PBiLoss: Popularity-Aware Regularization to Improve Fairness in Graph-Based Recommender Systems

1. Problem Statement

Graph-based recommender systems, particularly those utilizing Graph Neural Networks (GNNs) like LightGCN, have achieved state-of-the-art performance in capturing user-item interactions. However, they suffer significantly from popularity bias.

• The Issue: These models tend to over-recommend popular items (high-degree nodes) while under-recommending long-tail (less popular) items.
• The Cause: In GNNs, message passing inherently amplifies dominant interaction patterns. Popular items receive more frequent updates and stronger signals, creating a feedback loop that marginalizes niche content.
• Limitations of Existing Solutions:
  • Pre-processing: Distorts data distributions.
  • Post-processing: Limited in correcting bias already embedded in learned representations.
  • In-processing (Gradient-based): Often complicates optimization, creates unstable loss landscapes, or requires sensitive hyperparameter tuning.
  • Existing Regularization: Many methods rely on sensitive attributes or require complex architectural changes.

The authors propose a need for a method that is model-agnostic, integrated directly into the training objective, and capable of balancing fairness without sacrificing recommendation accuracy.

2. Methodology: PBiLoss

The core contribution is PBiLoss (Popularity Bias Loss), a novel regularization term designed to be added to standard training objectives (specifically Bayesian Personalized Ranking or BPR).

A. Core Concept

PBiLoss explicitly penalizes the model's inclination toward popular items during training. It aims to:

1. Demote the ranking of popular items that are irrelevant to a user.
2. Promote the ranking of unpopular items that are relevant to a user.

B. Distinguishing Popular vs. Unpopular Items

The paper proposes two methods to categorize items based on popularity (quantified by node degree):

1. Fixed Threshold ($\alpha$): Items with a degree $\ge \alpha$ are "popular"; others are "unpopular."
2. No Threshold (Probabilistic): Sampling probabilities are assigned based on degree. Popular items are sampled with probability proportional to their degree; unpopular items are sampled with probability proportional to the inverse of their degree.

C. Sampling Strategies

PBiLoss introduces two distinct sampling strategies to generate training triples $(u, i, j)$, where $u$ is a user, $i$ is a positive item, and $j$ is a negative item:

1. Popular Positive (PopPos):

   • Focus: Relevant (positive) interactions.
   • Mechanism: Samples pairs of relevant items where one is popular and one is unpopular.
   • Goal: Encourages the model to rank the unpopular relevant item higher than the popular relevant item, boosting long-tail exposure.
   • Formulation: $T_{PBi} = \{(u, i, j) | u \in U, i \in I^+_{u, unpop}, j \in I^+_{u, pop}\}$

2. Popular Negative (PopNeg):

   • Focus: Irrelevant (negative) interactions.
   • Mechanism: Samples pairs where the positive item is unpopular and the negative item is popular.
   • Goal: Explicitly penalizes the model for ranking popular items (which the user hasn't interacted with) too highly. This is often more effective because popularity bias in BPR is often driven by popular items appearing as "easy negatives."
   • Formulation: $T_{PBi} = \{(u, i, j) | u \in U, i \in I^+_{u, unpop}, j \in I_{pop} \setminus I^+_{u, pop}\}$

D. Final Objective Function

The total loss function combines the standard BPR loss with the PBiLoss term:
$$L_{total} = L_{BPR} + w \cdot L_{PBi}$$
Where $w$ is a hyperparameter controlling the strength of the fairness regularization.

3. Key Contributions

1. Novel Loss Function: PBiLoss integrates fairness constraints directly into the learning objective, avoiding the need for post-processing or complex architectural changes.
2. Innovative Sampling Strategies: Introduction of PopPos and PopNeg strategies to target different facets of popularity bias (relevant vs. irrelevant items).
3. Flexible Popularity Definition: Provides both a fixed threshold approach (deterministic) and a no-threshold approach (probabilistic), allowing adaptability to different data distributions.
4. Model-Agnostic Design: The method is a modular loss component that can be seamlessly integrated into existing GNN frameworks (e.g., LightGCN, r-AdjNorm, APDA, PopGo) without structural modifications.
5. Efficiency: The time complexity remains asymptotically equivalent to the base model ($O(|E|Ld)$), as the additional loss computation is a constant factor increase.

4. Experimental Results

The authors evaluated PBiLoss on three real-world datasets: Epinions, iFashion, and MovieLens, integrating it with four baseline models: LightGCN, r-AdjNorm, APDA, and PopGo.

Key Findings:

• Fairness Improvement: PBiLoss significantly reduced popularity bias.
  • PRU (Popularity-Rank Correlation for Users) and PRI (Popularity-Rank Correlation for Items) were reduced by up to 10% compared to baselines. Lower values indicate better fairness.
  • The PopNeg-FT (Popular Negative with Fixed Threshold) variant consistently achieved the best balance between fairness and accuracy.
• Accuracy Preservation: Crucially, the method improved fairness without degrading standard accuracy metrics (F1@10, NDCG@10, MAP@10). In many cases, accuracy metrics slightly improved or remained stable.
• Statistical Significance: A Friedman non-parametric test confirmed that the improvements in ranking metrics (NDCG) were statistically significant ($p < 0.05$).
• Hyperparameter Sensitivity:
  • Moderate values of the regularization weight $w$ provided the best trade-off.
  • Setting the popularity threshold $\alpha$ to the top 20% of items yielded optimal results.

5. Significance and Conclusion

• Practical Impact: PBiLoss offers a scalable, easy-to-implement solution for industrial recommendation systems. It addresses the "Matthew Effect" (rich get richer) in recommendations, ensuring long-tail content gets fair exposure.
• Theoretical Contribution: It demonstrates that fairness can be achieved through simple regularization of the loss function rather than complex architectural overhauls or sensitive attribute manipulation.
• Future Directions: The authors suggest extending the definition of popularity beyond node degree (e.g., temporal trends, content-based signals) and applying the method to sequential or multi-modal recommendation scenarios.

In summary, PBiLoss is a robust, model-agnostic framework that successfully mitigates popularity bias in graph-based recommenders, enhancing diversity and fairness while maintaining high recommendation accuracy.