How abundant are good interpolators?
Using large deviation principles, the paper analyzes the distribution of generalization errors among high-dimensional interpolating classifiers, revealing that good interpolators are exceedingly rare and that algorithmic solutions outperform most interpolators.
Key Findings
Methodology
This work employs high-dimensional asymptotic analysis combined with the large deviation principle (LDP) to quantify the performance distribution of random interpolating classifiers in binary linear classification. The authors consider the set of unit-norm classifiers θ that perfectly fit the training data with a fixed margin κ, under two data-generating models: Gaussian mixture and logistic models with Gaussian features. They formulate a variational problem to characterize the exponential decay rate of the probability that a randomly chosen interpolator achieves a given generalization error level. The analysis leverages tools from random matrix theory, geometric measure theory, and convex analysis to derive a deterministic rate function, which encapsulates the performance distribution at the exponential scale. Numerical simulations compare the typical performance point with that of gradient descent and linear programming algorithms, demonstrating that these efficient methods outperform the majority of interpolators, especially in the small sample ratio regime.
Key Results
- The paper derives an explicit large deviation rate function for the overlap between a random interpolator and the true signal direction, showing that the distribution concentrates exponentially around a 'typical' performance level x⋆. This x⋆ is obtained as the maximum of a variational problem involving the data distribution parameters, the margin κ, and the signal-to-noise ratio λ. The results indicate that in the low SNR and small α regime, most interpolators perform significantly worse than the optimal algorithms, with the performance gap narrowing as α and λ increase. Numerical solutions of the variational equations reveal that the performance of gradient descent and linear programming solutions exceeds that of a typical interpolator, especially in the overparameterized regime.
- The analysis extends to Bayesian posterior distributions, where the performance of the posterior mean estimator aligns with the typical interpolator performance, exhibiting exponential concentration. The derived formulas provide a comprehensive picture of the performance landscape, confirming that the probability of randomly selecting a good interpolator diminishes exponentially with the dimension, thus emphasizing the rarity of high-quality solutions in high-dimensional regimes.
- Overall, the results demonstrate that in high-dimensional linear classification, the set of 'good' interpolators is exponentially small, and algorithmic solutions like gradient descent and linear programming are statistically favored, providing a theoretical explanation for the empirical success of such algorithms in deep learning contexts.
Significance
This study advances the theoretical understanding of overparameterized models, especially in the context of deep learning, where models interpolate training data yet generalize well. By rigorously quantifying the rarity of good interpolators via large deviation analysis, it challenges the naive notion that all interpolators are equally bad. The findings highlight the crucial role of algorithmic bias—such as gradient descent—in selecting high-performance solutions from an exponentially small subset. This insight bridges the gap between empirical observations of benign overfitting and the mathematical theory of high-dimensional probability, offering a new lens to interpret the success of modern neural networks. Moreover, the explicit formulas for performance distribution open avenues for designing algorithms that explicitly target the performance peaks identified by the variational problems.
Technical Contribution
The core technical contribution lies in the rigorous derivation of a large deviation rate function for the performance of uniform random interpolators in high-dimensional linear classification. The authors formulate a variational problem involving the overlap with the true signal and derive explicit saddle point equations characterizing the maximum of the rate function. They extend classical tools from random matrix theory, geometric measure theory, and convex analysis to the high-dimensional asymptotic regime, establishing exponential concentration of performance around the typical value x⋆. The work also introduces novel analytical techniques to handle the dependence of the performance measure on data distribution parameters and margin κ, providing explicit formulas for the rate functions under Gaussian mixture and logistic models. Numerical solutions of the saddle point equations validate the theoretical predictions and demonstrate the sharpness of the large deviation bounds.
Novelty
This work is pioneering in applying large deviation principles to quantify the distribution of generalization errors among all interpolating classifiers in high dimensions. Unlike prior studies that focus on average-case analysis or specific algorithms, this paper characterizes the entire performance landscape via a variational rate function, revealing that high-quality interpolators are exponentially rare. The integration of geometric measure theory with high-dimensional probability to analyze the measure of interpolator sets is a key innovation. Furthermore, the explicit saddle point equations and their solutions for Gaussian mixture and logistic models provide new analytical tools for understanding the interplay between data distribution, margin, and performance in overparameterized regimes. This approach offers a fundamentally different perspective from traditional VC or Rademacher complexity bounds, emphasizing the probabilistic rarity of good solutions.
Limitations
- The analysis relies heavily on the high-dimensional limit (n/d→α), which may not fully capture finite-sample behaviors or low-dimensional scenarios. The models considered are linear and assume Gaussian or logistic features, limiting applicability to more complex, nonlinear models prevalent in deep learning.
- The derivation of the rate functions involves solving coupled nonlinear equations numerically, which may be computationally intensive for more complex data distributions or larger parameter spaces. The theoretical guarantees are asymptotic and may not directly translate to finite-dimensional settings.
- While the results show the exponential rarity of good interpolators, they do not directly prescribe practical algorithms for efficiently finding such solutions in finite samples. Future work is needed to connect these probabilistic insights with algorithmic design and empirical performance in real-world deep learning tasks.
Future Work
Future research could extend the large deviation framework to nonlinear models, such as neural networks, to understand the distribution of performance among interpolating solutions in more complex architectures. Investigating finite-sample corrections and developing algorithms that explicitly target the performance peaks identified by the variational analysis could bridge theory and practice. Additionally, exploring the impact of data heterogeneity, noise, and model misspecification on the performance distribution would deepen understanding of real-world scenarios. Integrating these probabilistic insights with optimization dynamics, such as stochastic gradient descent trajectories, may yield new principles for designing robust, high-performing models in overparameterized regimes.
AI Executive Summary
In recent years, overparameterized models—particularly deep neural networks—have revolutionized machine learning, achieving remarkable performance even when fitting training data perfectly. This phenomenon, often termed benign overfitting, challenges traditional notions that overfitting necessarily harms generalization. Despite extensive empirical success, the theoretical underpinnings of why and how such models generalize remain elusive.
This paper addresses a fundamental question: within the vast set of all possible interpolating classifiers—solutions that perfectly fit the training data—how many are genuinely good in terms of their ability to generalize? The authors focus on high-dimensional linear classifiers with fixed margin κ, considering data generated from Gaussian mixture models and logistic models with Gaussian features. Using the framework of large deviation principles (LDP), they rigorously quantify the probability that a randomly chosen interpolator achieves a certain generalization error.
The core insight is that, in the high-dimensional limit, the performance of most interpolators concentrates exponentially around a typical value x⋆, which is obtained by solving a variational problem involving the data distribution parameters, the signal-to-noise ratio λ, and the sample-to-dimension ratio α. This performance concentration implies that the vast majority of interpolators are suboptimal, with only a tiny exponentially small fraction achieving near-optimal performance. Numerical solutions of the variational equations reveal that the performance of efficient algorithms such as gradient descent and linear programming surpasses that of a typical interpolator, especially in regimes of small α.
Moreover, the analysis extends to Bayesian posterior distributions, showing that the performance of the posterior mean aligns with the typical interpolator, exhibiting exponential concentration. This confirms that high-dimensional performance landscapes are sharply peaked, with the best solutions being exceedingly rare among all interpolators.
These results have profound implications: they mathematically substantiate the empirical observation that algorithmic bias—like gradient descent—selects high-performance solutions from an exponentially small subset, explaining the benign overfitting phenomenon. The work bridges the gap between probabilistic theory and practical algorithm design, providing a rigorous foundation for understanding generalization in overparameterized models.
While the analysis is asymptotic and model-specific, it opens pathways for future research into nonlinear models, finite-sample effects, and algorithmic strategies that explicitly target performance peaks. Ultimately, this work advances the theoretical understanding of why modern deep learning models can interpolate data yet still generalize effectively, emphasizing the rarity of good solutions and the crucial role of algorithmic bias in high-dimensional learning.
Deep Dive
Abstract
Let $S$ be the set of unit norm linear classifiers $θ\in \mathbb{R}^d$ which correctly classify every point of a labeled dataset $(X_i,y_i)_{i=1}^n$, $X_i \in \mathbb{R}^d$, $y_i \in \{-1,+1\}$, with a possibly negative margin $κ$ fixed in advance. Under two natural data-generating distributions of the $(X,y)$ pairs -- a Gaussian mixture model and a logistic model with Gaussian features -- and in the proportional regime $n/d \to α$ with small enough $α$, we establish a large deviation principle on the event that a point $θ$ chosen uniformly at random from $S$ achieves a given generalization error, with high probability over the choice of the data. The associated large deviation rate function is deterministic and describes the proportion, at the exponential scale in $d$, of interpolating classifiers having a given desired performance. As a consequence, we establish the following concentration phenomenon: all but an exponentially small fraction of interpolating classifiers have approximately the same generalization performance given by the unique maximizer of this rate function. We numerically compare this maximizer to the performance of empirical risk minimization by gradient descent and to the performance of a natural linear program, both finding a point in $S$, and deduce that in the overparametrized regime of small $α$, these efficient procedures outperform the vast majority of interpolators, pointing to their nontrivial benign overfitting in this setting.