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

We consider gradient descent (GD) with a constant stepsize applied to logistic regression with linearly separable data, where the constant stepsize η is so large that the loss initially oscillates. We show that GD exits this initial oscillatory phase rapidly – in 𝒪(η) steps – and subsequently achieves an 𝒪̃(1 / (η t) ) convergence rate after t additional steps. Our results imply that, given a budget of T steps, GD can achieve an accelerated loss of 𝒪̃(1/T^2) with an aggressive stepsize η:= Θ( 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 𝒪̃(1/T^2) acceleration), nonlinear predictors in the neural tangent kernel regime, and online stochastic gradient descent (SGD) with a large stepsize, under suitable separability conditions.