# f-Divergence¶

The f-divergence is defined as1

$\operatorname{D}_f = \int f\left(\frac{p}{q}\right) q\mathrm d\mu,$

where $$p$$ and $$q$$ are two densities and $$\mu$$ is a reference distribution.

Requirements on the generating function

The generating function $$f$$ is required to

• be convex, and
• $$f(1) =0$$.

For $$f(x) = x \log x$$ with $$x=p/q$$, f-divergence is reduced to the KL divergence

\begin{align} &\int f\left(\frac{p}{q}\right) q\mathrm d\mu \\ =& \int \frac{p}{q} \log \left( \frac{p}{q} \right) \mathrm d\mu \\ =& \int p \log \left( \frac{p}{q} \right) \mathrm d\mu. \end{align}

For more special cases of f-divergence, please refer to wikipedia1. Nowozin 2016 also provides a concise review of f-divergence2.

1. Contributors to Wikimedia projects. F-divergence. In: Wikipedia [Internet]. 17 Jul 2021 [cited 4 Sep 2021]. Available: https://en.wikipedia.org/wiki/F-divergence

2. Nowozin S, Cseke B, Tomioka R. f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization. arXiv [stat.ML]. 2016. Available: http://arxiv.org/abs/1606.00709

Contributors: LM