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On Nonsmooth and Relatively Weakly Convex Minimization

  • Chen Jiang
  • Jiawen Bi
  • Shuzhong Zhang
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Abstract

Composite optimization plays a central role in modern machine learning and signal processing, as it offers a natural balance between data fidelity and structural properties. In this paper, we study composite optimization in the setting where both components are nonsmooth and nonconvex. We start with a deterministic Bregman proximal subgradient method that converges under subgradient upper-bound conditions. This approach relaxes the standard requirement on the convexity of the regularization term, thus accommodating a broader range of applications. To extend this to the stochastic regime, we develop a model-based minimization method under a relative Lipschitz condition and establish a convergence rate of 𝒪(ε−4). We also extend the framework with convergence guarantees to the setting where the distance generating function and its gradient are accessible only through a stochastic oracle.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026nonsmooth,
  title = {{On Nonsmooth and Relatively Weakly Convex Minimization}},
  author = {Chen Jiang and Jiawen Bi and Shuzhong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28530},
  eprint = {2608.28530},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28530},
}
RIS
TY  - UNPB
TI  - On Nonsmooth and Relatively Weakly Convex Minimization
AU  - Chen Jiang
AU  - Jiawen Bi
AU  - Shuzhong Zhang
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28530
UR  - https://arxiv.org/abs/2608.28530
AB  - Composite optimization plays a central role in modern machine learning and signal processing, as it offers a natural balance between data fidelity and structural properties. In this paper, we study composite optimization in the setting where both components are nonsmooth and nonconvex. We start with a deterministic Bregman proximal subgradient method that converges under subgradient upper-bound conditions. This approach relaxes the standard requirement on the convexity of the regularization term, thus accommodating a broader range of applications. To extend this to the stochastic regime, we develop a model-based minimization method under a relative Lipschitz condition and establish a convergence rate of 𝒪(ε^−4). We also extend the framework with convergence guarantees to the setting where the distance generating function and its gradient are accessible only through a stochastic oracle.
ER  -