Optimality of the phase-transition bound

Determine whether the exponent p in the polylogarithmic phase-transition-time bound for gradient descent with a large constant stepsize on linearly separable logistic regression, or the bound’s dependence on the stepsize eta, is optimal.

Background

The paper establishes an upper bound of the form tau <= C (ln eta)^p for the phase-transition time tau of constant-stepsize gradient descent applied to logistic regression on bounded, linearly separable data. The exponent p depends on the data margin and rank, while C depends on additional problem parameters.

The authors explicitly note that they prove no lower bounds and therefore do not resolve whether the exponent p or the dependence on the stepsize eta in their phase-transition-time upper bound can be improved. Establishing matching lower bounds or otherwise characterizing optimal dependence would determine the sharpness of the result.