Investigate the choice of the transformation g in the Bayesian update

Investigate and characterize the selection of the nonnegative decreasing transformation g: R → [0, ∞) used in the generalized Bayesian update \tilde{π}_{n+1}(x) ∝ g(l(x)) π_n(x) within the gradient-free optimization algorithm (Algorithm 13). Determine how the choice of g impacts convergence guarantees and algorithmic performance, and identify criteria or classes of g for which the theoretical analysis applies.

Background

The algorithm typically uses the exponential transformation exp(−l(x)) in the Bayesian update, which aligns with the established convergence analysis via epi-convergence and inhomogeneous gradient descent. The authors propose a generalization where any nonnegative decreasing transformation g(l(x)) can be employed, and they show that the mirror-descent interpretation still holds under this generalization.

However, the paper does not analyze which transformations g are most appropriate or under what conditions this generalization preserves the theoretical guarantees. The authors explicitly state that investigating the choice of g is deferred, identifying a concrete open direction to formalize criteria for g and assess practical trade-offs.