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arXiv · 2303.07160

Tighter Lower Bounds for Shuffling SGD: Random Permutations and Beyond

Jaeyoung Cha·Jaewook Lee·Chulhee Yun
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Abstract

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.

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Jaeyoung Cha, Jaewook Lee, Chulhee Yun. 2023-03-13. Tighter Lower Bounds for Shuffling SGD: Random Permutations and Beyond. https://arxiv.org/abs/2303.07160

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