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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 optimization error can be of order $(\\log n)/\\sqrt n$. 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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 optimization error can be of order $(\\log n)/\\sqrt n$. 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