# On a Unified and Simplified Proof for the Ergodic Convergence Rates of PPM, PDHG and ADMM

arXiv: https://arxiv.org/abs/2305.02165 · HTML: https://mlsanity.com/paper/2305.02165/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 May 2023.

## Built on by (most cited first, 5 of 5)

- [PDHG-Unrolled Learning-to-Optimize Method for Large-Scale Linear Programming](https://mlsanity.com/paper/2406.01908/) (2024)
- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [Auto-conditioned primal-dual hybrid gradient method and alternating direction method of multipliers](https://mlsanity.com/paper/2410.01979/) (2024)
- [On the Relationship Between CoCoA and ADMM for Distributed Empirical Risk Minimization](https://mlsanity.com/paper/2502.00470/) (2025)
- [Exploring New Frontiers in Vertical Federated Learning: the Role of Saddle Point Reformulation](https://mlsanity.com/paper/2602.15996/) (2026)

## Abstract (arXiv)

We present a unified viewpoint of proximal point method (PPM), primal-dual hybrid gradient (PDHG) and alternating direction method of multipliers (ADMM) for solving convex-concave primal-dual problems. This viewpoint shows the equivalence of these three algorithms upto a norm change, and it leads to a four-line simple proof of their mathcal O(1/k) ergodic rates. The simple proof technique is not limited to these three algorithms, but can also be utilized to analyze related algorithms, such as gradient descent, linearized PDHG, inexact algorithms, just to name a few.