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In this work, we investigate the margin-maximization bias exhibited bygradient-based algorithms in classifying linearly separable data. We present anin-depth analysis of the specific properties of the velocity field associatedwith (normalized) gradients, focusing on their role in margin maximization.Inspired by this analysis, we propose a novel algorithm called ProgressiveRescaling Gradient Descent (PRGD) and show that PRGD can maximize the margin atan {\em exponential rate}. This stands in stark contrast to all existingalgorithms, which maximize the margin at a slow {\em polynomial rate}.Specifically, we identify mild conditions on data distribution under whichexisting algorithms such as gradient descent (GD) and normalized gradientdescent (NGD) {\em provably fail} in maximizing the margin efficiently. Tovalidate our theoretical findings, we present both synthetic and real-worldexperiments. Notably, PRGD also shows promise in enhancing the generalizationperformance when applied to linearly non-separable datasets and deep neuralnetworks.

Achieving Margin Maximization Exponentially Fast via Progressive Norm Rescaling