Hidden Convexity of Fair PCA and Fast Solver via Eigenvalue Optimization (2503.00299v1)

Abstract: Principal Component Analysis (PCA) is a foundational technique in machine learning for dimensionality reduction of high-dimensional datasets. However, PCA could lead to biased outcomes that disadvantage certain subgroups of the underlying datasets. To address the bias issue, a Fair PCA (FPCA) model was introduced by Samadi et al. (2018) for equalizing the reconstruction loss between subgroups. The semidefinite relaxation (SDR) based approach proposed by Samadi et al. (2018) is computationally expensive even for suboptimal solutions. To improve efficiency, several alternative variants of the FPCA model have been developed. These variants often shift the focus away from equalizing the reconstruction loss. In this paper, we identify a hidden convexity in the FPCA model and introduce an algorithm for convex optimization via eigenvalue optimization. Our approach achieves the desired fairness in reconstruction loss without sacrificing performance. As demonstrated in real-world datasets, the proposed FPCA algorithm runs $8\times$ faster than the SDR-based algorithm, and only at most 85% slower than the standard PCA.

An Overview of the Paper "Hidden Convexity of Fair PCA and Fast Solver via Eigenvalue Optimization"

The paper "Hidden Convexity of Fair PCA and Fast Solver via Eigenvalue Optimization" addresses the issue of fairness in Principal Component Analysis (PCA), a core technique in machine learning for dimensionality reduction. The traditional PCA method, while effective for capturing variance in high-dimensional data, can produce biased results that may unfairly disadvantage certain subgroups within a dataset.

Problem Statement and Contributions

The key focus of this work is on Fair PCA (FPCA), initially introduced by Samadi et al. (2018), which aims at balancing the reconstruction loss across different subgroups. The original semidefinite relaxation (SDR)-based solution for FPCA was found to be computationally expensive. Thus, this paper's main contribution is in uncovering a hidden convexity within the FPCA model and leveraging this insight to propose a novel algorithm based on eigenvalue optimization.

This newly introduced method not only improves computational efficiency but also maintains the fairness criterion by equalizing the reconstruction loss across subgroups without degrading the overall performance. The authors report that their algorithm is up to 8 times faster than the SDR-based approach and has a performance slowdown of at most 85% compared to standard PCA, a significant improvement.

Methodological Innovations

The paper presents several innovative aspects:

1. Hidden Convexity Revelation: The authors identify that the FPCA problem can be reformulated as a convex optimization problem by exploring the joint numerical range of the involved matrices. This reformulation simplifies the problem and provides a geometric interpretation that is crucial for developing an efficient algorithm.
2. Eigenvalue Optimization Approach: The novel approach utilizes eigenvalue optimization, a change from the semidefinite programming solutions, focusing on minimizing the largest eigenvalues related to the fairness constraints. This makes the method computationally more feasible for large-scale data.
3. Theoretical Justification and Empirical Validation: The proposed method is theoretically grounded and empirically validated with extensive experiments on various real-world datasets. Quantitative results indicate accurate and fair solutions compared to traditional PCA and previous FPCA approaches.

Implications and Future Work

The implications of this paper are notable for both practical and theoretical aspects of machine learning:

• Practical Implications: The algorithm offers a robust solution for ensuring fairness in dimensionality reduction tasks, making it highly applicable to sensitive domains like finance, healthcare, and social sciences where fairness is a significant concern.
• Theoretical Insights: By uncovering the hidden convexity in FPCA, this paper adds a new perspective to understanding and solving fairness-related problems in machine learning models. It also paves the way for further exploration into other hidden convex structures within complex models.
• Future Research Directions: While the paper significantly advances FPCA models, the authors acknowledge the potential for extending their approach to multi-group scenarios. Further research could explore the generalization of this convex approach to problems involving more than two subgroups, possibly by expanding on geometric interpretations and optimization frameworks. Additionally, the paper suggests exploring the integration of this algorithm into other forms of PCA and dimensionality reduction techniques.

In summary, the paper presents a detailed analysis and a novel solution to achieve fairness in PCA, characterized by both methodological rigor and computational efficiency. This work is a valuable addition to existing literature, providing insights and tools for researchers and practitioners in the field striving to mitigate bias in machine learning systems.