Gradient Methods Never Overfit On Separable Data
This provides theoretical guarantees for practitioners using gradient methods on separable data, ensuring no overfitting occurs even with finite iterations, though it is incremental as it builds on prior asymptotic results.
The paper tackles the problem of whether gradient methods overfit non-asymptotically on separable data, showing that for linear predictors with exponentially-tailed losses, both empirical risk and generalization error decrease optimally until a point and then stabilize, with rates like O~(1/(γ^2 T)) and O~(1/(γ^2 m)).
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 $\tilde{\mathcal{O}}(1/γ^2 T)$ up till $T\approx m$, at which point the generalization error remains fixed at an essentially optimal level of $\tilde{\mathcal{O}}(1/γ^2 m)$ 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.