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DCG, but only the consistency of the regression loss is revealed. It is unknown
whether the conclusion can be extended to pairwise and listwise loss functions,
which are more popular than the pointwise regression loss.
Existing analyses are conducted with the setting of either document ranking or
subset ranking. However, as pointed out in the previous chapter, the setting of
two-layer ranking would make more sense. Unfortunately, there is still no work
on the consistency analysis for this setting.
According to the above discussions, there is still a long way to go along the
direction of statistical consistency for ranking.
Remark Note that there is some other work highly related to statistical consistency
of ranking, although not directly on the topic. For example, in [ 2 ], the regret bound
between the Kendall's τ and the pairwise 0-1 loss is derived when the ranking result
is produced by the voting of the pairwise classification results. In [ 1 ], the results in
[ 2 ] are extended to the case where quick sort is used to produce the ranking result. In
[ 11 ], the above findings are further extended to the case where NDCG is used as the
true loss for ranking instead of the Kendall's τ . Given these theoretical results, if we
further consider the previous findings [ 14 ] on the consistency of widely used surro-
gate loss functions like the hinge, exponential, and logistic losses with respect to the
0-1 classification loss, we may also come to the conclusion that in certain condi-
tions the pairwise ranking algorithms like Ranking SVM, RankBoost, and RankNet
(which minimize the hinge, exponential, and logistic losses respectively) are con-
sistent with the pairwise 0-1 loss, and thus the Kendall's τ or (1
NDCG).
18.6 Exercises
18.1 List different true loss functions used in previous work, and compare their pros
and cons as the true loss for ranking.
18.2 Can you list the properties that the true loss for ranking should possess, and
designsuchatrueloss?
18.3 The definitions of most ranking measures have not considered the sampling
of documents. As a result, the extension of them to the setting of two-layer
ranking might be difficult. Please show how to extend ranking measures to
consider the sampling of documents.
References
1. Ailon, N., Mohri, M.: An efficient reduction from ranking to classification. In: Proceedings of
the 21st Annual Conference on Learning Theory (COLT 2008), pp. 87-98 (2008)
2. Balcan, M.F., Bansal, N., Beygelzimer, A., Coppersmith, D., Langford, J., Sorkin, G.B.: Ro-
bust reductions from ranking to classification. In: Proceedings of the 20th Annual Conference
on Learning Theory (COLT 2007), pp. 139-153 (2007)
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