Biomedical Engineering Reference
In-Depth Information
where the gradient is numerically estimated or the objective function is
noisy.
5. Easy to implement and not very sensitive to numerical problems.
The Rprop algorithms are iterative optimization methods. Let
t
denote the
current iteration (epoch). In epoch
t
, each weight is changed according to
∂E
(
t
)
∂w
i
.
∆
w
i
(
t
+1)=
w
i
i
(
t
)
−
sign
(
t
)
.
(38)
The direction of the change depends on the sign of the partial derivative, but is
independent of its amount. The individual step sizes ∆
i
(
t
) are adapted based on
changes of sign of the partial derivatives of
E
(
w
) w.r.t. the corresponding weight:
∂E
(
t
−
1)
∂w
i
∂E
(
t
)
∂w
i
i
(
t
) is increased by a factor
η
+
>
1.
if
·
>
0, then ∆
∂E
(
t−
1)
∂w
i
∂E
(
t
)
∂w
i
(
t
) is decreased by a factor
η
−
∈
[0
,
1].
Additionally, some Rprop methods implement weight-backtracking, that is,
they partially retract “unfavorable” previous steps. Whether a weight change was
“unfavorable” is decided by a heuristic. In the following an improved version [64]
of the original algorithm is described in pseudo-code. The difference compared
to the original Rprop method is that the weight-backtracking heuristic considers
both the evolution of the partial derivatives and the overall error:
1. Step 1. Enter the iteration
n
.
2. Step 2. For each weighting factor
w
i
,if
∂E
(
t−
1)
∂w
i
i
if
·
≤
0, then ∆
∂E
(
t
)
∂w
i
·
>
0, then:
i
η
+
,
∆
max
)
,
min(∆
(
t
−
1)
·
sign
∂E
(
t
)
∂w
i
.
∆
(39)
w
i
(
t
+1)=
w
i
i
(
t
)
−
(
t
)
,
∂E
(
t
−
1)
∂w
i
∂E
(
t
)
∂w
i
i
η
−
,
∆
min
) if
3. Step 3. Elseif
·
<
0, then:
min(∆
(
t
−
1)
·
∂E
(
t
)
∂w
i
=0.
−
1), then
w
i
(
t
+1)=
w
i
E
(
t
)
>E
(
t
(
t
−
1),
∂E
(
t
−
1)
∂w
i
∂E
(
t
)
∂w
i
=0, and then
4. Step 4. Elseif
·
∂E
(
t
)
∂w
i
w
i
(
t
+1)=
w
i
i
(
t
)
−
sign
·
∆
(
t
)
.
(40)
If there is still
w
i
to be updated, go to step 2; otherwise go to
5. Step 5.
Step 6.
6. Step 6. If no more iteration is needed, exit. Otherwise,
n
=
n
+1go to
Step 2.
In the following, the energy change though the first segmentation example in
Figure 23 is plotted in Figure 26.
