Implicit Bias of Logistic Regression Aligns Early in Gradient Descent

Gradient descent, a cornerstone of machine learning optimization, exhibits an "implicit bias" that helps prevent overfitting, particularly the max-margin property in linear classifiers. While this max-margin convergence has been understood as an asymptotic phenomenon, occurring over a very long training period, empirical observations often show good generalization much earlier. This paper delves into the early-stage dynamics of gradient descent for logistic regression. The study reveals that the parameter vector begins to align with the max-margin direction significantly faster than the established asymptotic rates. Specifically, it shows a weak alignment within an exponentially faster number of iterations. This rapid alignment helps explain the common observation that "train longer, generalize better" can manifest even in the initial phases of learning, not just at the very end. By analyzing the radial and tangential flows of the parameter vector and focusing directly on the alignment dynamics with dataset geometry, the research provides a new theoretical understanding. This approach avoids complex asymptotic expansions, offering a clearer insight into the mechanism behind this early-stage alignment and its implications for model generalization.