Non-Asymptotic and Non-Lipschitzian Bounds on Optimal Values in Stochastic Optimization Under Heavy Tails

Jindong Tong, Hongcheng Liu, Johannes O. Royset
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:59812-59828, 2025.

Abstract

This paper focuses on non-asymptotic confidence bounds (CB) for the optimal values of stochastic optimization (SO) problems. Existing approaches often rely on two conditions that may be restrictive: The need for a global Lipschitz constant and the assumption of light-tailed distributions. Beyond either of the conditions, it remains largely unknown whether computable CBs can be constructed. In view of this literature gap, we provide three key findings below: (i) Based on the conventional formulation of sample average approximation (SAA), we derive non-Lipschitzian CBs for convex SP problems under heavy tails. (ii) We explore diametrical risk minimization (DRM)—a recently introduced modification to SAA—and attain non-Lipschitzian CBs for nonconvex SP problems in light-tailed settings. (iii) We extend our analysis of DRM to handle heavy-tailed randomness by utilizing properties in formulations for training over-parameterized classification models.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-tong25a, title = {Non-Asymptotic and Non-Lipschitzian Bounds on Optimal Values in Stochastic Optimization Under Heavy Tails}, author = {Tong, Jindong and Liu, Hongcheng and Royset, Johannes O.}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {59812--59828}, 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/tong25a/tong25a.pdf}, url = {https://proceedings.mlr.press/v267/tong25a.html}, abstract = {This paper focuses on non-asymptotic confidence bounds (CB) for the optimal values of stochastic optimization (SO) problems. Existing approaches often rely on two conditions that may be restrictive: The need for a global Lipschitz constant and the assumption of light-tailed distributions. Beyond either of the conditions, it remains largely unknown whether computable CBs can be constructed. In view of this literature gap, we provide three key findings below: (i) Based on the conventional formulation of sample average approximation (SAA), we derive non-Lipschitzian CBs for convex SP problems under heavy tails. (ii) We explore diametrical risk minimization (DRM)—a recently introduced modification to SAA—and attain non-Lipschitzian CBs for nonconvex SP problems in light-tailed settings. (iii) We extend our analysis of DRM to handle heavy-tailed randomness by utilizing properties in formulations for training over-parameterized classification models.} }
Endnote
%0 Conference Paper %T Non-Asymptotic and Non-Lipschitzian Bounds on Optimal Values in Stochastic Optimization Under Heavy Tails %A Jindong Tong %A Hongcheng Liu %A Johannes O. Royset %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-tong25a %I PMLR %P 59812--59828 %U https://proceedings.mlr.press/v267/tong25a.html %V 267 %X This paper focuses on non-asymptotic confidence bounds (CB) for the optimal values of stochastic optimization (SO) problems. Existing approaches often rely on two conditions that may be restrictive: The need for a global Lipschitz constant and the assumption of light-tailed distributions. Beyond either of the conditions, it remains largely unknown whether computable CBs can be constructed. In view of this literature gap, we provide three key findings below: (i) Based on the conventional formulation of sample average approximation (SAA), we derive non-Lipschitzian CBs for convex SP problems under heavy tails. (ii) We explore diametrical risk minimization (DRM)—a recently introduced modification to SAA—and attain non-Lipschitzian CBs for nonconvex SP problems in light-tailed settings. (iii) We extend our analysis of DRM to handle heavy-tailed randomness by utilizing properties in formulations for training over-parameterized classification models.
APA
Tong, J., Liu, H. & Royset, J.O.. (2025). Non-Asymptotic and Non-Lipschitzian Bounds on Optimal Values in Stochastic Optimization Under Heavy Tails. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:59812-59828 Available from https://proceedings.mlr.press/v267/tong25a.html.

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