<s>
Sequential	B-Algorithm
minimal	I-Algorithm
optimization	I-Algorithm
(	O
SMO	O
)	O
is	O
an	O
algorithm	O
for	O
solving	O
the	O
quadratic	B-Algorithm
programming	I-Algorithm
(	O
QP	O
)	O
problem	O
that	O
arises	O
during	O
the	O
training	O
of	O
support-vector	B-Algorithm
machines	I-Algorithm
(	O
SVM	B-Algorithm
)	O
.	O
</s>
<s>
SMO	O
is	O
widely	O
used	O
for	O
training	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
and	O
is	O
implemented	O
by	O
the	O
popular	O
LIBSVM	B-Language
tool	O
.	O
</s>
<s>
The	O
publication	O
of	O
the	O
SMO	O
algorithm	O
in	O
1998	O
has	O
generated	O
a	O
lot	O
of	O
excitement	O
in	O
the	O
SVM	B-Algorithm
community	O
,	O
as	O
previously	O
available	O
methods	O
for	O
SVM	B-Algorithm
training	O
were	O
much	O
more	O
complex	O
and	O
required	O
expensive	O
third-party	O
QP	O
solvers	O
.	O
</s>
<s>
Consider	O
a	O
binary	B-General_Concept
classification	I-General_Concept
problem	O
with	O
a	O
dataset	O
(	O
x1	O
,	O
y1	O
)	O
,	O
...	O
,	O
(	O
xn	O
,	O
yn	O
)	O
,	O
where	O
xi	O
is	O
an	O
input	O
vector	O
and	O
is	O
a	O
binary	O
label	O
corresponding	O
to	O
it	O
.	O
</s>
<s>
A	O
soft-margin	O
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
is	O
trained	O
by	O
solving	O
a	O
quadratic	B-Algorithm
programming	I-Algorithm
problem	O
,	O
which	O
is	O
expressed	O
in	O
the	O
dual	O
form	O
as	O
follows	O
:	O
</s>
<s>
where	O
C	O
is	O
an	O
SVM	B-Algorithm
hyperparameter	O
and	O
K(xi, xj )	O
is	O
the	O
kernel	O
function	O
,	O
both	O
supplied	O
by	O
the	O
user	O
;	O
and	O
the	O
variables	O
are	O
Lagrange	O
multipliers	O
.	O
</s>
<s>
The	O
first	O
approach	O
to	O
splitting	O
large	O
SVM	B-Algorithm
learning	O
problems	O
into	O
a	O
series	O
of	O
smaller	O
optimization	O
tasks	O
was	O
proposed	O
by	O
Bernhard	O
Boser	O
,	O
Isabelle	O
Guyon	O
,	O
Vladimir	O
Vapnik	O
.	O
</s>
<s>
In	O
1997	O
,	O
E	O
.	O
Osuna	O
,	O
R	O
.	O
Freund	O
,	O
and	O
F	O
.	O
Girosi	O
proved	O
a	O
theorem	O
which	O
suggests	O
a	O
whole	O
new	O
set	O
of	O
QP	O
algorithms	O
for	O
SVMs	B-Algorithm
.	O
</s>
<s>
The	O
SMO	O
algorithm	O
is	O
closely	O
related	O
to	O
a	O
family	O
of	O
optimization	O
algorithms	O
called	O
Bregman	B-Algorithm
methods	I-Algorithm
or	O
row-action	O
methods	O
.	O
</s>
