Extend convergence analysis to general Exponential Dispersion Model (EDM) kernels

Extend the deterministic convergence analysis of the gradient-free optimization algorithm that updates distributions via the generalized Bayesian rule and reprojection (Algorithm 1) from the case where π_{θ,γ} is a Gaussian kernel to the general class of Exponential Dispersion Model (EDM) kernels. Specifically, determine conditions under which the Laplace functionals h_n(θ) = −log ∫ exp(−l(x)) π_{θ,γ_n}(x) dx epi-converge to l and establish stability and convergence of the associated time-inhomogeneous gradient descent recursion θ_{n+1} = θ_n − γ_n ∇h_n(θ_n) when π_{θ,γ} belongs to an EDM family.

Background

The paper proves convergence of a gradient-free optimization algorithm by interpreting its reprojection step as an inhomogeneous gradient descent on smoothed approximations of the objective. These results are established for Gaussian mollifiers, which are standard approximations to the identity. Earlier sections motivate a broader class of instrumental distributions using regular exponential dispersion models (EDMs), noting their concentration properties and mirror-descent interpretation.

In the convergence section, the authors explicitly restrict the analysis to the Gaussian case and state that extending the theory to the general EDM setting remains to be completed. This extension would unify the algorithmic framework and theoretical guarantees across a wider family of kernels frequently used in statistics, such as Gamma and Wishart, beyond Gaussian approximations.