Tight Bounds for Logistic Regression with Large Stepsize Gradient Descent in Low Dimension

• URL: http://arxiv.org/abs/2602.12471v1
• Date: Thu, 12 Feb 2026 22:58:18 GMT
• Title: Tight Bounds for Logistic Regression with Large Stepsize Gradient Descent in Low Dimension
• Authors: Michael Crawshaw, Mingrui Liu,
• Abstract summary: We consider the optimization problem of minimizing the logistic gradient with descent to train a linear model for binary classification with separable data.<n>We show that GD with a sufficiently large learning rate $$ finds a point with loss smaller than $mathcalO (1/(T))$, as long as $T geq (n/+ 1/2)$, where $n$ is the dataset size.
• Score: 36.3266119975955
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 step size $η= Θ(γ^2 T)$ (where $γ$ 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 $η$ finds a point with loss smaller than $\mathcal{O}(1/(ηT))$, as long as $T \geq Ω(n/γ+ 1/γ^2)$, where $n$ is the dataset size. Our improved rate comes from a tighter bound on the time $τ$ 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 $τ$ matching our upper bound up to logarithmic factors, showing that our analysis is tight.

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