Faster Margin Maximization Rates for Generic and Adversarially Robust Optimization Methods

First-order optimization methods tend to inherently favor certain solutions over others when minimizing an underdetermined training objective that has multiple global optima. This phenomenon, known as implicit bias, plays a critical role in understanding the generalization capabilities of optimization algorithms. Recent research has revealed that in separable binary classification tasks gradient-descent-based methods exhibit an implicit bias for the $\ell_2$-maximal margin classifier. Similarly, generic optimization methods, such as mirror descent and steepest descent, have been shown to converge to maximal margin classifiers defined by alternative geometries. While gradient-descent-based algorithms provably achieve fast implicit bias rates, corresponding rates in the literature for generic optimization methods are relatively slow. To address this limitation, we present a series of state-of-the-art implicit bias rates for mirror descent and steepest descent algorithms. Our primary technique involves transforming a generic optimization algorithm into an online optimization dynamic that solves a regularized bilinear game, providing a unified framework for analyzing the implicit bias of various optimization methods. Our accelerated rates are derived by leveraging the regret bounds of online learning algorithms within this game framework. We then show the flexibility of this framework by analyzing the implicit bias in adversarial training, and again obtain significantly improved convergence rates.