The Last Iterate of the Projected Stochastic Subgradient Method Pays a Logarithm, Even in Dimension One

We prove that the projected stochastic subgradient method with the standard anytime stepsizes $η_t = γ/\sqrt t$ can suffer expected last-iterate optimization error of order $(\log n)/\sqrt n$ already on the real line. Combined with the classical in-expectation upper bound, this establishes the sharp