Metric Learning from Limited Pairwise Preference Comparisons (2403.19629v2)

Abstract: We study metric learning from preference comparisons under the ideal point model, in which a user prefers an item over another if it is closer to their latent ideal item. These items are embedded into $\mathbb{R}^d$ equipped with an unknown Mahalanobis distance shared across users. While recent work shows that it is possible to simultaneously recover the metric and ideal items given $\mathcal{O}(d)$ pairwise comparisons per user, in practice we often have a limited budget of $o(d)$ comparisons. We study whether the metric can still be recovered, even though it is known that learning individual ideal items is now no longer possible. We show that in general, $o(d)$ comparisons reveal no information about the metric, even with infinitely many users. However, when comparisons are made over items that exhibit low-dimensional structure, each user can contribute to learning the metric restricted to a low-dimensional subspace so that the metric can be jointly identified. We present a divide-and-conquer approach that achieves this, and provide theoretical recovery guarantees and empirical validation.

• The paper introduces a novel divide-and-conquer method for metric learning that leverages lower-dimensional subspaces to tackle limited pairwise comparisons.
• It establishes a rigorous theoretical framework with necessary and sufficient conditions for metric recovery and validates the method through synthetic experiments.
• The research demonstrates potential impact on AI by improving personalized recommendation systems and enabling efficient handling of high-dimensional sparse data.

Metric Learning from Limited Pairwise Preference Comparisons: Subspace-Clusterable Approach

Introduction

In exploring metric learning from limited pairwise preference comparisons, the research highlights fundamental challenges and introduces a novel divide-and-conquer strategy focusing on situations where direct learning from individual comparisons is unfeasible due to practical constraints on the number of comparisons per user. The ideal point model serves as the basis for considering pairwise preferences, where the preference indicates proximity to a user's latent ideal item. The innovative approach undertakes metric learning within lower-dimensional subspaces when the item space exhibits a subspace-clusterable structure, leading to significant theoretical insights and practical implications for metric learning with sparse user feedback.

Theoretical Foundation and Insights

The paper begins by establishing a critical impossibility result: when items are in general positions, learning the metric with fewer than d preference comparisons per user is generally infeasible. This foundational result underscores the inherent limitations in traditional approaches when faced with sparse data.

Subspace-Clusterable Structure and Divide-and-Conquer Strategy

The paper then shifts focus to scenarios where items exhibit a subspace-clusterable structure, allowing for the partitioning of the metric learning problem into more manageable sub-problems. This structure is vital for overcoming the aforementioned limitations by leveraging the intrinsic lower-dimensional properties of the item space. The divide-and-conquer strategy is meticulously developed, entailing:

• Identification of metrics within individual subspaces.
• Reconstruction of the global metric from these subspace metrics, facilitated by the positive-definite property of Mahalanobis distances.

This approach is rigorously validated through:

1. A theoretical framework establishing necessary and sufficient conditions for the identification of the unknown metric based on subspace-clusterability.
2. Providing recovery guarantees for the global metric based on the recovery accuracies within the subspaces.

Empirical Validation and Algorithmic Implementation

The practical efficiency of the proposed method is demonstrated through experiments with synthetic data, focusing on:

• Varying levels of user feedback sparsity.
• The impact of subspace dimensionality and the number of subspaces on metric recovery accuracy.
• Robustness to noise and the approximate subspace structure of items, highlighting the method's applicability in real-world settings where ideal assumptions may not hold.

The algorithmic implementation, detailed within the supplementary material, underscores the adaptability of the approach, including adjustments for handling approximately subspace-clusterable items and leveraging robust regression techniques for enhanced performance.

Implications for AI and Future Directions

The research's implications extend beyond metric learning, suggesting a paradigm shift in handling sparse and high-dimensional data in AI applications. By aligning with foundational mathematical principles, the approach opens new avenues for efficient data representation and preference learning.

The introspective on future work points towards:

• Extending theoretical guarantees to nearly subspace-clusterable settings.
• Development of end-to-end frameworks for discovering underlying subspace structures from raw data.
• Exploration of real-world datasets and applications, affirming the method's practical utility and adaptability.

Conclusion

This paper contributes significantly to metric learning, introducing a robust framework for addressing sparse user preferences under the ideal point model. By exploiting subspace-clusterable item structures, the research navigates the complexities of high-dimensional data and limited user feedback, presenting a promising pathway for future advancements in personalized recommendation systems and beyond. The insights provide a groundwork for subsequent exploration, pushing the boundaries of what is achievable in preference-based learning within AI.