AI 中文总结
研究\(\mathbb{R}^2\)上凸Lipschitz目标函数的次梯度法最后一次迭代，采用常数步长\(\eta = \Theta(1/\sqrt{n})\)，证明其优化误差为\(1/\sqrt{n}\)阶，解决了相关开放问题。
AI 中文摘要
我们研究了在\(\mathbb{R}^2\)上定义的凸Lipschitz目标函数的次梯度法（sGM）的最后一次迭代。我们证明，对于有限时间范围\(n\)和常数步长\(\eta = \Theta(1/\sqrt{n})\)，最后一次迭代实现了\(1/\sqrt{n}\)阶的优化误差，表明在平面中高维出现的额外\(\log{n}\)因子是不必要的。这解决了Koren和Segal在2020年提出的关于固定维度下sGM最后一次迭代的最优误差的一个开放问题。
英文摘要
We study the last iterate of the projected subGradient Method (sGM) for convex Lipschitz objectives defined on $\mathbb{R}^d$. We prove that, for a finite horizon $n$ and a constant stepsize $η=Θ(1/\sqrt n)$, the last iterate achieves an optimization error of order $d/\sqrt n$, showing that the extra $\log n$ factor appearing in high dimensions is unnecessary in every fixed dimension. We complement this result with a matching linear-in-$d$ lower bound and show that the sharp worst-case dimension-horizon dependence is of order $\min\{d,\log n\}/\sqrt n$. This solves, in particular, a COLT open problem posed by Koren and Segal in 2020 and shows that the correct dependence on the dimension is linear rather than logarithmic.
