this assumption may be violated…”

We thank the reviewer for highlighting this important concern. We clarify that our method does not rely on the homophily assumption (i.e., neighbors sharing the same label). Instead, the dual-view design in GD2 distinguishes information sources rather than relying on node similarity:

• The ego-view captures the prediction from a node’s own features.
• The structure-view aggregates signals from neighbors and reflects the class pattern implied by the neighborhood distribution—not necessarily the classes of neighbors.

This distinction is especially important in heterophilous graphs, where neighbors tend to belong to different classes. Prior works [1,2,3] have shown that nodes from the same class often share similar neighborhood patterns, even under heterophily. Thus, we expect that for a node with a clean label, its ego- and structure-view predictions should still be semantically aligned—i.e., they should agree on the class, despite structural label diversity.

To empirically verify this, we have conducted experiments on two heterophilous graphs (Roman-empirical and Amazon-ratings). As shown in Table 1, GD2 consistently outperforms baselines under label noise, even when neighbor-label similarity is low. This indicates the robustness of our discrepancy-based framework without relying on homophily.

[1] Simple and Asymmetric Graph Contrastive Learning without Augmentations, NeurIPS 2023

[2] Is Homophily a Necessity for Graph Neural Networks?, ICLR 2022

[3] Decoupled Self-supervised Learning for Graphs, NeurIPS 2022

Response to W2 & Q2

“The paper should further discuss the framework's limitations under extremely high noise ratios (e.g., p > 0.5)...”

Thank you for raising this important point. We agree that when the noise ratio becomes extremely high, label supervision becomes less reliable, and the risk of polluted structural information increases. Nonetheless, we would like to clarify two key aspects of our method that preserve its effectiveness in such settings.

First, our approach is based on a realistic and widely accepted assumption in noisy-label learning: within each true class, the number of correctly labeled nodes is greater than the number of nodes mislabeled into any single incorrect class. That is, although noisy labels exist globally, for any given ground-truth class, its clean samples are still the most concentrated compared to any other individual class. This class-wise imbalance ensures that the structure-view can retain useful semantic signals, even under significant noise.

Second, our method distinguishes clean and noisy labels based on the relative magnitude of cross-view prediction discrepancy, not on the binary presence of discrepancy. Importantly, this separation remains valid even when the neighborhood contains many noisy nodes.

To illustrate this, consider a 3-class classification task under uniform noise with p=0.6. Assume a node whose ground-truth class is 0, and 60% of its neighbors are truly class 0, while the remaining 40% are class 1. After label corruption, the observed neighbor label distribution becomes: 44% class 0, 36% class 1, 20% class 2.

Assuming the ego-view predicts the given label (one-hot), and the structure-view reflects the noisy neighbor label distribution, we have:

• If the node's given label is correct (class 0), then the ego-view is $(1,0,0)$, and the structure-view is $(0.44,0.36,0.20)$, giving a discrepancy of: $||(1, 0, 0) - (0.44, 0.36, 0.20)|| \approx 0.48.$
• If the label is corrupted to class 1, the discrepancy becomes: $||(0, 1, 0) - (0.44, 0.36, 0.20)|| \approx 0.64.$
• If corrupted to class 2, the discrepancy is even larger: $||(0, 0, 1) - (0.44, 0.36, 0.20)|| \approx 0.96.$

This example demonstrates that even with p=0.6, noisy-labeled nodes consistently yield higher discrepancy scores, enabling effective separation by GMM.

Finally, while our performance margins over baselines narrow under extreme noise, GD2 still consistently ranks among the top-performing methods, as shown in Table 1. This reflects a fundamental challenge shared by most robust GNNs, rather than a weakness specific to our design.

Response to W3 & Q3

“The GD² framework is complex... The authors should include an analysis of the computational complexity and runtime costs.”

We thank the reviewer for the suggestion. While GD² introduces dual-view inference, discrepancy modeling, and multi-view optimization, its overall computational cost remains moderate and scales linearly with graph size. We provide a detailed breakdown below, based on Algorithm 1.

1. View-specific representation (Line 4):

   Each epoch involves two GCN forward passes: one with identity matrix $I$ (ego-view), and one with adjacency $A$ (structure-view). For $L$ layers, input feature dimension $F$, and $N=∣V∣$ nodes, the total cost is: $O\big(2L(|E|F + NF^2)\big)$

2. Prediction and discrepancy (Lines 5–6):

   Two linear classifiers produce logits in $O(NFC)$. Then, computing the $\ell_2$ discrepancy for each training node costs $O(NC)$.

3. GMM fitting (Line 7):

   The discrepancy values are 1D scalars; fitting a 2-component GMM via EM costs only $O(N)$.

4. Mixed-view inference and pseudo-labeling (Lines 10–11):

   One additional GCN forward pass adds $O(L(|E|F + NF^2))$. Pseudo-label assignment costs at most $O(NC)$.

5. Loss computation and update (Lines 13–15):

   Three view-specific cross-entropy losses are computed over subsets of nodes, costing $O(NC)$, followed by standard backpropagation with cost similar to one GCN pass.

In total, each epoch has complexity: $O\big(3L(|E|F + NF^2) + NFC\big)$. Compared to a standard GCN with per-epoch cost $O(L(|E|F + NF^2) + NFC)$, GD2 introduces roughly 2–3× overhead due to three view-specific passes and auxiliary steps such as GMM fitting and pseudo-labeling. However, these extra components scale linearly with node count an remain lightweight in practice.

To further demonstrate practical efficiency, we report the total training time (in seconds) of GD2 against representative baselines across datasets in the table below.

We observe that GD2 has comparable runtime to ERASE, another strong noise-robust baseline, and is slower than TSS. However, GD2 significantly outperforms TSS, offering a favorable trade-off between robustness and efficiency.

Response to W1

“The paper should clarify how the interdependencies among connected nodes in graph affect the existing noise-agnostic works. To the best of my knowledge, NRGNN also took the neighborhood of nodes with similar labels into consideration and connected the labeled and unlabeled nodes for noise-agnostic learning.”

We thank the reviewer for this helpful comment. Indeed, several prior works—such as NRGNN and related methods—have utilized local structural signals (e.g., neighborhood similarity, label propagation) to enhance robustness against noisy labels. These methods implicitly leverage interdependencies among connected nodes to smooth predictions or refine graph structure.

However, our work differs in both purpose and formulation. Specifically, most prior approaches use interdependencies as an auxiliary tool to mitigate the effects of label noise—e.g., by refining edges, propagating softened labels, or enforcing similarity regularization. In contrast, our goal is to explicitly model the inconsistency between a node's own features and its structural context as a signal for detecting noise.

To this end, we construct two distinct views for each node: (1) the ego-view, reflecting prediction from a node’s own features;(2) the structure-view, reflecting prediction based on aggregated neighbors.

The cross-view discrepancy between them is not used to refine the graph or smooth labels, but to identify which nodes are likely to be mislabeled. This discrepancy serves as a dynamic and model-internal indicator during training, rather than a static preprocessing step.

Therefore, while both our method and prior works acknowledge the importance of interdependencies, our contribution lies in recasting those dependencies as a core signal for noise detection itself, rather than treating them as a denoising aid.

Response to W2

“The notations of the paper should be reorganized. For example, ‘s’ is redefined as clean-label component in line 158 while it was the footnote for structure-view in the notations above.”

Thank you for pointing this out. We have carefully revised the notation and related equations to ensure that each symbol has a unique and consistent meaning throughout the paper. All revisions will be reflected in the updated version of the paper.

Response to W3

“The main assumption and theoretical justification for now seems not sound for me… The result only provides the upper bound of probability of distribution discrepancy, but does not provide any direct correlation between the discrepancy and label noise.”

Thank you for the thoughtful critique. We believe there is a misunderstanding regarding the role of cross-view discrepancy in our framework. We clarify that our method does not rely on the binary presence or absence of discrepancy to determine whether a node is noisy. Instead, our assumption and algorithmic design are based on the distributional difference in discrepancy scores between clean-labeled and noisy-labeled nodes.

Concretely, we compute a cross-view prediction discrepancy—the distance between the ego-view and structure-view predictions. Our key insight is that this discrepancy is statistically higher for noisy-labeled nodes, as their ego-view (aligned with an incorrect label) tends to disagree more with the structure-view (which captures the semantic context via neighbor aggregation). This discrepancy serves as a soft, probabilistic signal, which we then model using a Gaussian Mixture Model (GMM) to separate clean from noisy samples based on their distributional patterns.

To further illustrate this point intuitively, we provide a simple example demonstrating that even under high noise levels, nodes with incorrect labels tend to exhibit higher cross-view discrepancy. Consider a 3-class classification problem under uniform label noise with p = 0.6. Assume a node whose ground-truth class is 0, and 60% of its neighbors are truly class 0, while the remaining 40% are class 1. Due to label corruption, the observed neighbor label distribution becomes:44% class 0, 36% class 1, 20% class 2.

For the sake of simplicity and interpretability, we assume the ego-view prediction is a one-hot vector aligned with the given label, and the structure-view prediction is the distribution of observed neighbor labels.

Under these assumptions:

• If the node's given label is correct (class 0), then the ego-view is $(1,0,0)$, and the structure-view is $(0.44,0.36,0.20)$, giving a discrepancy of: $||(1, 0, 0) - (0.44, 0.36, 0.20)|| \approx 0.48.$
• If the label is corrupted to class 1, the discrepancy becomes: $||(0, 1, 0) - (0.44, 0.36, 0.20)|| \approx 0.64.$
• If corrupted to class 2, the discrepancy is even larger: $||(0, 0, 1) - (0.44, 0.36, 0.20)|| \approx 0.96.$

This example demonstrates that nodes with incorrect labels exhibit higher cross-view discrepancy, even when neighborhood noise is significant. The separation between clean and noisy nodes does not depend on the absolute value of discrepancy for any individual node, but rather on the relative distributional differences between clean and noisy groups—which the GMM is designed to exploit.

Regarding the theoretical justification, we appreciate the reviewer’s concern. We clarify that the goal of our theoretical analysis is not to establish a necessary and sufficient condition that links cross-view discrepancy directly and exclusively to label correctness. Instead, the purpose is to demonstrate that, under mild assumptions, clean-labeled nodes tend to have bounded (i.e., low) cross-view discrepancy with high probability—providing a theoretical explanation for the observed distributional separability between clean and noisy samples.

Specifically, Theorem 1 in our paper derives an upper bound on the probability that a clean node exhibits high discrepancy, which formalizes the intuition that clean nodes are less likely to produce inconsistent ego- and structure-view predictions. We do not claim that noisy nodes are the only ones with large discrepancy, but rather that noisy nodes are more likely to lie in the high-discrepancy region, making the distributional signal exploitable for separation.

Response to W4 & Q2

“As the authors mention that ‘the structure-view captures topological patterns that remain relatively stable under noise,’ why not directly utilize the structure-view as clean representation for node label prediction? Please further conduct ablation study that only utilize the structure-view as clean representation for node label prediction in both training and inference.”

Thank you for the insightful question. While the structure-view captures topological patterns that are relatively stable under noise, using it alone would discard valuable node-specific information from the ego-view.

Our final prediction model uses a mixed-view representation that combines both views, but is trained only on the filtered clean subset, where the impact of noisy labels is significantly reduced. In this setting, incorporating the ego-view adds complementary information with minimal noise interference.

The core idea of our method is to use the discrepancy between ego- and structure-view predictions to identify noisy labels. After filtering, combining both views leads to more expressive and robust representations.

Empirically, we show that structure-view alone performs worse than our mixed-view model, validating the benefit of using both.

Response to W5

“The improvement on the results of the provided experiment seem marginal, with only less than 1% accuracy improvement.”

We respectfully disagree and would like to clarify. While some individual improvements appear small in absolute value, GD2 consistently ranks first or second across all datasets and noise settings, demonstrating strong overall robustness and competitiveness.

Notably, under more challenging settings such as asymmetric pair-0.4 noise on Computer, GD2 surpasses ERASE by 3.03%, which is a substantial margin in the context of noisy node classification. Even in cases where the gains are smaller, the improvements are highly consistent and statistically stable across multiple runs, indicating better robustness and generalization.

Overall, GD2 delivers reliable gains with minimal architectural complexity, validating the effectiveness of our discrepancy-based noise filtering approach.

Dear Reviewer,

We hope this message finds you well. As the discussion phase is entering its final days (less than three days remaining), we noticed that we have not yet received your follow-up comments regarding our paper.

We would sincerely appreciate it if you could let us know whether our rebuttal has addressed your concerns, or if there are any remaining issues we could further clarify. Your feedback would be greatly appreciated and would help us improve our work.

Thank you again for your time and thoughtful reviewing.

Response to W1

“It is unclear how the model can reliably distinguish clean vs noisy data at the beginning of training, when both ego- and neighbor-view predictions may be unstable.”

We appreciate the reviewer’s concern. Our method is designed in a bootstrapped and iterative manner rather than relying on accurate clean/noisy separation at the initial stage. Specifically, as shown in the Pseudo-code in the Appendix E, GD2 updates the clean node set at every training epoch based on the discrepancy between dual-view predictions. This process allows the model to gradually refine its belief about clean samples as training progresses, leveraging the co-training-style evolution of ego-view and neighbor-view representations. Such a progressive filtering mechanism is common and effective in robust learning frameworks under uncertain supervision.

Response to W2

“The experiments are conducted on a small number of benchmarks, lacking large-scale datasets.”

Thank you for the suggestion. We have included additional experiments on the large-scale ogbn-arxiv dataset. Our method achieves consistent gains under both symmetric and asymmetric label noise settings, showing its scalability and effectiveness beyond the original benchmark datasets.

Response to W3

“The method is only validated with GCNs; it is unknown whether the approach remains effective when applied to other widely used GNN architectures like GAT, GraphSAGE, or transformer-based models.”

Thank you for pointing this out. We would like to clarify that GD2 is inherently architecture-agnostic and does not rely on the specifics of any GNN variant. In fact, message-passing GNNs—including GCN, GAT, GraphSAGE, and Transformer-based models—can be abstracted as consisting of two core steps:

(1) Aggregation: gathering information from a node’s local neighbors or global context;

(2) Update: combining the aggregated message with the node’s current embedding.

Our dual-view construction leverages this decomposition:

• The ego-view is computed by skipping the aggregation step, relying only on the node’s own features and performing the update step independently.
• The structure-view is derived by removing the ego-node’s self-loop during aggregation (i.e., excluding its own embedding from the neighborhood), thus obtaining a purely structural context from neighbors.

In this way, GD2 defines two complementary perspectives via selective aggregation operations, which are supported in virtually all message-passing GNNs.

To demonstrate generalizability, we have included new experiments on Computer using GAT and SGFormer [1] as the backbone architecture (see Table below). GD2 consistently improves robustness on these architectures, confirming that our method is not confined to GCNs.

[1] SGFormer: Simplifying and Empowering Transformers for Large-Graph Representations. NIPS’23

Response to W4

“The clean node threshold ($\tau$) varies across datasets, but the paper does not offer a principled or simple strategy for choosing it automatically.”

Thank you for the valuable suggestion. The threshold $\tau$ determines how strictly GD2 filters out noisy nodes based on the agreement between ego-view and structure-view predictions.

As shown in Figure 3, GD2 achieves stable and reliable performance when $\tau≤0.7$, with only minor fluctuations. A noticeable performance drop occurs at $\tau = 0.9$, where overly conservative filtering removes too much supervision. This indicates that the method is not overly sensitive to $\tau$, as long as it is set within a moderate range.

In practice, $\tau$ can be selected using a small clean validation set, by maximizing validation accuracy or minimizing disagreement with clean labels. Alternatively, this setting is closely related to pseudo-label filtering in semi-supervised learning, where model confidence is used to select high-quality training signals. A similar principle can be applied here: cross-view agreement serves as a proxy for prediction confidence, and $\tau$ can be set to retain nodes with high agreement. This idea aligns with strategies used in recent works such as SoftMatch [1] and FreeMatch [2]. Inspired by these works, we believe designing adaptive or learnable thresholding mechanisms is a promising future direction.

[1] SoftMatch: Addressing the Quantity-Quality Tradeoff in Semi-supervised Learning, ICLR’23

[2] FreeMatch: Self-adaptive Thresholding for Semi-supervised Learning, ICLR’23