Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Consensus-Based Optimization for Saddle Point Problems

Hui Huang, Jinniao Qiu, Konstantin Riedl

TL;DR

Consensus-Based Optimization for Saddle Point Problems (CBO-SP) targets global Nash equilibria in nonconvex-nonconcave minimax problems using a derivative-free, population-based method. It couples two interacting particle populations to perform minimization in $x$ and maximization in $y$, and leverages a mean-field limit to obtain rigorous convergence guarantees. The authors establish well-posedness for both the particle system and the mean-field dynamics and prove exponential convergence to saddle points under regularity and well-preparedness assumptions, supported by a detailed proof framework and a Laplace-principle-based argument. Numerical experiments on 1D saddle-point benchmarks and a strongly monotone quadratic game demonstrate robust performance, highlighting CBO-SP’s potential for high-dimensional, derivative-free minimax optimization.

Abstract

In this paper, we propose consensus-based optimization for saddle point problems (CBO-SP), a novel multi-particle metaheuristic derivative-free optimization method capable of provably finding global Nash equilibria. Following the idea of swarm intelligence, the method employs a group of interacting particles, which perform a minimization over one variable and a maximization over the other. This paradigm permits a passage to the mean-field limit, which makes the method amenable to theoretical analysis and allows to obtain rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, which most notably include nonconvex-nonconcave objectives.

Consensus-Based Optimization for Saddle Point Problems

TL;DR

Consensus-Based Optimization for Saddle Point Problems (CBO-SP) targets global Nash equilibria in nonconvex-nonconcave minimax problems using a derivative-free, population-based method. It couples two interacting particle populations to perform minimization in and maximization in , and leverages a mean-field limit to obtain rigorous convergence guarantees. The authors establish well-posedness for both the particle system and the mean-field dynamics and prove exponential convergence to saddle points under regularity and well-preparedness assumptions, supported by a detailed proof framework and a Laplace-principle-based argument. Numerical experiments on 1D saddle-point benchmarks and a strongly monotone quadratic game demonstrate robust performance, highlighting CBO-SP’s potential for high-dimensional, derivative-free minimax optimization.

Abstract

In this paper, we propose consensus-based optimization for saddle point problems (CBO-SP), a novel multi-particle metaheuristic derivative-free optimization method capable of provably finding global Nash equilibria. Following the idea of swarm intelligence, the method employs a group of interacting particles, which perform a minimization over one variable and a maximization over the other. This paradigm permits a passage to the mean-field limit, which makes the method amenable to theoretical analysis and allows to obtain rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, which most notably include nonconvex-nonconcave objectives.
Paper Structure (20 sections, 14 theorems, 102 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 20 sections, 14 theorems, 102 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Organization
  4. Well-Posedness of CBO-SP and its Mean-Field Dynamics
  5. Well-Posedness of the Interacting Particle System
  6. Well-Posedness of the Mean-Field Dynamics
  7. Convergence to Saddle Points
  8. Proof Details for Section \ref{['sec:convergence']}
  9. Time-Evolution of the Variances $\mathrm{Var}^X$ and $\mathrm{Var}^Y$
  10. Time-Evolution of the Functionals ${\cal M}^X$ and ${\cal M}^Y$ from \ref{['eq:CM']}
  11. Time-Evolution of the Functionals ${\cal M}^X_*$ and ${\cal M}^Y_*$ from \ref{['eq:CM_star']}
  12. Proof of Theorem \ref{['theorem:convergece']}
  13. Implementation of CBO-SP and Numerical Experiments
  14. Numerical Algorithm and Implementation
  15. Illustrative Numerical Experiments for CBO-SP
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let ${\cal E}\in{\cal C}(\mathbb{R}^{d_1+d_2})$ be locally Lipschitz continuous. Then, for $N_1 , N_2 \in\mathbb{N}$ fixed, the system of SDEs eq:saddle_point_dynamics_micro admits a unique strong solution $\left(\mathbf{Z}_t\right)_{t\geq0}$ for any initial condition $\mathbf{Z}_0$ satisfying $\mat

Figures (1)

  • Figure 1: Illustration of the dynamics of CBO-SP when searching the global Nash equilibrium of four different saddle point functions plotted in (a)--(d) in $d=1$, where $R(x) = \sum_{k=1}^d x_k^2 + \frac{5}{2} (1-\cos(2\pi x_k))$ is the Rastrigin function. Each column visualizes the positions of the $N=20$ particles when running CBO-SP with parameters $\alpha=\beta=10^{15}$, $\lambda_1=\lambda_2=1$, $\sigma_1=\sigma_2=\sqrt{0.1}$ and time step size $\Delta t= 0.1$ at three different points in time ($t=0$, $t=2$ and $t=T=4$). The particles are sampled initially from $\rho_0\sim{\cal N}(2,4)\times{\cal N}(2,4)$.

Theorems & Definitions (31)

  • Definition 1.1
  • Remark 1.2
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['theorem:well-posedness_interacting particle dynamcis']}
  • Definition 2.3: Assumptions
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • ...and 21 more