  In this paper, we propose exact passive-aggressive (PA) online algorithms for
learning to rank. The proposed algorithms can be used even when we have
interval labels instead of actual labels for examples. The proposed algorithms
solve a convex optimization problem at every trial. We find exact solution to
those optimization problems to determine the updated parameters. We propose
support class algorithm (SCA) which finds the active constraints using the KKT
conditions of the optimization problems. These active constrains form support
set which determines the set of thresholds that need to be updated. We derive
update rules for PA, PA-I and PA-II. We show that the proposed algorithms
maintain the ordering of the thresholds after every trial. We provide the
mistake bounds of the proposed algorithms in both ideal and general settings.
We also show experimentally that the proposed algorithms successfully learn
accurate classifiers using interval labels as well as exact labels. Proposed
algorithms also do well compared to other approaches.
