Tighter Lower Bounds for Shuffling SGD: Random Permutations and Beyond

Jaeyoung Cha, Jaewook Lee, Chulhee Yun
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:3855-3912, 2023.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-cha23a, title = {Tighter Lower Bounds for Shuffling {SGD}: Random Permutations and Beyond}, author = {Cha, Jaeyoung and Lee, Jaewook and Yun, Chulhee}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {3855--3912}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/cha23a/cha23a.pdf}, url = {https://proceedings.mlr.press/v202/cha23a.html}, 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.} }
Endnote
%0 Conference Paper %T Tighter Lower Bounds for Shuffling SGD: Random Permutations and Beyond %A Jaeyoung Cha %A Jaewook Lee %A Chulhee Yun %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-cha23a %I PMLR %P 3855--3912 %U https://proceedings.mlr.press/v202/cha23a.html %V 202 %X 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.
APA
Cha, J., Lee, J. & Yun, C.. (2023). Tighter Lower Bounds for Shuffling SGD: Random Permutations and Beyond. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:3855-3912 Available from https://proceedings.mlr.press/v202/cha23a.html.

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