# (geodesic) smoothness of f-divergence with respect to the Wasserstein metric

We consider the f-divergence, which takes the form $$D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ.$$ For example, when $$f(t) = t \log t$$, we obtain the KL-divergence.

My question is that, as a function of $$P$$, under what regularity conditions on $$f$$ and $$Q$$ is the f-divergence $$D_f(P \| Q)$$ (geodesically) $$\mu$$-smooth with respect to the Wasserstein metric $$W_2$$?

For any function $$g(P)$$, the (geodesic) $$\mu$$-smoothness of $$g(P)$$ takes the form $$g(P') \leq g(P) + \langle\text{grad}\,g(P), \exp_P^{-1} (P') \rangle_P + \frac{\mu}{2}\cdot W_2^2(P, P')$$ or equivalently that the Hessian $$\text{Hess}(g)$$ is bounded from the above by $$\mu$$ for all $$P$$. Here, $$\text{grad}$$ and $$\text{Hess}$$ are the gradient and Hessian with respect to the $$W_2$$ metric, which is the geodesic distance in the space of probability distributions, and $$\exp_P^{-1} (P')$$ is the inverse of the exponential map.