  Perceptron is a classic online algorithm for learning a classification
function. In this paper, we provide a novel extension of the perceptron
algorithm to the learning to rank problem in information retrieval. We consider
popular listwise performance measures such as Normalized Discounted Cumulative
Gain (NDCG) and Average Precision (AP). A modern perspective on perceptron for
classification is that it is simply an instance of online gradient descent
(OGD), during mistake rounds, using the hinge loss function. Motivated by this
interpretation, we propose a novel family of listwise, large margin ranking
surrogates. Members of this family can be thought of as analogs of the hinge
loss. Exploiting a certain self-bounding property of the proposed family, we
provide a guarantee on the cumulative NDCG (or AP) induced loss incurred by our
perceptron-like algorithm. We show that, if there exists a perfect oracle
ranker which can correctly rank each instance in an online sequence of ranking
data, with some margin, the cumulative loss of perceptron algorithm on that
sequence is bounded by a constant, irrespective of the length of the sequence.
This result is reminiscent of Novikoff's convergence theorem for the
classification perceptron. Moreover, we prove a lower bound on the cumulative
loss achievable by any deterministic algorithm, under the assumption of
existence of perfect oracle ranker. The lower bound shows that our perceptron
bound is not tight, and we propose another, \emph{purely online}, algorithm
which achieves the lower bound. We provide empirical results on simulated and
large commercial datasets to corroborate our theoretical results.
