Gradient Methods Never Overfit On Separable Data

TL;DR

This paper proves that standard gradient methods for linear models on separable data do not overfit in finite time, with risks decreasing optimally until the dataset size is reached, after which they stabilize.

Contribution

It provides the first non-asymptotic analysis showing gradient methods never overfit on separable data, with optimal risk bounds up to dataset size.

Findings

Empirical risk and generalization error decrease at near-optimal rates.

Generalization error stabilizes at an optimal level after T approaches dataset size m.

Non-asymptotic bounds on margin violations are established and shown to be tight.

Abstract

A line of recent works established that when training linear predictors over separable data, using gradient methods and exponentially-tailed losses, the predictors asymptotically converge in direction to the max-margin predictor. As a consequence, the predictors asymptotically do not overfit. However, this does not address the question of whether overfitting might occur non-asymptotically, after some bounded number of iterations. In this paper, we formally show that standard gradient methods (in particular, gradient flow, gradient descent and stochastic gradient descent) never overfit on separable data: If we run these methods for $T$ iterations on a dataset of size $m$, both the empirical risk and the generalization error decrease at an essentially optimal rate of $\tilde{\mathcal{O}}(1/\gamma^2 T)$ up till $T\approx m$, at which point the generalization error remains fixed at an essentially optimal level of $\tilde{\mathcal{O}}(1/\gamma^2 m)$ regardless of how large $T$ is. Along the way, we present non-asymptotic bounds on the number of margin violations over the dataset, and prove their tightness.