Bias of Homotopic Gradient Descent for the Hinge Loss
This addresses a theoretical gap for practitioners using hinge loss in machine learning, though it is incremental as it extends existing smooth loss results to a non-smooth case.
The paper tackles the problem of gradient descent convergence for non-smooth hinge loss in linear classifiers on separable data, showing that a homotopic variant converges to the max-margin solution with explicit rates.
Gradient descent is a simple and widely used optimization method for machine learning. For homogeneous linear classifiers applied to separable data, gradient descent has been shown to converge to the maximal margin (or equivalently, the minimal norm) solution for various smooth loss functions. The previous theory does not, however, apply to non-smooth functions such as the hinge loss which is widely used in practice. Here, we study the convergence of a homotopic variant of gradient descent applied to the hinge loss and provide explicit convergence rates to the max-margin solution for linearly separable data.