Collaborative filtering with localised ranking

In recommendation systems, one is interested in the ranking of the predicted items as opposed to other losses such as the mean squared error. Although a variety of ways to evaluate rankings exist in the literature, here we focus on the Area Under the ROC Curve (AUC) as it widely used and has a strong theoretical underpinning. In practical recommendation, only items at the top of the ranked list are presented to the users. With this in mind we propose a class of objective functions which primarily represent a smooth surrogate for the real AUC, and in a special case we show how to prioritise the top of the list. This loss is differentiable and is optimised through a carefully designed stochastic gradient-descent-based algorithm which scales linearly with the size of the data. We mitigate sample bias present in the data by sampling observations according to a certain power-law based distribution. In addition, we provide computation results as to the efficacy of the proposed method using synthetic and real data.