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Extend exponential stability analysis to the isotropic diffusion CBO

Develop an exponential stability analysis for the finite-agent isotropic Consensus-Based Optimization dynamics in which each agent uses the diffusion term \(\sigma\|Z_t^n - \nu_f^\alpha(Z_t)\|_2\), establishing conditions and rates for almost sure and mean square exponential convergence analogous to the anisotropic case.

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Background

The paper’s stability results rely on anisotropic diffusion, which decouples dimensions and allows a graph-theoretic projection argument. In the isotropic variant, the diffusion couples all dimensions via the norm ∥Ztn−νfα(Zt)∥2\|Z_t^n - \nu_f^\alpha(Z_t)\|_2, preventing direct application of the same approach.

Mean-field analyses provide isotropic convergence rates under specific conditions, but a finite-N, finite-α\alpha exponential stability theory for the isotropic dynamics remains to be developed.

References

As a result, the approach used here for the anisotropic system cannot be directly applied, and extending the exponential stability analysis to the isotropic case is left for future work.

— Exponential stability of finite-$N$ consensus-based optimization (2510.19565 - Göttlich et al., 22 Oct 2025) in Section “Numerical Results”, Subsection “Convergence Behavior of 2D-Rastrigin” (isotropic variant)