How abundant are good interpolators?

Problem
This work addresses the gap in understanding the distribution of effective interpolating classifiers in high-dimensional settings, particularly in the context of overparameterization. The authors investigate the performance of unit norm linear classifiers that correctly classify a labeled dataset with a fixed negative margin. This study is particularly relevant as it is presented as a preprint, indicating that it has not yet undergone peer review.

Method
The authors consider two data-generating distributions: a Gaussian mixture model and a logistic model with Gaussian features. They analyze the set ( S ) of unit norm linear classifiers ( \theta \in \mathbb{R}^d ) that achieve correct classification for a dataset ( (X_i, y_i)_{i=1}^n ) with a margin ( \kappa ). The analysis is conducted in the proportional regime where ( n/d \to \alpha ) with ( \alpha ) being sufficiently small. A large deviation principle is established, providing a deterministic rate function that quantifies the proportion of interpolating classifiers achieving a specified generalization error. The authors also compare the performance of the maximizer of this rate function against empirical risk minimization via gradient descent and a linear programming approach, both of which are designed to find points in ( S ).

Results
The findings indicate that in the overparameterized regime (small ( \alpha )), the efficient procedures (gradient descent and linear programming) significantly outperform the majority of interpolators. Specifically, the empirical risk minimization methods yield better generalization performance than the vast majority of classifiers in ( S ), suggesting a concentration phenomenon where nearly all interpolators exhibit similar performance, characterized by the unique maximizer of the rate function. The authors provide numerical results that substantiate these claims, although specific performance metrics and comparisons to baseline methods are not detailed in the abstract.

Limitations
The authors acknowledge that their results are contingent on the assumptions of the Gaussian mixture and logistic models, which may not generalize to other data distributions. Additionally, the analysis is limited to linear classifiers, leaving open questions regarding the behavior of non-linear models. The paper does not address the computational complexity of the proposed methods or the scalability of the findings to larger datasets.

Why it matters
This research contributes to the understanding of classifier performance in high-dimensional spaces, particularly in the context of benign overfitting, which has implications for model selection and generalization in machine learning. The results suggest that efficient training methods can yield superior performance even in regimes where interpolators are abundant, challenging traditional notions of overfitting. This work is significant for future research on classifier design and optimization strategies, as it highlights the importance of training methodology in achieving robust generalization, as published in arXiv.