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Decentralized Min-Max Optimization with Gradient Tracking

Runze You, Kun Huang, Shi Pu

TL;DR

This paper addresses distributed min-max optimization over networks with heterogeneous per-agent variables and a NC-SC objective. It introduces two gradient-tracking-based methods, DGTA and DSGTA, which do not require bounded data heterogeneity or identical constraint sets, and proves convergence rates that match centralized results up to graph-dependent constants. Specifically, DGTA achieves an iteration complexity of $\mathcal{O}(\kappa^2\varepsilon^{-2})$, while DSGTA attains a sample complexity of $\mathcal{O}(\kappa^3\varepsilon^{-4})$ under NC-SC objectives, with additional insights for various batch sizes. A novel Lyapunov function underpins the analysis, and numerical experiments on distributed Wasserstein robustness demonstrate superior practical performance over existing decentralized methods. The results highlight the potential for efficient, coordination-light distributed min-max optimization in diverse multi-agent settings.

Abstract

This paper presents a novel distributed formulation of the min-max optimization problem. Such a formulation enables enhanced flexibility among agents when optimizing their maximization variables. To address the problem, we propose two distributed gradient methods over networks, termed Distributed Gradient Tracking Ascent (DGTA) and Distributed Stochastic Gradient Tracking Ascent (DSGTA). We demonstrate that DGTA achieves an iteration complexity of $\mathcal{O}(κ^2\varepsilon^{-2})$, and DSGTA attains a sample complexity of $\mathcal{O}(κ^3\varepsilon^{-4})$ for nonconvex strongly concave (NC-SC) objective functions. Both results match those of their centralized counterparts up to constant factors related to the communication network. Numerical experiments further demonstrate the superior empirical performance of the proposed algorithms compared to existing methods.

Decentralized Min-Max Optimization with Gradient Tracking

TL;DR

This paper addresses distributed min-max optimization over networks with heterogeneous per-agent variables and a NC-SC objective. It introduces two gradient-tracking-based methods, DGTA and DSGTA, which do not require bounded data heterogeneity or identical constraint sets, and proves convergence rates that match centralized results up to graph-dependent constants. Specifically, DGTA achieves an iteration complexity of , while DSGTA attains a sample complexity of under NC-SC objectives, with additional insights for various batch sizes. A novel Lyapunov function underpins the analysis, and numerical experiments on distributed Wasserstein robustness demonstrate superior practical performance over existing decentralized methods. The results highlight the potential for efficient, coordination-light distributed min-max optimization in diverse multi-agent settings.

Abstract

This paper presents a novel distributed formulation of the min-max optimization problem. Such a formulation enables enhanced flexibility among agents when optimizing their maximization variables. To address the problem, we propose two distributed gradient methods over networks, termed Distributed Gradient Tracking Ascent (DGTA) and Distributed Stochastic Gradient Tracking Ascent (DSGTA). We demonstrate that DGTA achieves an iteration complexity of , and DSGTA attains a sample complexity of for nonconvex strongly concave (NC-SC) objective functions. Both results match those of their centralized counterparts up to constant factors related to the communication network. Numerical experiments further demonstrate the superior empirical performance of the proposed algorithms compared to existing methods.
Paper Structure (18 sections, 10 theorems, 73 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 18 sections, 10 theorems, 73 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Main Contribution
  4. Notation
  5. Organization
  6. Setup and Algorithms
  7. Assumptions
  8. Algorithms
  9. Preliminary Analysis
  10. Main Results
  11. Numerical Experiment
  12. Conclusion
  13. Proofs
  14. Proof of Lemma \ref{['lem:dsgda_descent']}
  15. Proof of Lemma \ref{['lem:dsgda_delta']}
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let Assumptions a.finite and as:set_y hold. We have that $\Phi_i(\cdot)= \max_{y\in\mathcal{Y}_i}f_i(\cdot, y)$ is $2\kappa L$-smooth with $\nabla \Phi_i(x)= \nabla_x f(x, \hat{y}_i(x))$, where $\hat{y}_i(x):= \arg\max_{y\in\mathcal{Y}_i}f_i(x, y)$. In addition, $\hat{y}_i(\cdot)$ is $\kappa$-Lipsch

Figures (1)

  • Figure 1: Performance comparison of DSGTA, DGTA, and GT/DA under the same $\eta_y=0.01$ and varying stepsizes $\eta_x$. The horizontal axis ("Number of Gradients" and "Number of Stochastic Gradients") shows the iteration progress, where each gradient or stochastic gradient evaluation with respect to either variable $x$ or $y$ increments the count by one.

Theorems & Definitions (25)

  • Remark 2.1
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 15 more