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A First-Order Algorithm for Decentralised Min-Max Problems

Yura Malitsky, Matthew K. Tam

TL;DR

The paper addresses decentralized min-max problems with convex-concave structure by introducing a first-order algorithm that blends PG-EXTRA with the forward reflected backward method. The core idea is the primal-dual twice reflected (PDTR) framework, which can be interpreted as an extension of PG-EXTRA to min-max settings or as a distributed FoRB method, enabling non-decaying stepsizes and a single communication round per iteration. The authors develop product-space reformulations to handle distributed monotone inclusions and derive a concrete decentralized min-max algorithm that converges to saddle points under standard spectral and Lipschitz assumptions. This work advances distributed saddle-point computation by providing a simple, convergent method with solid theoretical guarantees and a clear connection to established primal-dual splitting techniques, with potential broad applicability in networked optimization tasks.

Abstract

In this work, we consider a connected network of finitely many agents working cooperatively to solve a min-max problem with convex-concave structure. We propose a decentralised first-order algorithm which can be viewed as a non-trivial combination of two algorithms: PG-EXTRA for decentralised minimisation problems and the forward reflected backward method for (non-distributed) min-max problems. In each iteration of our algorithm, each agent computes the gradient of the smooth component of its local objective function as well as the proximal operator of its nonsmooth component, following by a round of communication with its neighbours. Our analysis shows that the sequence generated by the method converges under standard assumptions with non-decaying stepsize.

A First-Order Algorithm for Decentralised Min-Max Problems

TL;DR

The paper addresses decentralized min-max problems with convex-concave structure by introducing a first-order algorithm that blends PG-EXTRA with the forward reflected backward method. The core idea is the primal-dual twice reflected (PDTR) framework, which can be interpreted as an extension of PG-EXTRA to min-max settings or as a distributed FoRB method, enabling non-decaying stepsizes and a single communication round per iteration. The authors develop product-space reformulations to handle distributed monotone inclusions and derive a concrete decentralized min-max algorithm that converges to saddle points under standard spectral and Lipschitz assumptions. This work advances distributed saddle-point computation by providing a simple, convergent method with solid theoretical guarantees and a clear connection to established primal-dual splitting techniques, with potential broad applicability in networked optimization tasks.

Abstract

In this work, we consider a connected network of finitely many agents working cooperatively to solve a min-max problem with convex-concave structure. We propose a decentralised first-order algorithm which can be viewed as a non-trivial combination of two algorithms: PG-EXTRA for decentralised minimisation problems and the forward reflected backward method for (non-distributed) min-max problems. In each iteration of our algorithm, each agent computes the gradient of the smooth component of its local objective function as well as the proximal operator of its nonsmooth component, following by a round of communication with its neighbours. Our analysis shows that the sequence generated by the method converges under standard assumptions with non-decaying stepsize.
Paper Structure (12 sections, 6 theorems, 72 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 12 sections, 6 theorems, 72 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Background and Literature Review
  4. Contributions and Structure
  5. Notation and Definitions
  6. PG-EXTRA as a primal-dual algorithm
  7. A New Primal-Dual Algorithm
  8. An Algorithm for decentralised monotone inclusions
  9. Product Space Reformulations
  10. Derivation of the Algorithm
  11. An Algorithm for decentralised min-max problems
  12. Conclusions and Outlook

Key Result

Lemma 3.2

Suppose $\tau,\sigma>0$ satisfy $\tau\sigma\|K\|^2<1$. Then the following assertions hold.

Figures (1)

  • Figure 1: (Left) A connected network of agents for the minimisation variable $\mathbf{x}$, and (right) a different connected network of agents for the maximisation variable $\mathbf{y}$. Dashed lines depict connections between pairs $(x_i,y_i)$.

Theorems & Definitions (20)

  • Definition 2.1: Mixing matrix
  • Remark 2.2
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Proposition 4.1
  • proof
  • ...and 10 more