We study convergence lower bounds of without-replacement stochastic gradient descent (SGD) for solving smooth (strongly-)convex finite-sum minimization problems. Unlike most existing results focusing on final iterate lower bounds in terms of the number of components $n$ and the number of epochs $K$, we seek bounds for arbitrary weighted average iterates that are tight in all factors including the condition number $\kappa$. For SGD with Random Reshuffling, we present lower bounds that have tighter $\kappa$ dependencies than existing bounds. Our results are the first to perfectly close the gap between lower and upper bounds for weighted average iterates in both strongly-convex and convex cases. We also prove weighted average iterate lower bounds for arbitrary permutation-based SGD, which apply to all variants that carefully choose the best permutation. Our bounds improve the existing bounds in factors of $n$ and $\kappa$ and thereby match the upper bounds shown for a recently proposed algorithm called GraB.

翻译：我们研究的是不替换的悬浮梯度下降的趋同界限,以解决平滑(强力)混凝土最小化问题。与大多数现有结果不同的是,以最终折流较低界限为重点,从元件数量和小巧数量来看,我们寻求任意加权平均迭代界限,所有因素,包括条件数$\kappa美元,都是紧凑的。对于随机重整的SGD,我们提出了比现有界限更紧的较低界限,比现有界限更紧的$\kappappa$。我们的结果是,在强凝固和convex两种情况下,均完全缩小了加权平均值的下限和上限之间的差距。我们还证明,加权平均值是任意调整基于 SGD的下限,这适用于所有仔细选择最佳调整的变式。我们的界限改进了现有的以美元和$\kapappa值计的系数,从而与最近提议的 GraB 算法显示的上限值相符。</s>