reverse KL-divergence: Bregman or not?

I am having a little trouble getting my head around the two "directions" of the Kullback-Leibler divergence:

Definition (Kullback-Leibler divergence) For discrete probability distributions $$P$$ and $$Q$$ defined on the same probability space, $$\chi$$, the Kullback-Leibler divergence from $$Q$$ to $$P$$ is defined to be $$D_{KL}(P||Q) := \sum_{x \in \chi}P(x)\log\bigg(\frac{P(x)}{Q(x)}\bigg).$$

Note that The KL-divergence is not really a true distance measure, since it is does not satisfy the Triangle Inequality and $$D_{KL}(P||Q)$$ does not in general equal $$D_{KL}(Q||P)$$. Hence the need to distinguish the KL-divergence from its dual, the so-called "reverse KL-divergence".

Fact. Both the KL-divergence and the reverse KL-divergence are examples of f-divergences.

Fact. The KL-divergence is an example of a Bregman divergence.

Question: Is the reverse Kullback-Leibler divergence also a Bregman divergence?

This is not obvious to me. Also, I have read conflicting information, with Amari (2009) arguing that the KL-divergence and its dual are are unique divergences belonging to the $$f$$-divergence and Bregman divergence classes, and others (e.g., Wang et al. 2019) saying that the reverse KL-divergence is no longer a Bregman. Which is correct?

Any pointers would be greatly appreciated.

References:

• Wang, Q., Li, Y. and Xiong, J. (2019) Divergence-Augmented Policy Optimization link
• Amari, S. (2009) $$\alpha$$-Divergence Is Unique, Belonging to Both $$f$$-Divergence and Bregman Divergence Classes link

Define the KL convergence as in the Amari's paper linked by you: $$KL(x||y):=D_{KL}(x||y):=\sum(y_i-x_i+x_i\ln\frac{x_i}{y_i}).$$ Then $$KL(x||y)=F(x)-F(y)-\nabla F(y)\cdot(x-y)$$ if $$F(x):=\sum(x_i\ln x_i-x_i)$$. So, the KL-divergence is a Bregman one.
On the other hand, the dual divergence, defined by $$LK(x||y):=KL(y||x)=\sum(x_i-y_i+y_i\ln\frac{y_i}{x_i}),$$ is not a Bregman one. Indeed, if it were a Bregman one, then for some appropriate function $$G$$ we would have $$LK(x||y)=G(x)-G(y)-\nabla G(y)\cdot(x-y)$$ and hence $$\nabla_x(LK(x||y))=\nabla G(x)-\nabla G(y),$$ whereas in fact $$\nabla_x(LK(x||y))=(1-y_i/x_i),$$ which cannot be of the form of a difference $$\nabla G(x)-\nabla G(y)$$ -- because otherwise we would have $$(1-v/u)+(1-w/v)=(1-w/u)$$ for all positive real $$u,v,w$$.
(Amari's proof contains formula (52), which contains functions $$\psi_\alpha$$ and $$\psi_{-\alpha}$$, supposedly defined by formula (49). However, the expression for $$\psi_\alpha$$ in (49) is undefined for $$\alpha=-1$$, whereas it is both of the values $$\pm1$$ of $$\alpha$$ that are needed in (52) for $$LK$$.)
• "LK", cute ${}{}{}$ Mar 18 '21 at 14:24