Exponential stability of finite-$N$ consensus-based optimization

TL;DR

The work analyzes finite-N Consensus-Based Optimization (CBO) dynamics, establishing both almost-sure and mean-square exponential stability for deterministic and stochastic versions without relying on mean-field limits. By projecting onto the orthogonal complement of the consensus manifold, the authors derive explicit rates: in the stochastic case the projected error decays at rate $\lambda+\tfrac{\sigma^2}{2}$ a.s. and at rate $2\lambda-\sigma^2$ in mean square (when $2\lambda>\sigma^2$), with discretization stability characterized by $\delta_{EM}$ for Euler-type schemes. The results extend to Euler–Maruyama discretizations, providing concrete step-size constraints under which finite-N CBO retains exponential convergence properties. Numerical experiments on 2D-Rastrigin and other benchmarks corroborate the theory, showing robust consensus formation and highlighting the impact of diffusion anisotropy, particle count, and problem dimensionality on convergence and exploration. The findings offer rigorous finite-N guarantees for CBO, bridging the gap between mean-field analyses and practical, finite-population applications.

Abstract

We study the finite-agent behavior of Consensus-Based Optimization (CBO), a recent metaheuristic for the global minimization of a function, that combines drift toward a consensus estimate with stochastic exploration. While previous analyses focus on asymptotic mean-field limits, we investigate the stability properties of CBO for finite population size \( N \). Following a hierarchical approach, we first analyze a deterministic formulation of the algorithm and then extend our results to the fully stochastic setting governed by a system of stochastic differential equations. Our analysis reveals that essential stability properties, including almost sure and mean square exponential convergence, persist in both regimes and provides sharp quantitative estimates on the rates of convergence.

Figures (6)

• Figure 1: Evolution of $N = 100$ particles on the 2D-Rastrigin function for the purely deterministic system (\ref{['eq:evolution_determinstic_system']}), the stochastic system (\ref{['eq:evolution_stochastic_system']}) with $\sigma = 1$ (satisfying the stability condition $2 \lambda > \sigma^2$), and the stochastic system (\ref{['eq:evolution_stochastic_system']}) with $\sigma = \sqrt{2.1}$ (violating the stability condition $2 \lambda > \sigma^2$). For each case, we display four snapshots over time: at initialization ($\text{time} = 0$), after 15 time steps ($\text{time} = 0.75$), after 40 time steps ($\text{time} = 2$), and at the end of the simulation ($\text{time} = 4.95$). All plots are rendered from a top-down perspective; the global minimum is located at $\mathbf{X}^{\star} = (0, 0)$.
• Figure 2: Monte-Carlo approximation over 1000 runs showing the evolution of the dynamics orthogonal to the consensus manifold $\mathcal{G}^{\star}$ for the purely deterministic system (\ref{['eq:evolution_determinstic_system']}), the stochastic system (\ref{['eq:evolution_stochastic_system']}) with $\sigma = 1$ (satisfying the stability condition $2 \lambda > \sigma^2$), and the stochastic system (\ref{['eq:evolution_stochastic_system']}) with $\sigma = \sqrt{2.1}$ (violating the stability condition). Dotted lines indicate the theoretical convergence rates from Theorem \ref{['thm:stochastic_system_exponential_stability']}.
• Figure 3: Monte-Carlo approximation over 1000 runs showing the evolution of the isotropic dynamics orthogonal to the consensus manifold $\mathcal{G}^{\star}$ for the purely deterministic system (\ref{['eq:evolution_determinstic_system']}), the isotropic stochastic system with $\sigma = \sqrt{0.5}$ (satisfying the isotropic stability condition $2 \lambda > D\sigma^2$), and the isotropic stochastic system with $\sigma = 1$ (violating the isotropic stability condition $2 \lambda > D\sigma^2$). Dotted lines indicate the theoretical convergence rates from fornasier_CBO_converge_globally.
• Figure 4: Monte-Carlo approximation over 1000 runs for varying $\alpha$ showing the evolution of the dynamics orthogonal to the consensus manifold $\mathcal{G}^{\star}$ for the (a) purely deterministic system (\ref{['eq:evolution_determinstic_system']}), the (b) stochastic system (\ref{['eq:evolution_stochastic_system']}) with $\sigma = 1$ (satisfying the stability condition $2 \lambda > \sigma^2$), and the (c) stochastic system (\ref{['eq:evolution_stochastic_system']}) with $\sigma = \sqrt{2.1}$ (violating the stability condition).
• Figure 5: Monte-Carlo approximation over 1000 runs for varying number of particles $N$ showing the evolution of the dynamics orthogonal to the consensus manifold $\mathcal{G}^{\star}$ for the (a) purely deterministic system (\ref{['eq:evolution_determinstic_system']}), the (b) stochastic system (\ref{['eq:evolution_stochastic_system']}) with $\sigma = 1$ (satisfying the stability condition $2 \lambda > \sigma^2$), and the (c) stochastic system (\ref{['eq:evolution_stochastic_system']}) with $\sigma = \sqrt{2.1}$ (violating the stability condition).

Theorems & Definitions (15)

• Definition 2.1: Equilibrium Point ode_lyapunov_stability
• Definition 2.2: Stability for ODE ode_lyapunov_stability
• Definition 2.3: Exponential Stability for ODE ode_lyapunov_stability
• Definition 2.4: Lyapunov functions for ODE ode_lyapunov_stability
• Theorem 2.5: Stability for ODEs ode_lyapunov_stability
• Theorem 2.6: Exponential Stability for ODE exponential_stability_ODE_2
• Lemma 2.7: Rank-One Update rank_one_update
• Lemma 2.8: Eigenvalues of $\hat{L}$
• proof
• Theorem 2.9: Global Exponential Stability of the Deterministic System