<s>
In	O
mathematics	O
,	O
specifically	O
statistics	O
and	O
information	O
geometry	O
,	O
a	O
Bregman	B-Algorithm
divergence	I-Algorithm
or	O
Bregman	B-Algorithm
distance	I-Algorithm
is	O
a	O
measure	O
of	O
difference	O
between	O
two	O
points	O
,	O
defined	O
in	O
terms	O
of	O
a	O
strictly	O
convex	O
function	O
;	O
they	O
form	O
an	O
important	O
class	O
of	O
divergences	O
.	O
</s>
<s>
When	O
the	O
points	O
are	O
interpreted	O
as	O
probability	O
distributions	O
–	O
notably	O
as	O
either	O
values	O
of	O
the	O
parameter	O
of	O
a	O
parametric	B-General_Concept
model	I-General_Concept
or	O
as	O
a	O
data	O
set	O
of	O
observed	O
values	O
–	O
the	O
resulting	O
distance	O
is	O
a	O
statistical	O
distance	O
.	O
</s>
<s>
The	O
most	O
basic	O
Bregman	B-Algorithm
divergence	I-Algorithm
is	O
the	O
squared	O
Euclidean	O
distance	O
.	O
</s>
<s>
Bregman	B-Algorithm
divergences	I-Algorithm
are	O
similar	O
to	O
metrics	O
,	O
but	O
satisfy	O
neither	O
the	O
triangle	O
inequality	O
(	O
ever	O
)	O
nor	O
symmetry	O
(	O
in	O
general	O
)	O
.	O
</s>
<s>
However	O
,	O
they	O
satisfy	O
a	O
generalization	O
of	O
the	O
Pythagorean	O
theorem	O
,	O
and	O
in	O
information	O
geometry	O
the	O
corresponding	O
statistical	O
manifold	O
is	O
interpreted	O
as	O
a	O
(	O
dually	O
)	O
flat	B-Algorithm
manifold	I-Algorithm
.	O
</s>
<s>
This	O
allows	O
many	O
techniques	O
of	O
optimization	O
theory	O
to	O
be	O
generalized	O
to	O
Bregman	B-Algorithm
divergences	I-Algorithm
,	O
geometrically	O
as	O
generalizations	O
of	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
Bregman	B-Algorithm
divergences	I-Algorithm
are	O
named	O
after	O
Russian	O
mathematician	O
Lev	O
M	O
.	O
Bregman	O
,	O
who	O
introduced	O
the	O
concept	O
in	O
1967	O
.	O
</s>
<s>
The	O
Bregman	B-Algorithm
distance	I-Algorithm
associated	O
with	O
F	O
for	O
points	O
is	O
the	O
difference	O
between	O
the	O
value	O
of	O
F	O
at	O
point	O
p	O
and	O
the	O
value	O
of	O
the	O
first-order	O
Taylor	O
expansion	O
of	O
F	O
around	O
point	O
q	O
evaluated	O
at	O
point	O
p	O
:	O
</s>
<s>
Linearity	O
:	O
If	O
we	O
think	O
of	O
the	O
Bregman	B-Algorithm
distance	I-Algorithm
as	O
an	O
operator	O
on	O
the	O
function	O
F	O
,	O
then	O
it	O
is	O
linear	O
with	O
respect	O
to	O
non-negative	O
coefficients	O
.	O
</s>
<s>
Mean	O
as	O
minimizer	O
:	O
A	O
key	O
result	O
about	O
Bregman	B-Algorithm
divergences	I-Algorithm
is	O
that	O
,	O
given	O
a	O
random	O
vector	O
,	O
the	O
mean	O
vector	O
minimizes	O
the	O
expected	O
Bregman	B-Algorithm
divergence	I-Algorithm
from	O
the	O
random	O
vector	O
.	O
</s>
<s>
Lack	O
of	O
triangle	O
inequality	O
:	O
Since	O
the	O
Bregman	B-Algorithm
divergence	I-Algorithm
is	O
essentially	O
a	O
generalization	O
of	O
squared	O
Euclidean	O
distance	O
,	O
there	O
is	O
no	O
triangle	O
inequality	O
.	O
</s>
<s>
The	O
only	O
symmetric	O
Bregman	B-Algorithm
divergences	I-Algorithm
on	O
are	O
squared	O
generalized	O
Euclidean	O
distances	O
(	O
Mahalanobis	O
distance	O
)	O
,	O
that	O
is	O
,	O
for	O
some	O
positive	O
definite	O
.	O
</s>
<s>
The	O
only	O
divergence	O
on	O
that	O
is	O
both	O
a	O
Bregman	B-Algorithm
divergence	I-Algorithm
and	O
an	O
f-divergence	O
is	O
the	O
Kullback	O
–	O
Leibler	O
divergence	O
.	O
</s>
<s>
If	O
,	O
then	O
any	O
Bregman	B-Algorithm
divergence	I-Algorithm
on	O
that	O
satisfies	O
the	O
data	B-General_Concept
processing	I-General_Concept
inequality	I-General_Concept
must	O
be	O
the	O
Kullback	O
–	O
Leibler	O
divergence	O
.	O
</s>
<s>
Given	O
a	O
Bregman	B-Algorithm
divergence	I-Algorithm
,	O
its	O
"	O
opposite	O
"	O
,	O
defined	O
by	O
,	O
is	O
generally	O
not	O
a	O
Bregman	B-Algorithm
divergence	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Kullback-Leiber	O
divergence	O
is	O
both	O
a	O
Bregman	B-Algorithm
divergence	I-Algorithm
and	O
an	O
f-divergence	O
.	O
</s>
<s>
Its	O
reverse	O
is	O
also	O
an	O
f-divergence	O
,	O
but	O
by	O
the	O
above	O
characterization	O
,	O
the	O
reverse	O
KL	O
divergence	O
cannot	O
be	O
a	O
Bregman	B-Algorithm
divergence	I-Algorithm
.	O
</s>
<s>
The	O
Itakura	B-Algorithm
–	I-Algorithm
Saito	I-Algorithm
distance	I-Algorithm
,	O
</s>
<s>
This	O
implies	O
that	O
natural	O
dual	O
concepts	O
in	O
computational	O
geometry	O
like	O
Voronoi	B-Architecture
diagrams	I-Architecture
and	O
Delaunay	B-Algorithm
triangulations	I-Algorithm
retain	O
their	O
meaning	O
in	O
distance	O
spaces	O
defined	O
by	O
an	O
arbitrary	O
Bregman	B-Algorithm
divergence	I-Algorithm
.	O
</s>
<s>
Bregman	B-Algorithm
divergences	I-Algorithm
can	O
be	O
interpreted	O
as	O
limit	O
cases	O
of	O
skewed	O
Jensen	O
divergences	O
(	O
see	O
Nielsen	O
and	O
Boltz	O
,	O
2011	O
)	O
.	O
</s>
<s>
Jensen	O
divergences	O
can	O
be	O
generalized	O
using	O
comparative	O
convexity	O
,	O
and	O
limit	O
cases	O
of	O
these	O
skewed	O
Jensen	O
divergences	O
generalizations	O
yields	O
generalized	O
Bregman	B-Algorithm
divergence	I-Algorithm
(	O
see	O
Nielsen	O
and	O
Nock	O
,	O
2017	O
)	O
.	O
</s>
<s>
Bregman	B-Algorithm
divergences	I-Algorithm
can	O
also	O
be	O
defined	O
between	O
matrices	O
,	O
between	O
functions	O
,	O
and	O
between	O
measures	O
(	O
distributions	O
)	O
.	O
</s>
<s>
Bregman	B-Algorithm
divergences	I-Algorithm
between	O
matrices	O
include	O
the	O
Stein	O
's	O
loss	O
and	O
von	O
Neumann	O
entropy	O
.	O
</s>
<s>
Bregman	B-Algorithm
divergences	I-Algorithm
between	O
functions	O
include	O
total	O
squared	O
error	O
,	O
relative	O
entropy	O
,	O
and	O
squared	O
bias	O
;	O
see	O
the	O
references	O
by	O
Frigyik	O
et	O
al	O
.	O
</s>
<s>
Similarly	O
Bregman	B-Algorithm
divergences	I-Algorithm
have	O
also	O
been	O
defined	O
over	O
sets	O
,	O
through	O
a	O
submodular	B-Algorithm
set	I-Algorithm
function	I-Algorithm
which	O
is	O
known	O
as	O
the	O
discrete	O
analog	O
of	O
a	O
convex	O
function	O
.	O
</s>
<s>
The	O
submodular	B-Algorithm
Bregman	B-Algorithm
divergences	I-Algorithm
subsume	O
a	O
number	O
of	O
discrete	O
distance	O
measures	O
,	O
like	O
the	O
Hamming	O
distance	O
,	O
precision	O
and	O
recall	O
,	O
mutual	O
information	O
and	O
some	O
other	O
set	O
based	O
distance	O
measures	O
(	O
see	O
Iyer	O
&	O
Bilmes	O
,	O
2012	O
)	O
for	O
more	O
details	O
and	O
properties	O
of	O
the	O
submodular	B-Algorithm
Bregman	O
.	O
)	O
</s>
<s>
For	O
a	O
list	O
of	O
common	O
matrix	O
Bregman	B-Algorithm
divergences	I-Algorithm
,	O
see	O
Table	O
15.1	O
in	O
.	O
</s>
<s>
In	O
machine	O
learning	O
,	O
Bregman	B-Algorithm
divergences	I-Algorithm
are	O
used	O
to	O
calculate	O
the	O
bi-tempered	O
logistic	O
loss	O
,	O
performing	O
better	O
than	O
the	O
softmax	B-Algorithm
function	I-Algorithm
with	O
noisy	O
datasets	O
.	O
</s>
<s>
Bregman	B-Algorithm
divergence	I-Algorithm
is	O
used	O
in	O
the	O
formulation	O
of	O
mirror	B-Algorithm
descent	I-Algorithm
,	O
which	O
includes	O
optimization	O
algorithms	O
used	O
in	O
machine	O
learning	O
such	O
as	O
gradient	B-Algorithm
descent	I-Algorithm
and	O
the	O
hedge	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
