A sequential smoothing majorant stochastic approximation method for nonconvex nonconcave minimax problems
Abstract
We propose a sequential smoothing majorant stochastic approximation (SMSA) method for nonconvex-nonconcave minimax optimization problems. To overcome the nonconcavity of the inner maximization problem, we introduce a new majorant stochastic approximation with an O(βN2) accuracy bound, where βN is the sample coverage radius. The accuracy bound significantly improves upon the O(βN) approximation bound for the standard stochastic approximation. Moreover, we establish nonasymptotic bounds for both global optimal values and minimizer sets, and prove consistency for the Clarke stationary points, as βN ↓ 0 almost surely. We show that the generated sequence by the SMSA method is bounded, and that the returned point is an approximate Clarke stationary point of the majorant stochastic approximation models. Numerical experiments on a synthetic toy example and robust logistic regression on two UCI datasets demonstrate improved approximation fidelity and lower mean robust test losses relative to the standard sampled approximation.
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BibTeX
@misc{luo2026sequential,
title = {{A sequential smoothing majorant stochastic approximation method for nonconvex nonconcave minimax problems}},
author = {Zhouxing Luo and Wei Liu and Xin Liu and Xiaojun Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11171},
eprint = {2609.11171},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11171},
}
RIS
TY - UNPB
TI - A sequential smoothing majorant stochastic approximation method for nonconvex nonconcave minimax problems
AU - Zhouxing Luo
AU - Wei Liu
AU - Xin Liu
AU - Xiaojun Chen
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11171
UR - https://arxiv.org/abs/2609.11171
AB - We propose a sequential smoothing majorant stochastic approximation (SMSA) method for nonconvex-nonconcave minimax optimization problems. To overcome the nonconcavity of the inner maximization problem, we introduce a new majorant stochastic approximation with an O(β_N^2) accuracy bound, where β_N is the sample coverage radius. The accuracy bound significantly improves upon the O(β_N) approximation bound for the standard stochastic approximation. Moreover, we establish nonasymptotic bounds for both global optimal values and minimizer sets, and prove consistency for the Clarke stationary points, as β_N ↓ 0 almost surely. We show that the generated sequence by the SMSA method is bounded, and that the returned point is an approximate Clarke stationary point of the majorant stochastic approximation models. Numerical experiments on a synthetic toy example and robust logistic regression on two UCI datasets demonstrate improved approximation fidelity and lower mean robust test losses relative to the standard sampled approximation.
ER -