<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
online	B-Algorithm
machine	I-Algorithm
learning	I-Algorithm
is	O
a	O
method	O
of	O
machine	O
learning	O
in	O
which	O
data	O
becomes	O
available	O
in	O
a	O
sequential	O
order	O
and	O
is	O
used	O
to	O
update	O
the	O
best	O
predictor	O
for	O
future	O
data	O
at	O
each	O
step	O
,	O
as	O
opposed	O
to	O
batch	O
learning	O
techniques	O
which	O
generate	O
the	O
best	O
predictor	O
by	O
learning	O
on	O
the	O
entire	O
training	O
data	O
set	O
at	O
once	O
.	O
</s>
<s>
Online	O
learning	O
is	O
a	O
common	O
technique	O
used	O
in	O
areas	O
of	O
machine	O
learning	O
where	O
it	O
is	O
computationally	O
infeasible	O
to	O
train	O
over	O
the	O
entire	O
dataset	O
,	O
requiring	O
the	O
need	O
of	O
out-of-core	B-Application
algorithms	O
.	O
</s>
<s>
Online	O
learning	O
algorithms	O
may	O
be	O
prone	O
to	O
catastrophic	B-Algorithm
interference	I-Algorithm
,	O
a	O
problem	O
that	O
can	O
be	O
addressed	O
by	O
incremental	B-Algorithm
learning	I-Algorithm
approaches	O
.	O
</s>
<s>
In	O
the	O
setting	O
of	O
supervised	B-General_Concept
learning	I-General_Concept
,	O
a	O
function	O
of	O
is	O
to	O
be	O
learned	O
,	O
where	O
is	O
thought	O
of	O
as	O
a	O
space	O
of	O
inputs	O
and	O
as	O
a	O
space	O
of	O
outputs	O
,	O
that	O
predicts	O
well	O
on	O
instances	O
that	O
are	O
drawn	O
from	O
a	O
joint	O
probability	O
distribution	O
on	O
.	O
</s>
<s>
A	O
common	O
paradigm	O
in	O
this	O
situation	O
is	O
to	O
estimate	O
a	O
function	O
through	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
or	O
regularized	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
(	O
usually	O
Tikhonov	O
regularization	O
)	O
.	O
</s>
<s>
The	O
choice	O
of	O
loss	O
function	O
here	O
gives	O
rise	O
to	O
several	O
well-known	O
learning	O
algorithms	O
such	O
as	O
regularized	O
least	B-Algorithm
squares	I-Algorithm
and	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
.	O
</s>
<s>
For	O
many	O
formulations	O
,	O
for	O
example	O
nonlinear	O
kernel	B-Algorithm
methods	I-Algorithm
,	O
true	O
online	O
learning	O
is	O
not	O
possible	O
,	O
though	O
a	O
form	O
of	O
hybrid	O
online	O
learning	O
with	O
recursive	O
algorithms	O
can	O
be	O
used	O
where	O
is	O
permitted	O
to	O
depend	O
on	O
and	O
all	O
previous	O
data	O
points	O
.	O
</s>
<s>
Mini-batch	O
techniques	O
are	O
used	O
with	O
repeated	O
passing	O
over	O
the	O
training	O
data	O
to	O
obtain	O
optimized	O
out-of-core	B-Application
versions	O
of	O
machine	O
learning	O
algorithms	O
,	O
for	O
example	O
,	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
When	O
combined	O
with	O
backpropagation	B-Algorithm
,	O
this	O
is	O
currently	O
the	O
de	O
facto	O
training	O
method	O
for	O
training	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
.	O
</s>
<s>
The	O
simple	O
example	O
of	O
linear	O
least	B-Algorithm
squares	I-Algorithm
is	O
used	O
to	O
explain	O
a	O
variety	O
of	O
ideas	O
in	O
online	O
learning	O
.	O
</s>
<s>
Consider	O
the	O
setting	O
of	O
supervised	B-General_Concept
learning	I-General_Concept
with	O
being	O
a	O
linear	O
function	O
to	O
be	O
learned	O
:	O
</s>
<s>
The	O
recursive	O
least	B-Algorithm
squares	I-Algorithm
(	O
RLS	O
)	O
algorithm	O
considers	O
an	O
online	O
approach	O
to	O
the	O
least	B-Algorithm
squares	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
by	O
initialising	O
and	O
,	O
the	O
solution	O
of	O
the	O
linear	O
least	B-Algorithm
squares	I-Algorithm
problem	I-Algorithm
given	O
in	O
the	O
previous	O
section	O
can	O
be	O
computed	O
by	O
the	O
following	O
iteration	O
:	O
</s>
<s>
or	O
by	O
,	O
this	O
becomes	O
the	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
algorithm	O
.	O
</s>
<s>
This	O
setting	O
is	O
a	O
special	O
case	O
of	O
stochastic	B-Algorithm
optimization	I-Algorithm
,	O
a	O
well	O
known	O
problem	O
in	O
optimization	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
consider	O
the	O
case	O
of	O
online	O
least	B-Algorithm
squares	I-Algorithm
linear	O
regression	O
.	O
</s>
<s>
This	O
method	O
can	O
thus	O
be	O
looked	O
as	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
To	O
use	O
the	O
above	O
for	O
arbitrary	O
convex	O
functions	O
and	O
regularisers	O
,	O
one	O
uses	O
online	B-Algorithm
mirror	I-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
The	O
optimal	O
regularization	O
in	O
hindsight	O
can	O
be	O
derived	O
for	O
linear	O
loss	O
functions	O
,	O
this	O
leads	O
to	O
the	O
AdaGrad	B-Algorithm
algorithm	O
.	O
</s>
<s>
Continual	B-Algorithm
learning	I-Algorithm
capabilities	O
are	O
essential	O
for	O
software	O
systems	O
and	O
autonomous	O
agents	O
interacting	O
in	O
an	O
ever	O
changing	O
real	O
world	O
.	O
</s>
<s>
generally	O
leads	O
to	O
catastrophic	B-Algorithm
forgetting	I-Algorithm
.	O
</s>
<s>
The	O
prototypical	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
algorithm	O
is	O
used	O
for	O
this	O
discussion	O
.	O
</s>
<s>
The	O
first	O
interpretation	O
consider	O
the	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
method	O
as	O
applied	O
to	O
the	O
problem	O
of	O
minimizing	O
the	O
expected	O
risk	O
defined	O
above	O
.	O
</s>
<s>
sample	O
of	O
stochastic	O
estimates	O
of	O
the	O
gradient	O
of	O
the	O
expected	O
risk	O
and	O
therefore	O
one	O
can	O
apply	O
complexity	O
results	O
for	O
the	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
method	O
to	O
bound	O
the	O
deviation	O
,	O
where	O
is	O
the	O
minimizer	O
of	O
.	O
</s>
<s>
The	O
second	O
interpretation	O
applies	O
to	O
the	O
case	O
of	O
a	O
finite	O
training	O
set	O
and	O
considers	O
the	O
SGD	O
algorithm	O
as	O
an	O
instance	O
of	O
incremental	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
method	O
.	O
</s>
<s>
Since	O
the	O
gradients	O
of	O
in	O
the	O
incremental	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
iterations	O
are	O
also	O
stochastic	O
estimates	O
of	O
the	O
gradient	O
of	O
,	O
this	O
interpretation	O
is	O
also	O
related	O
to	O
the	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
method	O
,	O
but	O
applied	O
to	O
minimize	O
the	O
empirical	O
risk	O
as	O
opposed	O
to	O
the	O
expected	O
risk	O
.	O
</s>
<s>
Vowpal	B-Algorithm
Wabbit	I-Algorithm
:	O
Open-source	O
fast	O
out-of-core	B-Application
online	O
learning	O
system	O
which	O
is	O
notable	O
for	O
supporting	O
a	O
number	O
of	O
machine	O
learning	O
reductions	O
,	O
importance	O
weighting	O
and	O
a	O
selection	O
of	O
different	O
loss	O
functions	O
and	O
optimisation	O
algorithms	O
.	O
</s>
<s>
It	O
uses	O
the	O
hashing	B-Algorithm
trick	I-Algorithm
for	O
bounding	O
the	O
size	O
of	O
the	O
set	O
of	O
features	O
independent	O
of	O
the	O
amount	O
of	O
training	O
data	O
.	O
</s>
<s>
Classification	O
:	O
Perceptron	B-Algorithm
,	O
SGD	B-Algorithm
classifier	I-Algorithm
,	O
Naive	B-General_Concept
bayes	I-General_Concept
classifier	I-General_Concept
.	O
</s>
<s>
Clustering	O
:	O
Mini-batch	B-Algorithm
k-means	I-Algorithm
.	O
</s>
<s>
Feature	O
extraction	O
:	O
Mini-batch	B-General_Concept
dictionary	I-General_Concept
learning	I-General_Concept
,	O
Incremental	B-Application
PCA	I-Application
.	O
</s>
