Training and testing of recommender systems on data missing not at random

Users typically rate only a small fraction of all available items. We show that the absence of ratings carries useful information for improving the top-k hit rate concerning all items, a natural accuracy measure for recommendations. As to test recommender systems, we present two performance measures that can be estimated, under mild assumptions, without bias from data even when ratings are missing not at random (MNAR). As to achieve optimal test results, we present appropriate surrogate objective functions for efficient training on MNAR data. Their main property is to account for all ratings - whether observed or missing in the data. Concerning the top-k hit rate on test data, our experiments indicate dramatic improvements over even sophisticated methods that are optimized on observed ratings only.