Learning Large Scale Ordinal Ranking Model via Divide-and-Conquer Technique

Structured prediction, where outcomes have a precedence order, lies at the heart of machine learning for information retrieval, movie recommendation, product review prediction, and digital advertising. Ordinal ranking, in particular, assumes that the structured response has a linear ranked order. Due to the extensive applicability of these models, substantial research has been devoted to understanding them, as well as developing efficient training techniques. One popular and widely cited technique of training ordinal ranking models is to exploit the linear precedence order and systematically reduce it to a binary classification problem. This facilitates the usage of readily available, powerful binary classifiers, but necessitates an expansion of the original training data, where the training data increases by $K-1$ times of its original size, with K being the number of ordinal classes. Due to prevalent nature of problems with large number of ordered classes, the reduction leads to datasets which are too large to train on single machines. While approximation methods like stochastic gradient descent are typically applied here, we investigate exact optimization solutions that can scale. In this paper, we present a divide-and-conquer (DC) algorithm, which divides large scale binary classification data into a cluster of machines and trains logistic models in parallel, and combines them at the end of the training phase to create a single binary classifier, which can then be used as an ordinal ranker. It requires no synchronization between the parallel learning algorithms during the training period, which makes training on large datasets feasible and efficient. We prove consistency and asymptotic normality property of the learned models using our proposed algorithm. We provide empirical evidence, on various ordinal datasets, of improved estimation and prediction performance of the model learnt using our algorithm, over several standard divide-and-conquer algorithms.