# Moreau Envelope ADMM for Decentralized Weakly Convex Optimization

arXiv: https://arxiv.org/abs/2308.16752 · HTML: https://mlsanity.com/paper/2308.16752/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 28 references that are in the map.
- First posted on arXiv 31 Aug 2023.

## Builds on (most cited first, 11 of 11)

- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Achieving Geometric Convergence for Distributed Optimization over Time-Varying Graphs](https://mlsanity.com/paper/1607.03218/) (2016)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Global Convergence of ADMM in Nonconvex Nonsmooth Optimization](https://mlsanity.com/paper/1511.06324/) (2015)
- [Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems](https://mlsanity.com/paper/1410.1390/) (2014)
- [Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval](https://mlsanity.com/paper/1705.02356/) (2017)
- [Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results](https://mlsanity.com/paper/1709.05747/) (2017)
- [Convergence Rate of Distributed ADMM over Networks](https://mlsanity.com/paper/1601.00194/) (2016)
- [Communication-Efficient Online Federated Learning Framework for Nonlinear Regression](https://mlsanity.com/paper/2110.06556/) (2021)
- [Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints](https://mlsanity.com/paper/2101.08519/) (2021)
- [Privacy-Preserving Average Consensus via State Decomposition](https://mlsanity.com/paper/1902.09576/) (2019)

## Abstract (arXiv)

This paper proposes a proximal variant of the alternating direction method of multipliers (ADMM) for distributed optimization. Although the current versions of ADMM algorithm provide promising numerical results in producing solutions that are close to optimal for many convex and non-convex optimization problems, it remains unclear if they can converge to a stationary point for weakly convex and locally non-smooth functions. Through our analysis using the Moreau envelope function, we demonstrate that MADM can indeed converge to a stationary point under mild conditions. Our analysis also includes computing the bounds on the amount of change in the dual variable update step by relating the gradient of the Moreau envelope function to the proximal function. Furthermore, the results of our numerical experiments indicate that our method is faster and more robust than widely-used approaches.