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3.3.7 LambdaRank
In [ 7 ], another way of introducing position-based weights to the pairwise loss func-
tion is proposed. In particular, the evaluation measures (which are position based)
are directly used to define the gradient with respect to each document pair in the
training process. The hope is that the corresponding implicit loss function can be
more consistent with the way that we evaluate the final ranking results.
Here we use an example to illustrate the basic idea. Suppose we have only two
relevant documents x 1 and x 2 , and their current ranks are as shown in Fig. 3.11 .
Suppose we are using NDCG@1 as the evaluation measure. Then, it is clear that if
we can move either x 1 or x 2 to the top position of the ranked list, we will achieve
the maximum NDCG@1.
It is clearly more convenient to move x 1 up since the effort will be much smaller
than that for x 2 . So, we can define (but not compute) the “gradient” with regards to
the ranking score of x 1 (denoted as s 1 =
f(x 1 ) ) as larger than that with regards to
the ranking score of x 2 (denoted as s 2 =
f(x 2 ) ). In other words, we can consider that
there is an underlying implicit loss function L in the optimization process, which
∂s 1
> ∂L
∂s 2
The above “gradient” is called the lambda function, and this is why the algorithm
is named LambdaRank. When we use NDCG to guide the training process, a specific
form of the lambda function is given in [ 7 ]:
2 y u
2 y v
f(x v )) η r(x u )
η(x v ) ,
Z m
exp (f (x u )
where r(
) represents the rank position of a document in the previous iteration of
For each pair of documents x u and x v , in each round of optimization, their scores
are updated by
λ respectively. In [ 15 ], the lambda functions for MAP
and MRR are derived, and the local optimality of the LambdaRank algorithm is
discussed. According to the experimental results, the LambdaRank algorithm can
lead to local optimum of the objective function, and can have a competitive ranking
λ and
Fig. 3.11 Optimizing
NDCG@1 by LambdaRank
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