# Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists

arXiv: https://arxiv.org/abs/1912.00137 · HTML: https://mlsanity.com/paper/1912.00137/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2019.

## Built on by (most cited first, 16 of 16)

- [ProxSkip: Yes! Local Gradient Steps Provably Lead to Communication Acceleration! Finally!](https://mlsanity.com/paper/2202.09357/) (2022)
- [EF-BV: A Unified Theory of Error Feedback and Variance Reduction Mechanisms for Biased and Unbiased Compression in Distributed Optimization](https://mlsanity.com/paper/2205.04180/) (2022)
- [MURANA: A Generic Framework for Stochastic Variance-Reduced Optimization](https://mlsanity.com/paper/2106.03056/) (2021)
- [DADAO: Decoupled Accelerated Decentralized Asynchronous Optimization](https://mlsanity.com/paper/2208.00779/) (2022)
- [DISA: A Dual Inexact Splitting Algorithm for Distributed Convex Composite Optimization](https://mlsanity.com/paper/2209.01850/) (2022)
- [Decentralized Inexact Proximal Gradient Method With Network-Independent Stepsizes for Convex Composite Optimization](https://mlsanity.com/paper/2302.03238/) (2023)
- [On the Convergence of FedProx with Extrapolation and Inexact Prox](https://mlsanity.com/paper/2410.01410/) (2024)
- [Near-Linear Time Projection onto the ℓ1,∞ Ball; Application to Sparse Autoencoders](https://mlsanity.com/paper/2307.09836/) (2023)
- [Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers](https://mlsanity.com/paper/2503.19190/) (2025)
- [Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs](https://mlsanity.com/paper/2603.26940/) (2026)
- [Tikhonov Regularization of Circle-Valued Signals](https://mlsanity.com/paper/2108.02602/) (2021)
- [Sparse Index Tracking: Simultaneous Asset Selection and Capital Allocation via ℓ0-Constrained Portfolio](https://mlsanity.com/paper/2309.10152/) (2023)
- [A new Linear Time Bi-level ℓ1,∞ projection ; Application to the sparsification of auto-encoders neural networks](https://mlsanity.com/paper/2407.16293/) (2024)
- [Contractivity and linear convergence in bilinear saddle-point problems: An operator-theoretic approach](https://mlsanity.com/paper/2410.14592/) (2024)
- [Theoretical Foundations of Communication-Efficient, Robust, and Practical Distributed and Federated Optimization](https://mlsanity.com/paper/2608.06563/) (2026)
- [Lightweight Interpretable RGB-Guided Hyperspectral Super-Resolution under Real Cross-resolution Misalignment](https://mlsanity.com/paper/2609.01060/) (2026)

## Abstract (arXiv)

Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they consist of simple operations, handling the terms in the objective function separately. In this overview, we demystify a selection of recent proximal splitting algorithms: we present them within a unified framework, which consists in applying splitting methods for monotone inclusions in primal-dual product spaces, with well-chosen metrics. Along the way, we easily derive new variants of the algorithms and revisit existing convergence results, extending the parameter ranges in several cases. In particular, we emphasize that when the smooth term in the objective function is quadratic, e.g., for least-squares problems, convergence is guaranteed with larger values of the relaxation parameter than previously known. Such larger values are usually beneficial for the convergence speed in practice.