Quasi-projection inequality for non-KL f-divergences

Determine whether there exist f-divergences other than the Kullback–Leibler divergence for which a relaxed projection inequality with a multiplicative constant holds: for some c in (0,1), establish whether D_f(μ,ν_θ) ≥ c D_f(μ,\barν_θ) for all admissible moment-matching measures μ, moment-matching points \barν_θ in the associated f-exponential family, and comparison points ν_θ in that family.

Background

The paper proves that, among C² f-divergences, the Kullback–Leibler divergence is uniquely characterized by the sharp entropy-adapted projection inequality with multiplicative constant one. Specifically, for a finite moment-constrained setting, the moment-matching element of the associated f-exponential family is the best f-divergence approximation uniformly over the relevant moment set only for KL, up to the usual equivalence of generators.

The conclusion asks whether this rigidity persists when the inequality is weakened by allowing a constant factor c between zero and one. Such a quasi-projection property would still be useful because it could transfer approximation and convergence estimates from the best element of an f-exponential family to the moment-matching reconstruction, including in applications such as moment-based distributional reinforcement learning.