# Variable Smoothing for Weakly Convex Composite Functions

arXiv: https://arxiv.org/abs/2003.07612 · HTML: https://mlsanity.com/paper/2003.07612/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2020.

## Built on by (most cited first, 12 of 12)

- [Moreau Envelope for Nonconvex Bi-Level Optimization: A Single-loop and Hessian-free Solution Strategy](https://mlsanity.com/paper/2405.09927/) (2024)
- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [A Unified Convergence Theorem for Stochastic Optimization Methods](https://mlsanity.com/paper/2206.03907/) (2022)
- [FedCanon: Non-Convex Composite Federated Learning with Efficient Proximal Operation on Heterogeneous Data](https://mlsanity.com/paper/2504.11903/) (2025)
- [AdAUC: End-to-end Adversarial AUC Optimization Against Long-tail Problems](https://mlsanity.com/paper/2206.12169/) (2022)
- [Coordinate Descent Methods for DC Minimization: Optimality Conditions and Global Convergence](https://mlsanity.com/paper/2109.04228/) (2021)
- [Coordinate Descent Methods for Fractional Minimization](https://mlsanity.com/paper/2201.12691/) (2022)
- [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025)
- [ADMM for Structured Fractional Minimization](https://mlsanity.com/paper/2411.07496/) (2024)
- [Decentralized Nonconvex Composite Federated Learning with Gradient Tracking and Momentum](https://mlsanity.com/paper/2504.12742/) (2025)
- [Sample Complexity Analysis for Constrained Bilevel Reinforcement Learning](https://mlsanity.com/paper/2602.00282/) (2026)
- [A proximal subgradient method for nonconvex stochastic optimization under the Kurdyka-Lojasiewicz condition](https://mlsanity.com/paper/2608.05460/) (2026)

## Abstract (arXiv)

We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that introduce less bias than the standard convex regularizers. We develop a variable smoothing algorithm, based on the Moreau envelope with a decreasing sequence of smoothing parameters, and prove a complexity of O(ε-3) to achieve an ε-approximate solution. This bound interpolates between the O(ε-2) bound for the smooth case and the O(ε-4) bound for the subgradient method. Our complexity bound is in line with other works that deal with structured nonsmoothness of weakly convex functions.