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Tight Bounds for Logistic Regression with Large Stepsize Gradient Descent in Low Dimension
[Submitted on 12 Feb 2026 (v1), last revised 15 Jun 2026 (this v · 2026-06-16 · via cs.LG updates on arXiv.org

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Abstract:We consider the optimization problem of minimizing the logistic loss with gradient descent to train a linear model for binary classification with separable data. With a budget of $T$ iterations, it was recently shown that an accelerated $1/T^2$ rate is possible by choosing a large stepsize $\eta = \Theta(\gamma^2 T)$ (where $\gamma$ is the dataset's margin) despite the resulting non-monotonicity of the loss. In this paper, we provide a tighter analysis of gradient descent for this problem when the data is two-dimensional: we show that GD with a sufficiently large learning rate $\eta$ finds a point with loss smaller than $\mathcal{O}(1/(\eta \gamma^2 T))$, as long as $T \geq \Omega(n/\gamma + 1/\gamma^2)$, where $n$ is the dataset size. Our improved rate comes from a tighter bound on the time $\tau$ that it takes for GD to transition from unstable (non-monotonic loss) to stable (monotonic loss), via a fine-grained analysis of the oscillatory dynamics of GD in the subspace orthogonal to the max-margin classifier. We also provide a lower bound of $\tau$ matching our upper bound up to logarithmic factors, showing that our analysis is tight.

Submission history

From: Michael Crawshaw [view email]
[v1] Thu, 12 Feb 2026 22:58:18 UTC (246 KB)
[v2] Mon, 15 Jun 2026 16:08:07 UTC (239 KB)

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