Fast Differentiable Sorting and Ranking

The sorting operation is one of the most basic and commonly used building blocks in computer programming. In machine learning, it is commonly used for robust statistics. However, seen as a function, it is piecewise linear and as a result includes many kinks at which it is non-differentiable. More problematic is the related ranking operator, commonly used for order statistics and ranking metrics. It is a piecewise constant function, meaning that its derivatives are null or undefined. While numerous works have proposed differentiable proxies to sorting and ranking, they do not achieve the $O(n \log n)$ time complexity one would expect from sorting and ranking operations. In this paper, we propose the first differentiable sorting and ranking operators with $O(n \log n)$ time and $O(n)$ space complexity. Our proposal in addition enjoys exact computation and differentiation. We achieve this feat by constructing differentiable sorting and ranking operators as projections onto the permutahedron, the convex hull of permutations, and using a reduction to isotonic optimization. Empirically, we confirm that our approach is an order of magnitude faster than existing approaches and showcase two novel applications: differentiable Spearman's rank correlation coefficient and soft least trimmed squares.