Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency

Abstract

We consider \emph{gradient descent} (GD) with a constant stepsize applied to logistic regression with linearly separable data, where the constant stepsize $\eta$ is so large that the loss initially oscillates. We show that GD exits this initial oscillatory phase rapidly — in $O(\eta)$ steps, and subsequently achieves an $\tilde{O}(1 / (\eta t) )$ convergence rate after $t$ additional steps. Our results imply that, given a budget of $T$ steps, GD can achieve an \emph{accelerated} loss of $\tilde{O}(1/T^2)$ with an aggressive stepsize $\eta:= \Theta( T)$, without any use of momentum or variable stepsize schedulers. Our proof technique is versatile and also handles general classification loss functions (where exponential tails are needed for the $\tilde{O}(1/T^2)$ acceleration), nonlinear predictors in the \emph{neural tangent kernel} regime, and online \emph{stochastic gradient descent} (SGD) with a large stepsize, under suitable separability conditions.