Achieving Margin Maximization Exponentially Fast via Progressive Norm Rescaling

摘要

In this work, we investigate the margin-maximization bias exhibited by gradient-based algorithms in classifying linearly separable data. We present an in-depth analysis of the specific properties of the velocity field associated with (normalized) gradients, focusing on their role in margin maximization. Inspired by this analysis, we propose a novel algorithm called Progressive Rescaling Gradient Descent (PRGD) and show that PRGD can maximize the margin at an *exponential rate*. This stands in stark contrast to all existing algorithms, which maximize the margin at a slow *polynomial rate*. Specifically, we identify mild conditions on data distribution under which existing algorithms such as gradient descent (GD) and normalized gradient descent (NGD) *provably fail* in maximizing the margin efficiently. To validate our theoretical findings, we present both synthetic and real-world experiments. Notably, PRGD also shows promise in enhancing the generalization performance when applied to linearly non-separable datasets and deep neural networks.