arXiv:2606.24879v1 Announce Type: cross Abstract: We study the last iterate of the stochastic subgradient method for one-dimensional convex Lipschitz objectives. For a fixed horizon $n$, we consider the standard fixed stepsizes $\eta =\Theta(1/\sqrt n)$. We prove that, for such stepsize policies, under additive i.i.d. subgradient noise with uniformly bounded variance, the last iterate features an optimization error of order $1/\sqrt n$, thereby removing the extra $(\log n)$ factor present in existing generic bounds. On the other hand, we show that without the i.i.d. assumption, the optimizatio
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