Exact Passive-Aggressive Algorithms for Ordinal Regression Using Interval Labels

In this article, we propose exact passive-aggressive (PA) online algorithms for ordinal regression. The proposed algorithms can be used even when we have interval labels instead of actual labels for example. The proposed algorithms solve a convex optimization problem at every trial. We find an exact solution to those optimization problems to determine the updated parameters. We propose a support class algorithm (SCA) that finds the active constraints using the Karush-Kuhn-Tucker (KKT) conditions of the optimization problems. These active constraints form a 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. The proposed algorithms also do well compared to other approaches.