Stochastic Optimization of Sorting Networks via Continuous Relaxations

TL;DR

This paper introduces NeuralSort, a continuous relaxation of sorting operators that enables gradient-based optimization of permutation-based problems in machine learning.

Contribution

NeuralSort provides a differentiable relaxation of sorting, allowing end-to-end training and stochastic optimization over permutations using reparameterized gradients.

Findings

Enables gradient-based optimization of sorting operations.

Applies to learning semantic orderings of high-dimensional objects.

Extends k-nearest neighbors with differentiable sorting.

Abstract

Sorting input objects is an important step in many machine learning pipelines. However, the sorting operator is non-differentiable with respect to its inputs, which prohibits end-to-end gradient-based optimization. In this work, we propose NeuralSort, a general-purpose continuous relaxation of the output of the sorting operator from permutation matrices to the set of unimodal row-stochastic matrices, where every row sums to one and has a distinct arg max. This relaxation permits straight-through optimization of any computational graph involve a sorting operation. Further, we use this relaxation to enable gradient-based stochastic optimization over the combinatorially large space of permutations by deriving a reparameterized gradient estimator for the Plackett-Luce family of distributions over permutations. We demonstrate the usefulness of our framework on three tasks that require learning semantic orderings of high-dimensional objects, including a fully differentiable, parameterized extension of the k-nearest neighbors algorithm.