aiqbench analysis
This research demonstrates how early stopping during gradient descent can achieve minimax-optimal performance in Gaussian mixture classification, effectively overcoming the statistical suboptimality inherent to max-margin interpolating classifiers. By establishing a novel calibration result that handles label-flipping noise and model misspecification, the study proves that this approach yields optimal convergence rates even for covariance spectra with fast decay. The findings highlight a critical advantage of early stopping over linear interpolation in high-dimensional settings where standard bounds typically suffer from slower square-root rate degradation.
Minimax Optimal Early-Stopped Gradient Descent for Gaussian Mixture Classification
arXiv:2608.06250v1 Announce Type: cross Abstract: In overparameterised classification, training data can be linearly separable even when the underlying distribution is not. In this setting, gradient descent (GD) on the logistic loss diverges in norm while converging in direction to a max-margin interpolating classifier, whose implicit bias can be statistically suboptimal. In this work, we show that early stopping can overcome this suboptimality: in a Gaussian mixture model with label-flipping noise, GD stopped at an appropriate oracle time achieves minimax-optimal excess zero-one risk for covariance spectra with fast and continuous decay, including polynomial and exponential spectral decays. Our analysis combines a sharp upper bound for the early-stopped iterate with a matching statistical lower bound over arbitrary classifiers, yielding optimal rates that are validated by experiments. A central technical contribution is a new calibration result that converts excess logistic risk into excess zero-one risk; it handles the model misspecification induced by the label-flipping noise, and removes the square-root rate in standard bounds. We also establish a lower bound for linear interpolators, showing that interpolation can require exponentially more samples than early stopping to achieve the same excess risk.
Abstract:In overparameterised classification, training data can be linearly separable even when the underlying distribution is not. In this setting, gradient descent (GD) on the logistic loss diverges in norm while converging in direction to a max-margin interpolating classifier, whose implicit bias can be statistically suboptimal. In this work, we show that early stopping can overcome this suboptimality: in a Gaussian mixture model with label-flipping noise, GD stopped at an appropriate oracle time achieves minimax-optimal excess zero-one risk for covariance spectra with fast and continuous decay, including polynomial and exponential spectral decays. Our analysis combines a sharp upper bound for the early-stopped iterate with a matching statistical lower bound over arbitrary classifiers, yielding optimal rates that are validated by experiments. A central technical contribution is a new calibration result that converts excess logistic risk into excess zero-one risk; it handles the model misspecification induced by the label-flipping noise, and removes the square-root rate in standard bounds. We also establish a lower bound for linear interpolators, showing that interpolation can require exponentially more samples than early stopping to achieve the same excess risk.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2608.06250 [stat.ML]
(or arXiv:2608.06250v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2608.06250 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 278 words
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