A Stochastic LineseArch Method (SLAM) for Minimizing Expectation Residuals: Weakening Boundedness of Gradient
Abstract
We establish convergence guarantees for the Stochastic LineseArch Method (SLAM) proposed and analyzed in Wang et al. (arXiv:2512.14979) for nonconvex composite stochastic optimization, retaining the iteration and sample complexity orders of the original analysis assuming neither the boundedness of true gradients nor the Lipschitz continuity of the convex nonsmooth term r. ChatGPT was used to assist in developing a key intermediate result. Under uniformly Lipschitz continuous and unbiased sampled gradients with uniformly bounded variance with mini-batch sizes proportional to the iteration budget, SLAM achieves an iteration complexity 𝒪(ε−2) and a sample complexity 𝒪(ε−4) for obtaining an iterate whose expected proximal-gradient residual norm is at most ε.
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BibTeX
@misc{wang2026stochastic,
title = {{A Stochastic LineseArch Method (SLAM) for Minimizing Expectation Residuals: Weakening Boundedness of Gradient}},
author = {Qi Wang and Uday V. Shanbhag and Yue Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37947},
eprint = {2610.37947},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37947},
}
RIS
TY - UNPB
TI - A Stochastic LineseArch Method (SLAM) for Minimizing Expectation Residuals: Weakening Boundedness of Gradient
AU - Qi Wang
AU - Uday V. Shanbhag
AU - Yue Xie
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37947
UR - https://arxiv.org/abs/2610.37947
AB - We establish convergence guarantees for the Stochastic LineseArch Method (SLAM) proposed and analyzed in Wang et al. (arXiv:2512.14979) for nonconvex composite stochastic optimization, retaining the iteration and sample complexity orders of the original analysis assuming neither the boundedness of true gradients nor the Lipschitz continuity of the convex nonsmooth term r. ChatGPT was used to assist in developing a key intermediate result. Under uniformly Lipschitz continuous and unbiased sampled gradients with uniformly bounded variance with mini-batch sizes proportional to the iteration budget, SLAM achieves an iteration complexity 𝒪(ε^−2) and a sample complexity 𝒪(ε^−4) for obtaining an iterate whose expected proximal-gradient residual norm is at most ε.
ER -