An efficient reduction of ranking to classification

Abstract: This paper describes an efficient reduction of the learning problem of ranking to binary classification. The reduction guarantees an average pairwise misranking regret of at most that of the binary classifier regret, improving a recent result of Balcan et al which only guarantees a factor of 2. Moreover, our reduction applies to a broader class of ranking loss functions, admits a simpler proof, and the expected running time complexity of our algorithm in terms of number of calls to a classifier or preference function is improved from $\Omega(n^2)$ to $O(n \log n)$. In addition, when the top $k$ ranked elements only are required ($k \ll n$), as in many applications in information extraction or search engines, the time complexity of our algorithm can be further reduced to $O(k \log k + n)$. Our reduction and algorithm are thus practical for realistic applications where the number of points to rank exceeds several thousands. Much of our results also extend beyond the bipartite case previously studied. Our rediction is a randomized one. To complement our result, we also derive lower bounds on any deterministic reduction from binary (preference) classification to ranking, implying that our use of a randomized reduction is essentially necessary for the guarantees we provide.