Transfer learning via Regularized Linear Discriminant Analysis (2501.02411v3)

Abstract: Linear discriminant analysis is a widely used method for classification. However, the high dimensionality of predictors combined with small sample sizes often results in large classification errors. To address this challenge, it is crucial to leverage data from related source models to enhance the classification performance of a target model. We propose to address this problem in the framework of transfer learning. In this paper, we present novel transfer learning methods via regularized random-effects linear discriminant analysis, where the discriminant direction is estimated as a weighted combination of ridge estimates obtained from both the target and source models. Multiple strategies for determining these weights are introduced and evaluated, including one that minimizes the estimation risk of the discriminant vector and another that minimizes the classification error. Utilizing results from random matrix theory, we explicitly derive the asymptotic values of these weights and the associated classification error rates in the high-dimensional setting, where $p/n \rightarrow \gamma$, with $p$ representing the predictor dimension and $n$ the sample size. We also provide geometric interpretations of various weights and a guidance on which weights to choose. Extensive numerical studies, including simulations and analysis of proteomics-based 10-year cardiovascular disease risk classification, demonstrate the effectiveness of the proposed approach.

• The paper introduces TL-RDA, a method for integrating auxiliary and limited target data to approximate the Bayes optimal predictor.
• It develops a weighted summation framework that minimizes bias-variance trade-offs under high-dimensional settings.
• The approach achieves lower asymptotic error rates compared to traditional and pooled RDA methods in simulation scenarios.

Transfer Learning via Regularized Linear Discriminant Analysis

The paper "Transfer Learning via Regularized Linear Discriminant Analysis" by Hongzhe Zhang, Arnab Auddy, and Hongzhe Lee proposes a novel approach termed TL-RDA (Transfer Learning Regularized Discriminant Analysis) to enhance predictive modeling in scenarios where multiple auxiliary datasets are available, but the target dataset is scarce. This work addresses the complexities inherent in the bias-variance tradeoff when dealing with limited data by leveraging auxiliary information in the framework of regularized discriminant analysis (RDA).

Key Contributions

1. Framework and Model Setup: The authors introduce a random classification weights setup, where the difference in the means of covariates between two classes is characterized by a random vector with zero mean and constant variance. The correlation structure of this random difference vector ($\delta$) establishes the connection between auxiliary and target datasets. This setup informs the transfer learning process by exploiting commonalities across datasets through shared correlation structures.
2. Estimation of Bayes Optimal Predictor: The TL-RDA method aims to estimate the Bayes optimal predictor by combining naive RDA estimates from both auxiliary and target datasets through a weighted summation. The weights are meticulously designed to minimize the distance between the TL-RDA estimator and the Bayes optimal direction.
3. Asymptotic Analysis: This work extends to a high-dimensional context, where the number of features ($p$) is proportional to the sample size ($n$). The paper provides explicit asymptotic error rates for TL-RDA, showing that it achieves the lowest error rate among all estimators relying on weighted summations, including those using only target dataset information.
4. Technical Framework: The authors employ a comprehensive statistical and mathematical framework informed by recent advances in random matrix theory, facilitating the analysis of the TL-RDA estimator under high-dimensional data settings. Theoretical results include convergence analyses, spectral properties, and optimizations that guide the choice of weights.

Analysis and Implications

• Numerical Results and Claims:

The paper substantiates its claims with numerical results demonstrating the superior predictive accuracy of TL-RDA across various simulation scenarios. It juxtaposes its method with traditional RDA and pooled RDA, showing consistent improvements.

• Practical and Theoretical Implications:

Practically, this work provides a robust methodology for improving classification tasks across domains with related but distinct datasets. Theoretically, it expands the landscape of transfer learning by incorporating regularized discriminant analysis within a high-dimensional asymptotic framework.

Future Directions

The paper opens avenues for further research into adaptive strategies for weight selection in more complex settings, including those with heterogeneous population covariance matrices. Exploration of non-linear extensions or implementations within federated learning environments could further extend the utility and applicability of TL-RDA.

In conclusion, the paper offers a rigorous and insightful approach to transfer learning using regularized discriminant analysis, underscoring the importance of integrating auxiliary information in predictive modeling tasks particularly in high-dimensional settings. Its methodological advancements set a solid foundation for both practical applications and further scholarly inquiry in the field of statistical learning and artificial intelligence.