Convergence of the Gauss-Bregman inductive center beyond separable generators

Prove the convergence of the Gauss-Bregman (A, m_{∇F}) inductive center for arbitrary Legendre-type multivariate generators F that are not necessarily separable. Specifically, for any finite weighted set of natural parameters {θ1, …, θn} with weights w in the open simplex, establish that the double sequence initialized at θ0 = ∑_{i=1}^n w_i θ_i and θ0* = (∇F)^{-1}(∑_{i=1}^n w_i ∇F(θ_i), and updated by θ_{t+1} = (θ_t + θ_t*)/2 and θ_{t+1}* = (∇F)^{-1}((∇F(θ_t)+∇F(θ_t*))/2), converges to a common limit θ_GB for all inputs without assuming separability of F.

Background

The paper introduces the Gauss-Bregman inductive center as the limit of a double sequence that alternates an arithmetic mean in the natural parameter space and a quasi-arithmetic mean induced by the gradient ∇F of a Legendre-type convex function, generalizing Nakamura’s arithmetic–harmonic inductive mean to arbitrary (A, m_{∇F}) means.

Convergence of this inductive scheme is proven under a separability assumption on the generator F (i.e., when F can be written as a sum of univariate convex functions). Extending the convergence proof to general, non-separable multivariate F would establish broad applicability of the Gauss-Bregman center as a fast proxy for Jeffreys centroids across exponential families.