Incremental Gradient Descent with Small Epoch Counts is Surprisingly Slow on Ill-Conditioned Problems

Yujun Kim, Jaeyoung Cha, Chulhee Yun
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:29913-29968, 2025.

Abstract

Recent theoretical results demonstrate that the convergence rates of permutation-based SGD (e.g., random reshuffling SGD) are faster than uniform-sampling SGD; however, these studies focus mainly on the large epoch regime, where the number of epochs $K$ exceeds the condition number $\kappa$. In contrast, little is known when $K$ is smaller than $\kappa$, and it is still a challenging open question whether permutation-based SGD can converge faster in this small epoch regime (Safran and Shamir, 2021). As a step toward understanding this gap, we study the naive deterministic variant, Incremental Gradient Descent (IGD), on smooth and strongly convex functions. Our lower bounds reveal that for the small epoch regime, IGD can exhibit surprisingly slow convergence even when all component functions are strongly convex. Furthermore, when some component functions are allowed to be nonconvex, we prove that the optimality gap of IGD can be significantly worse throughout the small epoch regime. Our analyses reveal that the convergence properties of permutation-based SGD in the small epoch regime may vary drastically depending on the assumptions on component functions. Lastly, we supplement the paper with tight upper and lower bounds for IGD in the large epoch regime.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-kim25a, title = {Incremental Gradient Descent with Small Epoch Counts is Surprisingly Slow on Ill-Conditioned Problems}, author = {Kim, Yujun and Cha, Jaeyoung and Yun, Chulhee}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {29913--29968}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/kim25a/kim25a.pdf}, url = {https://proceedings.mlr.press/v267/kim25a.html}, abstract = {Recent theoretical results demonstrate that the convergence rates of permutation-based SGD (e.g., random reshuffling SGD) are faster than uniform-sampling SGD; however, these studies focus mainly on the large epoch regime, where the number of epochs $K$ exceeds the condition number $\kappa$. In contrast, little is known when $K$ is smaller than $\kappa$, and it is still a challenging open question whether permutation-based SGD can converge faster in this small epoch regime (Safran and Shamir, 2021). As a step toward understanding this gap, we study the naive deterministic variant, Incremental Gradient Descent (IGD), on smooth and strongly convex functions. Our lower bounds reveal that for the small epoch regime, IGD can exhibit surprisingly slow convergence even when all component functions are strongly convex. Furthermore, when some component functions are allowed to be nonconvex, we prove that the optimality gap of IGD can be significantly worse throughout the small epoch regime. Our analyses reveal that the convergence properties of permutation-based SGD in the small epoch regime may vary drastically depending on the assumptions on component functions. Lastly, we supplement the paper with tight upper and lower bounds for IGD in the large epoch regime.} }
Endnote
%0 Conference Paper %T Incremental Gradient Descent with Small Epoch Counts is Surprisingly Slow on Ill-Conditioned Problems %A Yujun Kim %A Jaeyoung Cha %A Chulhee Yun %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-kim25a %I PMLR %P 29913--29968 %U https://proceedings.mlr.press/v267/kim25a.html %V 267 %X Recent theoretical results demonstrate that the convergence rates of permutation-based SGD (e.g., random reshuffling SGD) are faster than uniform-sampling SGD; however, these studies focus mainly on the large epoch regime, where the number of epochs $K$ exceeds the condition number $\kappa$. In contrast, little is known when $K$ is smaller than $\kappa$, and it is still a challenging open question whether permutation-based SGD can converge faster in this small epoch regime (Safran and Shamir, 2021). As a step toward understanding this gap, we study the naive deterministic variant, Incremental Gradient Descent (IGD), on smooth and strongly convex functions. Our lower bounds reveal that for the small epoch regime, IGD can exhibit surprisingly slow convergence even when all component functions are strongly convex. Furthermore, when some component functions are allowed to be nonconvex, we prove that the optimality gap of IGD can be significantly worse throughout the small epoch regime. Our analyses reveal that the convergence properties of permutation-based SGD in the small epoch regime may vary drastically depending on the assumptions on component functions. Lastly, we supplement the paper with tight upper and lower bounds for IGD in the large epoch regime.
APA
Kim, Y., Cha, J. & Yun, C.. (2025). Incremental Gradient Descent with Small Epoch Counts is Surprisingly Slow on Ill-Conditioned Problems. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:29913-29968 Available from https://proceedings.mlr.press/v267/kim25a.html.

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