Ranking a set of retrieved documents according to their relevance to a given query has become a popular problem at the intersection of web search, machine learning, and information retrieval. Recent work on ranking focused on a number of different paradigms, namely, pointwise, pairwise, and list-wise approaches. Each of those paradigms focuses on a different aspect of the dataset while largely ignoring others. The current paper shows how a combination of them can lead to improved ranking performance and, moreover, how it can be implemented in log-linear time.

The basic idea of the algorithm is to use isotonic regression with adaptive bandwidth selection per relevance grade. This results in an implicitly-defined loss function which can be minimized efficiently by a subgradient descent procedure. Experimental results show that the resulting algorithm is competitive on both commercial search engine data and publicly available LETOR data sets.