<s>
Proximal	B-Algorithm
gradient	I-Algorithm
(	O
forward	O
backward	O
splitting	O
)	O
methods	O
for	O
learning	O
is	O
an	O
area	O
of	O
research	O
in	O
optimization	O
and	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
which	O
studies	O
algorithms	O
for	O
a	O
general	O
class	O
of	O
convex	O
regularization	O
problems	O
where	O
the	O
regularization	O
penalty	O
may	O
not	O
be	O
differentiable	O
.	O
</s>
<s>
Proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
offer	O
a	O
general	O
framework	O
for	O
solving	O
regularization	O
problems	O
from	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
with	O
penalties	O
that	O
are	O
tailored	O
to	O
a	O
specific	O
problem	O
application	O
.	O
</s>
<s>
Such	O
customized	O
penalties	O
can	O
help	O
to	O
induce	O
certain	O
structure	O
in	O
problem	O
solutions	O
,	O
such	O
as	O
sparsity	O
(	O
in	O
the	O
case	O
of	O
lasso	B-Algorithm
)	O
or	O
group	O
structure	O
(	O
in	O
the	O
case	O
of	O
group	O
lasso	B-Algorithm
)	O
.	O
</s>
<s>
The	O
proximal	O
operator	O
can	O
be	O
seen	O
as	O
a	O
generalization	O
of	O
a	O
projection	B-Algorithm
.	O
</s>
<s>
One	O
important	O
technique	O
related	O
to	O
proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
is	O
the	O
Moreau	O
decomposition	O
,	O
which	O
decomposes	O
the	O
identity	O
operator	O
as	O
the	O
sum	O
of	O
two	O
proximity	O
operators	O
.	O
</s>
<s>
The	O
Moreau	O
decomposition	O
can	O
be	O
seen	O
to	O
be	O
a	O
generalization	O
of	O
the	O
usual	O
orthogonal	O
decomposition	O
of	O
a	O
vector	O
space	O
,	O
analogous	O
with	O
the	O
fact	O
that	O
proximity	O
operators	O
are	O
generalizations	O
of	O
projections	B-Algorithm
.	O
</s>
<s>
This	O
is	O
the	O
case	O
for	O
group	O
lasso	B-Algorithm
.	O
</s>
<s>
Consider	O
the	O
regularized	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
problem	O
with	O
square	O
loss	O
and	O
with	O
the	O
norm	O
as	O
the	O
regularization	O
penalty	O
:	O
</s>
<s>
where	O
The	O
regularization	O
problem	O
is	O
sometimes	O
referred	O
to	O
as	O
lasso	B-Algorithm
(	O
least	B-Algorithm
absolute	I-Algorithm
shrinkage	I-Algorithm
and	I-Algorithm
selection	I-Algorithm
operator	I-Algorithm
)	O
.	O
</s>
<s>
Sparse	O
solutions	O
are	O
of	O
particular	O
interest	O
in	O
learning	B-General_Concept
theory	I-General_Concept
for	O
interpretability	O
of	O
results	O
:	O
a	O
sparse	O
solution	O
can	O
identify	O
a	O
small	O
number	O
of	O
important	O
factors	O
.	O
</s>
<s>
which	O
is	O
known	O
as	O
the	O
soft	B-Algorithm
thresholding	I-Algorithm
operator	O
.	O
</s>
<s>
To	O
finally	O
solve	O
the	O
lasso	B-Algorithm
problem	O
we	O
consider	O
the	O
fixed	O
point	O
equation	O
shown	O
earlier	O
:	O
</s>
<s>
This	O
fixed	O
point	O
method	O
has	O
decoupled	O
the	O
effect	O
of	O
the	O
two	O
different	O
convex	O
functions	O
which	O
comprise	O
the	O
objective	O
function	O
into	O
a	O
gradient	O
descent	O
step	O
(	O
)	O
and	O
a	O
soft	B-Algorithm
thresholding	I-Algorithm
step	O
(	O
via	O
)	O
.	O
</s>
<s>
There	O
have	O
been	O
numerous	O
developments	O
within	O
the	O
past	O
decade	O
in	O
convex	O
optimization	O
techniques	O
which	O
have	O
influenced	O
the	O
application	O
of	O
proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
in	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
.	O
</s>
<s>
The	O
problem	O
of	O
lasso	B-Algorithm
(	O
)	O
regularization	O
involves	O
the	O
penalty	O
term	O
,	O
which	O
is	O
not	O
strictly	O
convex	O
.	O
</s>
<s>
Proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
provide	O
a	O
general	O
framework	O
which	O
is	O
applicable	O
to	O
a	O
wide	O
variety	O
of	O
problems	O
in	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
.	O
</s>
<s>
Group	O
lasso	B-Algorithm
is	O
a	O
generalization	O
of	O
the	O
lasso	B-Algorithm
method	I-Algorithm
when	O
features	O
are	O
grouped	O
into	O
disjoint	O
blocks	O
.	O
</s>
<s>
Where	O
the	O
lasso	B-Algorithm
penalty	O
has	O
a	O
proximity	O
operator	O
which	O
is	O
soft	B-Algorithm
thresholding	I-Algorithm
on	O
each	O
individual	O
component	O
,	O
the	O
proximity	O
operator	O
for	O
the	O
group	O
lasso	B-Algorithm
is	O
soft	B-Algorithm
thresholding	I-Algorithm
on	O
each	O
group	O
.	O
</s>
<s>
In	O
contrast	O
to	O
lasso	B-Algorithm
,	O
the	O
derivation	O
of	O
the	O
proximity	O
operator	O
for	O
group	O
lasso	B-Algorithm
relies	O
on	O
the	O
Moreau	O
decomposition	O
.	O
</s>
<s>
Here	O
the	O
proximity	O
operator	O
of	O
the	O
conjugate	O
of	O
the	O
group	O
lasso	B-Algorithm
penalty	O
becomes	O
a	O
projection	B-Algorithm
onto	O
the	O
ball	O
of	O
a	O
dual	O
norm	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
group	O
lasso	B-Algorithm
problem	O
,	O
where	O
features	O
are	O
grouped	O
into	O
disjoint	O
blocks	O
,	O
it	O
may	O
be	O
the	O
case	O
that	O
grouped	O
features	O
are	O
overlapping	O
or	O
have	O
a	O
nested	O
structure	O
.	O
</s>
<s>
Such	O
generalizations	O
of	O
group	O
lasso	B-Algorithm
have	O
been	O
considered	O
in	O
a	O
variety	O
of	O
contexts	O
.	O
</s>
<s>
For	O
overlapping	O
groups	O
one	O
common	O
approach	O
is	O
known	O
as	O
latent	O
group	O
lasso	B-Algorithm
which	O
introduces	O
latent	O
variables	O
to	O
account	O
for	O
overlap	O
.	O
</s>
