There is a distance-generating function $\omega(x) : X \rightarrow R $ in the definition of Bregman distance, which is then defined as $V(x,y) = \omega(y) − \omega(x) − \langle\omega^\prime(x), y − x\rangle$. One possible example of $\omega(x)$ is $\frac12\|x\|_2^2$. But is there an example of $\omega(x)$ bounded on the whole space $\mathbb{R}^n$ (bounded both below and above)?
Asked
Active
Viewed 139 times
0
-
Not sure why you deleted this, but anyway: The behaviour of the smallest term $M_{1,k}$ in the sum $$S_k=\sum_{i=1}^kM_{i,k}$$ says little about the behaviour of $S_k$ (and note that $M_{1,k}$ behaves roughly like $e^{-\sqrt{k}}$, not like $e^{-\sqrt{k}/2}$)... But, one has $$S_{k+1}=\left(1-\frac1{2\sqrt{k+1}}\right)(S_k+1)$$ hence the change of variable $$S_k=2\sqrt{k}-c+R^{(c)}k$$ yields $$R^{(c)}{k+1}=\left(1-\frac1{2\sqrt{k+1}}\right)R^{(c)}_k+T^{(c)}_k$$ – Did Mar 06 '17 at 12:24
-
... with $$T^{(c)}_k=\frac{(c-1)(\sqrt{k+1}+\sqrt{k})-4\sqrt{k+1}+2}{2\sqrt{k+1}(\sqrt{k+1}+\sqrt{k})}$$ The choice $$c=3$$ yields $$T^{(3)}_k\ll\frac1{\sqrt{k}}$$ hence $$\lim R^{(3)}_k=0$$ that is, $$S_k=2\sqrt{k}-3+o(1)$$ – Did Mar 06 '17 at 12:24
-
@Did, sorry, I was afraid that someone was at the typing process, but did not know how to check it. The reason of deleting was that I've simulated it on my computer and implicate that this sum is divergent, so it is not of my interest anymore. Nonetheless, I recovered the question, and you can upload your approach now, I will certainly mark it. I've received the same behavior $\sim 2\sqrt{k}$ via simulations. Thank you for your work – Andrei Kulunchakov Mar 06 '17 at 12:29