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：我们研究无放回随机梯度下降法（SGD）在求解光滑（强）凸有限和最小化问题时的收敛下界。与多数现有结果关注基于分量数$n$和轮次数$K$的最终迭代下界不同，我们寻求任意加权平均迭代的下界，该下界在所有因子（包括条件数$\kappa$）中均为紧界。针对随机重排SGD，我们提出的下界在$\kappa$依赖性上比现有结果更紧。我们的结果首次在强凸和凸情况下完美弥合加权平均迭代下界与上界之间的差距。我们还证明了基于任意排列的SGD的加权平均迭代下界，该结果适用于所有精心选择最优排列的变体。我们的下界在$n$和$\kappa$因子上改进了现有结果，从而与最近提出的GraB算法的上界相匹配。