An Improved Analysis of the Last Iterate of the SubGradient Method in the Plane

We study the last iterate of the subGradient Method (sGM) for convex Lipschitz objectives defined on $\mathbb R^2$.

We prove that for a finite horizon $n$ and a constant stepsize $\eta = \Theta(1/\sqrt{n})$, the last iterate achieves an optimization error of order $1/\sqrt{n}$, showing that the extra $\log{n}$ factor appearing in high dimensions is unnecessary in the plane.

This makes progress on an open problem posed by Koren and Segal in 2020 on the optimal error of the last iterate of sGM in fixed dimension.