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Fig. 3.2
The network
structure for the preference
function
the true ranker used in the final test is defined by (
3.2
). It is not obvious whether
the optimal preference function learned by loss minimization can really lead to the
optimal ranker. To answer this concern, in [
12
], it has been proven that the loss
with respect to the ranker can be bounded by the loss with respect to the preference
function, as follows:
u<v
(
1
+
u,v
:
y
u,v
=
1
L(h
−
;
h(x
σ
−
1
(u)
,x
σ
−
1
(v)
))
x
u
,x
v
,y
u,v
)
L(σ, y)
≤
u,v
:
y
u,v
=
1
1
(3.3)
3.2.2 SortNet: Neural Network-Based Sorting Algorithm
The goal of [
28
] is also to learn a preference function from training data. Specifi-
cally, a neural network with the structure as shown in Fig.
3.2
is used as the prefer-
ence function.
As can be seen from Fig.
3.2
, there are two outputs of the network,
P(x
u
x
v
)
and
P(x
u
≺
x
v
)
. Since the network approximates a preference function, the
following constraints on these two outputs are enforced:
P(x
u
x
v
)
=
P(x
v
≺
x
u
).
(3.4)
Specifically, these two outputs are generated as follows:
sigmoid
i,i
)
P(x
u
x
v
)
=
(w
i,
h
i
(x
u
,x
v
)
+
w
i
,
h
i
(x
u
,x
v
)
+
b
sigmoid
i,i
(w
i,
≺
h
i
(x
v
,x
u
)
+
w
i
,
≺
h
i
(x
v
,x
u
)
+
b
≺
)
=
=
P(x
v
≺
x
u
),
(3.5)
