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

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