The Implicit Bias of Gradient Descent on Separable Data

Daniel Soudry, Elad Hoffer, Mor Shpigel Nacson, Nathan Srebro

Abstract:

We show that gradient descent on an unregularized logistic regression
problem, for almost all separable datasets, converges to the same direction as the max-margin solution. The result generalizes also to other monotone decreasing loss functions with an infimum at infinity, and we also discuss a multi-class generalizations to the cross entropy loss. Furthermore,
we show this convergence is very slow, and only logarithmic in the
convergence of the loss itself. This can help explain the benefit
of continuing to optimize the logistic or cross-entropy loss even
after the training error is zero and the training loss is extremely
small, and, as we show, even if the validation loss increases. Our
methodology can also aid in understanding implicit regularization
in more complex models and with other optimization methods.

TL;DR: The normalized solution of gradient descent on logistic regression (or a similarly decaying loss) slowly converges to the L2 max margin solution on separable data.

Keywords: gradient descent, implicit regularization, generalization, margin, logistic regression, loss functions, optimization, exponential tail, cross-entropy

Code: [ paper-submissions/MaxMargin](https://github.com/paper-submissions/MaxMargin) + [ 1 community implementation](https://paperswithcode.com/paper/?openreview=r1q7n9gAb)

Community Implementations: [ 2 code implementations](https://www.catalyzex.com/paper/the-implicit-bias-of-gradient-descent-on/code)